Research ArticleCredibility Test for Frequency Estimation of Sinusoid UsingChebyshevrsquos Inequality
Hu Guobing12 Xu LiZhong2 Gao Yan1 and Wu Shanshan1
1Nanjing College of Information Technology Nanjing 210023 China2Hohai University Nanjing 210098 China
Correspondence should be addressed to Hu Guobing guobinghu163com
Received 16 August 2014 Revised 19 November 2014 Accepted 19 November 2014 Published 4 December 2014
Academic Editor Kui Fu Chen
Copyright copy 2014 Hu Guobing et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Estimation of sinusoid frequency is a key research problem related to radar sonar and communication systems The results ofnumerous investigations on frequency estimation have been reported in the literature Nevertheless to the best of our knowledgenone of themhave dealt with credibility evaluation which is used to decidewhether an individual frequency estimate of the sinusoidis accurate or not In this study the credibility problem is modeled as a hypothesis test based on Chebyshevrsquos inequality (CI) Thecorrelation calculated from the received signal and the reference signal generated according to the frequency estimate is used as atest statistic A threshold is determined based on CI and the analytical expression for the frequency estimation credibility detectionperformance is derived Simulations show that the proposed method performs well even at low signal-to-noise ratios
1 Introduction
Frequency estimation of a sinusoid signal is an importantproblem concerning applications related to commercial andmilitary signal processing systems For some methods esti-mation of frequency is the precondition for estimating theother parameters of sinusoid signals [1] as well as frequencyestimation of modulated signals [2 3] Many algorithms havebeen proposed for estimation of frequency from receivedsignals [4ndash12] Performance evaluation of frequency estima-tion algorithms is also a key operation in practical signalprocessing systems and can be considered from two pointsof view Algorithm designers focus on the overall statisticalperformance that can be evaluated by comparing the meansquare error with the Cramer-Rao lower bound (CRLB)On the other hand users consider the credibility (or con-fidence) of individual frequency estimation important Ina noncooperative context especially at low signal-to-noise(SNR) ratios it is important to decide whether an individualfrequency estimate is accurate or not when the true value ofthe frequency is unknown at the receiver side For examplein the pulse sorting system pulse radio frequency is one ofthe five key parameters that are pulse duration (PD) pulseamplitude (PA) pulse radio frequency (RF) angle of arrival
(AOA) and time of arrival (TOA) Therefore the sortingperformance may be affected by the accuracy and credibilityof pulse radio frequency As another example in electronicintelligence (Elint) it is helpful for the users to determine theparameter limit and to remove the outlier estimates by usingthe credibility checking of each estimate [13]
Recently investigations conducted have been devotedto the confidence evaluation of blind modulation recogni-tion results to enhance the reliability of the overall signalprocessing units and conserve both software and hard-ware resources The confidence measurement of modulationrecognition becomes the key output information of thesignal processing system in military applications and is usedto identify unknown radar signals [14] As IEEE 19906for cognitive radio (CR) [15] expressed the modulationrecognition confidence rating is regarded as additional outputinformation in some civilian signal processing devices anexample is Agilentrsquos option MR1 for E3238S signal detectionand monitoring systems Nevertheless the detailed methodof evaluating the reliability of modulation recognition isnot described in both [10 11] Fehske et al [16] definedthe half value of the maximum and the second maximumoutput of the back propagation- (BP-) based classifier asthe confidence metric of modulation classifying results for
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 279325 10 pageshttpdxdoiorg1011552014279325
2 Mathematical Problems in Engineering
CR Lin and Liu [17] proposed a confidence measurementmethod based on the information entropy to measure theconfidence of modulation recognition results in single-inputand single-output (SISO) as well as multiple-input multiple-output (MIMO) channels of CR Still credibility evaluation offrequency estimation of a sinusoid remains an inadequatelyaddressed research problem
In this paper we propose a method to automaticallydecide the credibility of an individual frequency estimateof a sinusoid without any a priori knowledge about theparameters of the received signal Section 2 of this paperpresents the sinusoid signal model and a hypothesis testfor credibility evaluation The null hypothesis is definedas that when the absolute frequency estimation error ofthe sinusoid is less than a quarter of a discrete frequencyinterval and the alternative one is the contrary In Section 3the statistic is defined by calculating the magnitude of thecorrelation between the received signal and the referencesignal generated according to a frequency estimator If theabsolute estimation error is less than a quarter of the discretefrequency interval it shows that the mean of the correlationunder the null hypothesis is different from that under thealternative hypothesis By using this property Section 4presents a decision rule for the credibility test for frequencyestimation of sinusoid based on Chebyshevrsquos inequality (CI)Finally Section 5 summarizes the proposed algorithm andSection 6 reports the simulation results
2 Signal Model and Basic Assumptions
21 Signal Model A complex sinusoid contaminated bynoises can be described by the following signal model
where119860 1198910 and 120579 denote respectively the amplitude carrier
frequency and initial phase of the sinusoid signal 119904(119899) Δ119905is the discrete sampling interval and 119873 corresponds to thelength of the samples The additive noise 119908(119899) is supposed tobe a white complex Gaussian process with a zero mean andvariance 1205902 whose real and imaginary parts are independentof each other
22 Hypothesis Model for Credibility Evaluation In the non-cooperative environment both the modulation format andsignal parameters are unknown at the receiver sideGenerallyprocessing of the sinusoid is performed in two main stepsmodulation recognition and carrier frequency estimationFor a certain processing cycle credibility evaluation ofthe frequency estimator is aimed at detecting whether theindividual frequency estimate is accurate or not In practicefor the widely used fast Fourier transform- (FFT-) basedestimators if the signal-to-noise ratio (SNR) is greater thanthe moderate threshold the maximum absolute bias of thefrequency estimation (|Δ119891|) is less than a quarter of the
discrete sampling frequency interval (Δ119865) [18] Therefore wedescribe the credibility assessment as the following hypothe-sis test
11986701003816100381610038161003816Δ119891
1003816100381610038161003816le 025Δ119865
11986711003816100381610038161003816Δ119891
1003816100381610038161003816gt 025Δ119865
(2)
3 Statistic Selection and Analysis
31 Feature Analysis Assuming that the observed signal is asingle-tone sinusoid the reference signal can be constructedby the sinusoid model as follows
119910 (119899) = exp (minus1198952120587 1198910119899Δ119905) 0 le 119899 le 119873 minus 1 (3)
where 1198910is estimated by the maximum likelihood (ML)
method or the other suboptimal estimators The correlationbetween the observed signal 119909(119899) and the reference signal119910(119899) can be expressed as
For the appropriate frequency estimator if the SNRs areabove a certain threshold [5 14] the absolute estimation erroris less than the discrete sampling frequency interval Δ119865 thatis 0 le 120575 le 1 Consider the magnitude of the mean of 119885 as
|119864 (119885)| = |119904| = radic1205832
119911119877+ 1205832
119911119868= 119873119860 sinc (120575) (9)
Mathematical Problems in Engineering 3
00 01 02 03 04 05 06 07 08 09 1000
01
02
03
04
05
06
07
08
09
10
r
120575
Figure 1 Relationship between 119903 and 120575
Under hypothesis1198670 assuming that the absolute error is
relatively small that is |Δ119891| le 025Δ119865 and 120575 le 025 it followsthat
Under hypothesis 1198671 because of the low SNR or other
possible reasons the maximum magnitude of the observedsignalsrsquo spectrums might not be located in the correct pointor the discrete frequency interval and the estimation errorΔ119891is larger than Δ1198654 that is 120575 ge 025 Thus
1003816100381610038161003816119864 (1198851)1003816100381610038161003816= 119873119860 sinc (120575) le 09003119873119860 (11)
In order to explain the relationship between the frequencyestimation error and the magnitude of the mean of 119885 wedefine the ratio as
119903 =
|119864(119885)|120575
|119864(119885)|0
asymp sinc (120575) (12)
where |119864(119885)|120575is the magnitude of the mean of 119885 at a
given 120575 According to the sinc function the ratio decreasesmonotonically with the frequency estimating error factor 120575supposing 0 le 120575 le 1 As Δ119891 increases 120575 increases andthe ratio 119903 decreases This relationship is shown in Figure 1For 120575 le 025 the ratio 119903 gt 09003 and it decreases slowlydeparting from 1 with the increase of 120575 For 120575 gt 025 theratio decreases dramatically Hence we can infer the absoluteerror of certain frequency estimation depending on the ratio119903 and obtain the credibility metric of the estimation
In order to facilitate convenient handling of the formuladerivation the statistic used to decide 119867
0and 119867
1can be
defined as follows
119881 =
|119885|2
120590119911
(13)
Clearly the random variable 119881 under 119867119894 119894 = 0 1
follows a standard noncentral law and its contribution can berepresented as [19]
119901119881|119867119894
(V) =
1
2
exp [minus12
(V2 + 1205822
119894)] 1198680(radic120582119894V) V ge 0
0 V lt 0
(14)
where 1198680(119909) is the zero-order modified Bessel function and
120582119894= (1205832
119911119877119894+ 1205832
119911119868119894)1205902
119911denotes the noncentrality
Under the assumption relating to1198670 that is 120575 le 025 we
obtain
1205820ge 120582119905=
2 [119873119860 sinc (025)]2
1198731205902
asymp 16211119873 SNR (15)
where 120582119905is the noncentrality parameter when 120575 = 025
Similarly under the1198671assumption we obtain
1205821= 2 [sinc(120575)]2119873 SNR lt 120582
119905 (16)
From the above discussion we find that under each assump-tion 119867
0or 1198671 the statistic 119881 follows a standard noncentral
law with different parameters Therefore the hypothesistest defined by (2) can be transformed to a parameter testinvolving 120582 that is given by
1198670 120582 ge 120582
119905 119867
1 120582 lt 120582
119905 (17)
32 Decision Rule and the Threshold We first consider thedegradation form of (17) given as
1198671015840
0 120582 = 120582
119905 119867
1015840
1 120582 = 120582
119905 (18)
Consequently letting 119888 = 2 + 120582 119887 = 120582119888 the statistic 119881 canbe approximated to a Gaussian distribution by119882 = (119881119888)
13
[20] whose mean and variance are given by
119864 (119882) = 1 minus
2 (1 + 119887)
9119888
119863 (119882) =
2 (1 + 119887)
9119888
(19)
respectively Thus the standard variable
119880 =
119882 minus 119864 (119882)
radic119863 (119882)
sim 119873 (0 1) (20)
and the hypothesis test expressed by (17) can be simplified by
1198671015840
0 120583119880= 0 119867
1015840
1 120583119880
= 0 (21)
where 120583119880denotes the mean of 119880
As the means of 119880 between the two assumptions aredifferent CI can be used as the decision rule to perform thehypothesis test Hence we write CI as
119875 |119880| le 120576 ge 1 minus
1
1205762 (22)
where 120576 gt 0 is a certain real number As 119880 approximatelyfollows a standard Gaussian law the probability of thesamples located in the range of threefold the standard error is
4 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7000
025
050
075
100
p(u)
u
minus10 minus9
minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
= minus3719 120572 = 0001
120575 = 01
120575 = 025
120575 = 04
120574
Figure 2 Distribution of the statistic 119880 for 120575 = 01 025 04 atSNR = 0 dB The carrier frequency is set to 19081MHz the lengthof received samples is 1024 the initial phase is 1205876 and the numberof simulations is 1000
greater than 09973 and we can adjust 120576 to obtain the desiredprobability Hence we decide1198671015840
1if
minus120576 le 119880 le 120576 (23)
Consequently the hypothesis test of (2) can be rewritten as
1198670 120583119880ge 0 119867
1 120583119880lt 0 (24)
Therefore we decide1198671if
119880 ge 120574 = minus120576 (25)
where 120574 is the thresholdThe probability of false alarm can bedefined as
119875119903119880 lt 120574119867
0 asymp Φ (120574) = 120572 (26)
where Φ(119909) = (1radic2120587) int
119909
minusinfin119890minus11990522119889119905 is the distribution
function of a standard Gaussian variable and 120572 is a givenfalse alarm Taking the inverse function of (26) leads to thethreshold
120574 = Φminus1(120572) (27)
whereΦminus1(119909) denotes the inverse of the distribution functionof a standard Gaussian probability density function (pdf)
Figure 2 shows the histogram and the fitted pdfs of thestatistic 119880 for different frequency estimator factors It can beseen that under a certain SNR the proposed statistic basedonCI approximated to the standardGaussian and the selectedthreshold can be used to distinguish119867
0and119867
1assumptions
effectively
4 Algorithm Summary
The proposed credibility test algorithm is composed of thefollowing steps
(1) Estimation of the frequency of the signal by using acertain estimation method
(2) Construction of the reference signal 119910(119899) with (3) byusing the estimated frequency
1198910
(3) Computation of the correlation between the receivedsignal and the reference signal and contraction of thestatistic 119880 with (20)
(4) Computation of the test threshold 120574 from the proba-bility of false alarm with (27)
(5) Evaluation of the credibility of frequency estimationby comparing the statistic 119880 to the threshold 120574 with(25)The hypothesis119867
0is chosen if119880 lt 120574 Otherwise
1198671is accepted
5 Performance Analysis
Under 11986710158400assumption with 120582 = 120582
119905 the statistic defined in
(20) is expressed as
119880120582119905
1198671015840
0
=
(119881119888119905)13
minus [1 minus (29) (1 + 119887119905) 119888119905]
radic(29) (1 + 119887119905) 119888119905
sim 119873 (0 1) (28)
where 119888119905= 2 + 120582
119905and 119887 = 120582
119905119888119905 respectively
For 120582 ge 120582119905 the statistic is
119880120582
1198670=
(119881119888)13
minus [1 minus (29) (1 + 119887) 119888]
radic(29) (1 + 119887) 119888
sim 119873 (0 1) (29)
We note that for 120582 gt 120582119905 it can be rewritten as
119880120582119905
1198670=
(119881119888119905)13
minus [1 minus (29) (1 + 119887119905) 119888119905]
radic(29) (1 + 119887119905) 119888119905
=
(119881119888)13
minus 12058913
[1 minus (29) (1 + 119887119905) 119888119905]
12058913radic(29) (1 + 119887
119905) 119888119905
= ((
119881
119888
)
13
minus [1 minus
(29) (1 + 119887)
119888
]
+ [1 minus
(29) (1 + 119887)
119888
]
minus 12058913
[1 minus
(29) (1 + 119887119905)
119888119905
])
times (12058913radic(29) (1 + 119887
119905)
119888119905
)
minus1
(30)
where 120589 = 119888119905119888 and 120589 ge 1 According to the property of the
Gaussian distribution the statistic is given by
119880120582119905
1198670sim 119873(120583
120582119905
1198670 120590120582119905
1198670) (31)
Mathematical Problems in Engineering 5
where
120583120582119905
1198670= 119864 (119880
120582119905
1198670)
=
1 minus (29) (1 + 119887) 119888 minus 12058913
[1 minus (29) (1 + 119887119905) 119888119905]
12058913radic(29) (1 + 119887
119905) 119888119905
(32)
and the variance is
120590120582119905
1198670= radic119863(119880
120582119905
1198670) = 120589minus13
radic
(1 + 119887) 119888
(1 + 119887119905) 119888119905
(33)
Hence we obtain the probability of false alarm as
119875FA = 119875 119880120582119905
0le 120574 | 119867
0
= int
120574
minusinfin
1
radic2120587120590120582119905
1198670
exp[
[
minus
(119909 minus 120583120582119905
1198670)
2
2 (120590120582119905
1198670)
2
]
]
119889119909
asymp Φ(
120574 minus 120583120582119905
1198670
120590120582119905
1198670
)
(34)
Similarly under1198671assumption with 120582 lt 120582
119905 the statistic
119880120582119905
1198671=
(119881119888119905)13
minus [1 minus (29) (1 + 119887119905) 119888119905]
radic(29) (1 + 119887119905) 119888119905
= ((
119881
119888
)
13
minus [1 minus
(29) (1 + 119887)
119888
]
+ [1 minus
(29) (1 + 119887)
119888
]
minus 12058513
[1 minus
(29) (1 + 119887119905)
119888119905
])
times (12058513radic(29) (1 + 119887
119905)
119888119905
)
minus1
(35)
where 120585 = 119888119905119888 and 0 le 120585 le 1 From (35) 119880120582119905
1198671sim 119873(120583
120582119905
1198671 120590120582119905
1198671)
with the mean and variance are respectively given by
Figure 3 Effect of the false alarm probability on the detectionprobability obtained by the credibility test of a frequency estimate
Thus the detection probability can be expressed by
119875119863= Pr 119880120582119905
1198671le 120574 | 119867
1
= int
120574
minusinfin
1
radic2120587120590120582119905
1198671
exp[
[
minus
(119909 minus 120583120582119905
1198671)
2
2 (120590120582119905
1198671)
2
]
]
119889119909
asymp Φ(
120574 minus 120583120582119905
1198671
120590120582119905
1198671
)
(37)
From (34) and (37) we observe that the detection perfor-mance depends largely on the parameter 120582 which is thefunction of the number of samples received the SNR and thefrequency estimation error factor
6 Simulation Results
Monte Carlo simulations were carried out to study the behav-ior of the proposed algorithm in different environments Thesimulations were aimed at detecting whether the absoluteerror of a certain estimated frequency is greater than a quarterof the discrete sampling frequency Ten thousand MonteCarlo trials were performed for each condition Monte Carlotrials were run in each of the following conditions (1) thesinusoid signal contaminated by additive white Gaussiannoises is received (2) SNR = 10 log
1011986021205902 (3) the sampling
frequency is set to 100MHz except in 66
61 Influence of False Alarm Figure 3 shows the detectionbehavior for the credibility test by using the CI method withrespect to the false alarm and the comparisonswith the valuestheoretically calculated using (37) The carrier frequency ofthe sinusoid is 19801MHz while the sample size is 1024 The
Figure 4 Effect of the frequency estimation error factor on thedetection probability obtained by the credibility test of a frequencyestimate
initial phase is 1205876 and the frequency estimation error factoris 05 The SNR is varied from minus15 dB to 6 dB in steps of 3 dBThe values of probability of false alarm are fixed as 01 001and 0001 respectively
In the three cases the probability of detection (119875119863) is close
to 1 at an SNR greater than 0 dB The detection probability isincreased by a higher value of probability of false alarmWith120572 = 01 the probability of detection is close to 1 for an SNRof minus6 dB Moreover the derived theoretical 119875
119863is close to the
simulated value
62 Influence of the Frequency Estimation Error Figure 4shows the probability of detection with respect to the carrierfrequency estimation error and the comparisonswith the the-oretically calculated values The carrier frequency estimationerror is respectively equal to 03 04 and 05 for a sinusoidwith 1024 samples and the probability of false alarm equals001 The rest of the parameter settings of the signal in thesimulation remain the same as that stated in Section 61 Thedetection probability is increased by increasing the carrierfrequency estimation error factor For 120575 = 05 and an SNRclose to minus9 dB the probability of detection is approximatelyequal to 1
In fact a larger frequency estimation error factorimproves the mean of the proposed statistic 119880 and the dis-crimination between nonnull and null is easier In additionit can be observed from Figure 3 that the derived theoretical119875119863is very close to the simulated values in the three cases
63 Influence of the Number of Received Samples N Figure 5shows the detection performance of the credibility test versusthe received samples of the signal and the comparisons withthe theoretical calculated values The number of samples isequal to 512 1024 and 2048 respectively The carrier fre-quency estimation error is set to 04The rest of the parameter
00
01
02
03
04
05
06
07
08
09
10
SNR (dB)0 3 6
PD
minus15 minus12 minus9 minus6 minus3
N = 512 simulatedN = 512 theoreticalN = 1024 simulated
N = 1024 theoreticalN = 2048 simulatedN = 2048 theoretical
Figure 5 Effect of the number of received samples on the detectionprobability obtained by the credibility test of a frequency estimate
settings are the same as that stated in Section 62 In thethree cases with different sample numbers the detectionprobability is close to 1 for an SNR of 0 dBThe credibility testperformance is enhanced by the greater number of receivedsamples at the same SNR When the number of samplesis 2048 the detection probability is close to 1 for an SNRof minus6 dB We can observe from Figure 5 that the derivedtheoretical119875
119863is very close to the simulated values in the three
cases
64 Influence of Frequency Discretization We evaluated theperformance of our method in the case of different carrierfrequencies Figure 6 shows the performance of detectionversus the carrier frequency selected in the simulations Wedefined the discretization factor as 119889 = (119891
0minus 1198960Δ119865) where
1198960
= 199 and five frequencies were generated by setting119889 = 0 01 03 05 07 Clearly if 119889 = 0 the frequencyequals 1953125MHz which is just located on the discretefrequency bin When 119889 = 0 it indicates that the selectedfrequencies are not located on the discrete frequency binThe parameters are 119873 = 1024 120575 = 04 120579 = 1205876 and120572 = 001 Figure 6 shows that the detection probability doesnot depend on the frequencies we selected by using differentdiscretization factors and the same levels of performance areachieved in the five cases and are approximately identical tothe theoretically calculated performance in all the cases
65 Influence of the Initial Phase We investigated the impactof the initial phase on the performance of our methodFigure 7 shows the performance of detection versus the initialfrequency used in the simulations We set three initial phasesas 12058712 1205876 and 1205873 and the sample size as 1024 The restof the parameters set in each simulation are the same asthat mentioned in Section 63 Figure 7 shows that the initialphase of the sinusoid does not affect the detection probability
Figure 7 Effect of the initial phase of the signal on the detectionprobability obtained by the credibility test of a frequency estimate
and the same levels of performance are achieved in the twocases and are identical to the theoretical performances as thestatistic defined in (13) is not relevant to the phase because ofthe use of the modulus operation
66 Influence of the Sampling Frequency We now clarifythe impact of the sampling frequency on the performanceof the proposed method Figure 8 shows the performanceof detection versus the sampling frequency selected in thesimulations The other parameters are 119891
0= 19081MHz
119873 = 1024 120575 = 04 120579 = 1205876 and 120572 = 001 From the figurewe observe that the performance of detection is not affected
0 3 600
01
02
03
04
05
06
07
08
09
10
SNR (dB)
PD
minus15 minus12 minus9 minus6 minus3
Theoreticalfs = 120MHzfs = 100MHz
fs = 80MHz
Figure 8 Effect of the sampling frequency of the signal on thedetection probability obtained by the credibility test of a frequencyestimate
0 3 6
00
01
02
03
04
05
06
07
08
09
10
SNR (dB)minus12 minus9 minus6 minus3
PD (OR rule)PD (AND rule)
PD (CI only)PD (AM only)
PD
Figure 9 Probability of detection and probability of false alarmcomparison between the proposed scheme and others method
by the sampling frequency and is identical to the theoreticalperformance
67 Comparison to Other Methods Comparative experi-ments were also carried out for the proposed statisticAboutanios-Mulgrew (AM) method-based [21] detector anda combination of them by OR and AND rules respectivelyWe have described the AM-based detector that can alsobe used to test the credibility of the frequency estimatein the appendix The parameters are 119891
0= 19081MHz
119873 = 1024 120575 = 04 120579 = 1205876 and 120572 = 001 Thecurves of probabilities of detection for different SNRs areshown in Figure 9 We can observe that under an SNR
8 Mathematical Problems in Engineering
less than minus3 dB the probabilities of detection of the AM-basedmethod are greater than those of the CI-basedmethodFor the fusion method the probabilities of detection usingthe ldquoANDrdquo rule are slightly greater than those of the AM-based method and vice versa for the ldquoORrdquo rule that isthe probability of detection is the smallest when the SNR isless than minus3 dB When the SNRs are greater than minus3 dB theprobabilities of detection of the four detectors all approach1 Therefore under lower SNRs the more efficient fusionmethods combining the CI and AM methods need to befurther developed
It should be noted that calculating the threshold isdifficult because the exact probability distribution function ofthe AM-based statistic 119879AM defined in the appendix is hardto be expressed analytically Therefore here we introducethe bootstrap-based method [22 23] to obtain the thresh-old without depending on the analytical probability of thestatistic based on AM but the computational load increasesmany times because of repeated resampling operations in thebootstrap-based hypothesis test procedures
68 Case Study In this section we use the CI-based credibil-ity test statistic to evaluate three commonly used frequencyestimators the Rife algorithm [4] the maximum likelihoodestimator [6] by Newton iteration and the Aboutanious-Mulgrew estimator [21]The carrier frequency of the sinusoidis 19081MHz the initial phase is 1205876 and the sample size is1024 The range of SNR is from minus18 dB to minus6 dB The valueof probability of false alarm is set to 005 If we let 119863
119894(119894 =
0 1) be the event associated with the decision of choosing119867119894(119894 = 0 1) 119899
119894119895stands for the number of times that deciding
119863119895when hypothesis 119867
119894is correct Therefore Type I error
probability and Type II error probability can be calculatedusing 119899
01(11989900
+ 11989901) and 119899
10(11989910
+ 11989911) respectively The
error probability of the test can be expressed by 119875119864= (11989901+
11989910)10000 while 119875
119863= 11989911(11989911+ 11989910) indicates the ability
of the proposed method to detect the unreliable frequencyestimate which means that the maximum absolute bias of acertain frequency estimate (|Δ119891|) is less than a quarter of thediscrete sampling frequency interval (Δ119865)
Table 1 illustrates the performance of the credibility testfor the ML frequency estimator of the sinusoid by using theproposed CI-based statistic It should be remarked that thecredibility test performance is enhanced by increasing theSNR If the SNR is minus6 dB all the 10000 frequency estimatesbased on ML are reliable and the error probability of theproposed test approximates to 0 When the SNR decreases tominus12 dB 61 estimates are unreliable among 10000 simulationsBy using the proposed statistic all the 61 unreliable estimatescan be detected indicating a detection rate of about 100Among the 9939 reliable estimates there are 21 times whenestimates are mistakenly decided as unreliable in otherwords the error probability is about 021 When the SNR isdecreased to minus15 dB the error probability is about 081 andthe detection rate is greater than 95
Table 2 depicts the performance of the credibility test forthe Rife-based sinusoid frequency estimator The credibilitytest performance is enhanced by a larger SNR For an SNR ofminus6 dB all the 10000 times the frequency estimates obtained
Table 1 Credibility test performance of ML frequency estimator
by using the Rife algorithm are reliable (1198670) and the error
probability is approximately equal to 0 When the SNRdeclines to minus12 dB there are 71 unreliable estimates among10000 simulations By using the CI-based credibility teststatistic these 70 unreliable estimates can be detected andthe detection rate is about 986 On the other hand amongthe 9929 reliable estimates there are 25 times when theestimates are mistakenly decided as unreliable that is theerror probability is about 026 With the SNR decreased tominus15 dB the error probability is about 085 and the detectionrate is greater than 90
Table 3 provides the performance of the credibility test forthe AM-based sinusoid frequency estimator The credibilitytest performance is enhanced by a larger SNR Similar to thatdata in Tables 1 and 2 the test performance is improved bya larger SNR With the SNR decreased to minus15 dB the errorprobability is about 069 and the detection rate is greaterthan 80
Mathematical Problems in Engineering 9
Overall from Tables 2 and 3 we can observe that theproposed credibility test statistic can be used for effectivelyidentifying unreliable frequency estimates by using ML AMand Rife algorithms with a high detection probability and alow error probability when the SNR is greater than minus15 dBCompared to the estimator performances summarized inTables 1 and 2 the estimator performance summarized inTable 3 is the lowest because AM-based estimation has abetter performance than that of bothML and Rife algorithmsand the frequency estimation factors are small
In practice for the detected unreliable estimate we canabandon the results or estimate the frequency of the signalagain thereby enhancing the reliability of the entire signalprocessing unit For example if the FFT-based coarse fre-quency estimator is used in practice andwe set the parametersas 119891119904= 100MHz 119891
0= 1955MHz 119873 = 1000 and the
number of DFT119872 = 1000 there are two cases
(1) Case 1 When the SNRs (eg SNR = minus23 dB) arebelow the SNR threshold the maximum spectrumline cannot be detected correctly In this case theestimate error is larger than 025Δ119865 and the estimatewill be judged as incredible by the proposed CI-basedstatistic If we perform resampling over the frequencygrid by setting 119872 = 2000 the maximum spectrumline still cannot be detected As the absolute estima-tion errors are the same under the two scenariosthe resampling action cannot enhance the credibilityof the estimate and this kind of incredible estimateshould be abandoned
(2) Case 2When an SNR is greater than the SNR thresh-old an incredible frequency estimate is detectedbecause of the displacement of the maximum spec-trum line by the discretization In this case ifwe enlarge 119872 to 2000 by resampling in the fre-quency domain 119891
0can be just moved to the dis-
crete frequency bin Hence the estimation error canbe decreased and the credibility can be enhancedHowever in practice the true frequency cannot beknown by the user hence it is difficult to adjustthe resampling parameter to match the estimationexactly In this case we need to select the moreaccurate estimating method or abandon the resultobtained by original estimation
7 Conclusion
The paper presented a CI-based credibility test algorithm forfrequency estimation of a sinusoid A credibility assessmenttesting model is defined to analyze the mathematical charac-teristics of the correlations between the observed signal andthe reference signal under different hypotheses The test isperformed with the proposed threshold that is based on CIExperimental results revealed that a good performance wasachieved even at low SNRs Credibility tests were performedwith the ML frequency estimator using the proposed thresh-old based on CI as well with the Rife frequency estimator andthe Aboutanious-Mulgrew estimator Experimental results
revealed that a good performance was achieved by theproposed method even at low SNRs
Appendix
Thefrequency estimation credibility testmethod based on theAM estimator includes
(1) estimation of the frequency of the signal by using acertain estimation method
(2) construction of the reference signal 119910(119899) with (3) byusing the estimated frequency
1198910
(3) computation of the correlation between the receivedsignal and the reference signal and estimation ofthe frequency displacement Δ
119891 by the Aboutanios-Mulgrew frequency estimator [21]
(4) evaluation of the credibility of frequency estimationby comparing the statistic 119879AM = ||Δ
119891| minus 025Δ119865| to
the threshold th that can be calculated by a given falsealarm 120572 The hypothesis 119867
0is chosen if 119879AM lt th
Otherwise1198671is accepted
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This study is financially supported by the Natural ScienceFoundation of Jiangsu Province (Project no BK2011837) theNational Natural Science Foundation of China 2012 (Projectno 61201208) and the 333 Research Project of JiangsuProvince (Project no BRA2013171)
References
[1] D C Rife and R R Boorstyn ldquoSingle-tone parameter esti-mation from discrete-time observationrdquo IEEE Transactions onInformation Theory vol 20 no 5 pp 591ndash598 1974
[2] S Peleg and B Porat ldquoLinear FM signal parameter estima-tion from discrete-time observationsrdquo IEEE Transactions onAerospace and Electronic Systems vol 27 no 4 pp 607ndash6161991
[3] M Ghogho A Swami and T Durrani ldquoBlind estimation offrequency offset in the presence of unknown multipathrdquo inProceedings of the IEEE International Conference on PersonalWireless Communications pp 104ndash108 December 2000
[4] D C Rife and G A Vincent ldquoUse of the discrete Fouriertransform in the measurement of frequencies and levels oftonesrdquoThe Bell System Technical Journal vol 49 no 2 pp 197ndash228 1970
[5] J-R Liao andC-M Chen ldquoPhase correction of discrete Fouriertransform coefficients to reduce frequency estimation bias ofsingle tone complex sinusoidrdquo Signal Processing vol 94 no 1pp 108ndash117 2014
[6] T Abatzoglou ldquoA fast maximum likelihood algorithm forfrequency estimation of a sinusoid based on Newtonrsquos methodrdquo
10 Mathematical Problems in Engineering
IEEE Transactions on Acoustics Speech and Signal Processingvol 33 no 1 pp 77ndash89 1985
[7] Y Liu Z Nie Z Zhao and Q H Liu ldquoGeneralization ofiterative Fourier interpolation algorithms for single frequencyestimationrdquo Digital Signal Processing vol 21 no 1 pp 141ndash1492011
[8] G Wei C Yang and F-J Chen ldquoClosed-form frequencyestimator based on narrow-band approximation under noisyenvironmentrdquo Signal Processing vol 91 no 4 pp 841ndash851 2011
[9] C Yang and G Wei ldquoA noniterative frequency estimator withrational combination of three spectrum linesrdquo IEEE Transac-tions on Signal Processing vol 59 no 10 pp 5065ndash5070 2011
[10] C Candan and S Koc ldquoBeamspace approach for detection ofthe number of coherent sourcesrdquo in Proceedings of the IEEERadar Conference Ubiquitous Radar (RADAR rsquo12) pp 0913ndash0918 May 2012
[11] Y Cao G Wei and F-J Chen ldquoA closed-form expandedautocorrelationmethod for frequency estimation of a sinusoidrdquoSignal Processing vol 92 no 4 pp 885ndash892 2012
[12] F Qian S Leung Y Zhu W Wong D Pao and W LauldquoDamped sinusoidal signals parameter estimation in frequencydomainrdquo Signal Processing vol 92 no 2 pp 381ndash391 2012
[13] G Richard ELINT The Interception and Analysis of RadarSignals John Wiley amp Sons Artech House Dedham MassUSA 2nd edition 2006
[14] W Su J A Kosinski and M Yu ldquoDual-use of modulationrecognition techniques for digital communication signalsrdquo inProceedings of the IEEE Long Island Systems Applications andTechnology Conference (LISAT rsquo06) pp 1ndash6 Long Island NYUSA May 2006
[15] L Pucker ldquoReview of contemporaryspectrum sensingtechnologiesrdquo IEEE-SA P19006 Standards Group 2009 httpgrouperieeeorggroupsdyspan6documentswhite papersP19006 Sensor Surveypdf
[16] A Fehske J Gaeddert and JH Reed ldquoAnew approach to signalclassification using spectral correlation and neural networksrdquoin Proceedings of the 1st IEEE International Symposium on NewFrontiers in Dynamic Spectrum Access Networks (DySPAN rsquo05)pp 144ndash150 Baltimore Md USA November 2005
[17] W S Lin andK J R Liu ldquoModulation forensics for wireless dig-ital communicationsrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo08) pp 1789ndash1792 April 2008
[18] G-B Hu L-Z Xu and M Jin ldquoReliability testing for blindprocessing results of LFM signals based on NP criterionrdquo ActaElectronica Sinica vol 41 no 4 pp 739ndash743 2013
[19] A D Whalen Detection of Signals in Noise Academic PressNew York NY USA 2nd edition 1995
[20] M Sankaran ldquoOn the non-central chi-square distributionrdquoBiometrika vol 46 no 1-2 pp 235ndash237 1959
[21] E Aboutanios and B Mulgrew ldquoIterative frequency estimationby interpolation on Fourier coefficientsrdquo IEEE Transactions onSignal Processing vol 53 no 4 pp 1237ndash1242 2005
[22] AM Zoubir andD R Iskander Bootstrap Techniques for SignalProcessing Cambridge University Press Cambridge UK 2004
[23] A M Zoubir and D R Iskandler ldquoBootstrap methods andapplicationsrdquo IEEE Signal Processing Magazine vol 24 no 4pp 10ndash19 2007
CR Lin and Liu [17] proposed a confidence measurementmethod based on the information entropy to measure theconfidence of modulation recognition results in single-inputand single-output (SISO) as well as multiple-input multiple-output (MIMO) channels of CR Still credibility evaluation offrequency estimation of a sinusoid remains an inadequatelyaddressed research problem
In this paper we propose a method to automaticallydecide the credibility of an individual frequency estimateof a sinusoid without any a priori knowledge about theparameters of the received signal Section 2 of this paperpresents the sinusoid signal model and a hypothesis testfor credibility evaluation The null hypothesis is definedas that when the absolute frequency estimation error ofthe sinusoid is less than a quarter of a discrete frequencyinterval and the alternative one is the contrary In Section 3the statistic is defined by calculating the magnitude of thecorrelation between the received signal and the referencesignal generated according to a frequency estimator If theabsolute estimation error is less than a quarter of the discretefrequency interval it shows that the mean of the correlationunder the null hypothesis is different from that under thealternative hypothesis By using this property Section 4presents a decision rule for the credibility test for frequencyestimation of sinusoid based on Chebyshevrsquos inequality (CI)Finally Section 5 summarizes the proposed algorithm andSection 6 reports the simulation results
2 Signal Model and Basic Assumptions
21 Signal Model A complex sinusoid contaminated bynoises can be described by the following signal model
where119860 1198910 and 120579 denote respectively the amplitude carrier
frequency and initial phase of the sinusoid signal 119904(119899) Δ119905is the discrete sampling interval and 119873 corresponds to thelength of the samples The additive noise 119908(119899) is supposed tobe a white complex Gaussian process with a zero mean andvariance 1205902 whose real and imaginary parts are independentof each other
22 Hypothesis Model for Credibility Evaluation In the non-cooperative environment both the modulation format andsignal parameters are unknown at the receiver sideGenerallyprocessing of the sinusoid is performed in two main stepsmodulation recognition and carrier frequency estimationFor a certain processing cycle credibility evaluation ofthe frequency estimator is aimed at detecting whether theindividual frequency estimate is accurate or not In practicefor the widely used fast Fourier transform- (FFT-) basedestimators if the signal-to-noise ratio (SNR) is greater thanthe moderate threshold the maximum absolute bias of thefrequency estimation (|Δ119891|) is less than a quarter of the
discrete sampling frequency interval (Δ119865) [18] Therefore wedescribe the credibility assessment as the following hypothe-sis test
11986701003816100381610038161003816Δ119891
1003816100381610038161003816le 025Δ119865
11986711003816100381610038161003816Δ119891
1003816100381610038161003816gt 025Δ119865
(2)
3 Statistic Selection and Analysis
31 Feature Analysis Assuming that the observed signal is asingle-tone sinusoid the reference signal can be constructedby the sinusoid model as follows
119910 (119899) = exp (minus1198952120587 1198910119899Δ119905) 0 le 119899 le 119873 minus 1 (3)
where 1198910is estimated by the maximum likelihood (ML)
method or the other suboptimal estimators The correlationbetween the observed signal 119909(119899) and the reference signal119910(119899) can be expressed as
For the appropriate frequency estimator if the SNRs areabove a certain threshold [5 14] the absolute estimation erroris less than the discrete sampling frequency interval Δ119865 thatis 0 le 120575 le 1 Consider the magnitude of the mean of 119885 as
|119864 (119885)| = |119904| = radic1205832
119911119877+ 1205832
119911119868= 119873119860 sinc (120575) (9)
Mathematical Problems in Engineering 3
00 01 02 03 04 05 06 07 08 09 1000
01
02
03
04
05
06
07
08
09
10
r
120575
Figure 1 Relationship between 119903 and 120575
Under hypothesis1198670 assuming that the absolute error is
relatively small that is |Δ119891| le 025Δ119865 and 120575 le 025 it followsthat
Under hypothesis 1198671 because of the low SNR or other
possible reasons the maximum magnitude of the observedsignalsrsquo spectrums might not be located in the correct pointor the discrete frequency interval and the estimation errorΔ119891is larger than Δ1198654 that is 120575 ge 025 Thus
1003816100381610038161003816119864 (1198851)1003816100381610038161003816= 119873119860 sinc (120575) le 09003119873119860 (11)
In order to explain the relationship between the frequencyestimation error and the magnitude of the mean of 119885 wedefine the ratio as
119903 =
|119864(119885)|120575
|119864(119885)|0
asymp sinc (120575) (12)
where |119864(119885)|120575is the magnitude of the mean of 119885 at a
given 120575 According to the sinc function the ratio decreasesmonotonically with the frequency estimating error factor 120575supposing 0 le 120575 le 1 As Δ119891 increases 120575 increases andthe ratio 119903 decreases This relationship is shown in Figure 1For 120575 le 025 the ratio 119903 gt 09003 and it decreases slowlydeparting from 1 with the increase of 120575 For 120575 gt 025 theratio decreases dramatically Hence we can infer the absoluteerror of certain frequency estimation depending on the ratio119903 and obtain the credibility metric of the estimation
In order to facilitate convenient handling of the formuladerivation the statistic used to decide 119867
0and 119867
1can be
defined as follows
119881 =
|119885|2
120590119911
(13)
Clearly the random variable 119881 under 119867119894 119894 = 0 1
follows a standard noncentral law and its contribution can berepresented as [19]
119901119881|119867119894
(V) =
1
2
exp [minus12
(V2 + 1205822
119894)] 1198680(radic120582119894V) V ge 0
0 V lt 0
(14)
where 1198680(119909) is the zero-order modified Bessel function and
120582119894= (1205832
119911119877119894+ 1205832
119911119868119894)1205902
119911denotes the noncentrality
Under the assumption relating to1198670 that is 120575 le 025 we
obtain
1205820ge 120582119905=
2 [119873119860 sinc (025)]2
1198731205902
asymp 16211119873 SNR (15)
where 120582119905is the noncentrality parameter when 120575 = 025
Similarly under the1198671assumption we obtain
1205821= 2 [sinc(120575)]2119873 SNR lt 120582
119905 (16)
From the above discussion we find that under each assump-tion 119867
0or 1198671 the statistic 119881 follows a standard noncentral
law with different parameters Therefore the hypothesistest defined by (2) can be transformed to a parameter testinvolving 120582 that is given by
1198670 120582 ge 120582
119905 119867
1 120582 lt 120582
119905 (17)
32 Decision Rule and the Threshold We first consider thedegradation form of (17) given as
1198671015840
0 120582 = 120582
119905 119867
1015840
1 120582 = 120582
119905 (18)
Consequently letting 119888 = 2 + 120582 119887 = 120582119888 the statistic 119881 canbe approximated to a Gaussian distribution by119882 = (119881119888)
13
[20] whose mean and variance are given by
119864 (119882) = 1 minus
2 (1 + 119887)
9119888
119863 (119882) =
2 (1 + 119887)
9119888
(19)
respectively Thus the standard variable
119880 =
119882 minus 119864 (119882)
radic119863 (119882)
sim 119873 (0 1) (20)
and the hypothesis test expressed by (17) can be simplified by
1198671015840
0 120583119880= 0 119867
1015840
1 120583119880
= 0 (21)
where 120583119880denotes the mean of 119880
As the means of 119880 between the two assumptions aredifferent CI can be used as the decision rule to perform thehypothesis test Hence we write CI as
119875 |119880| le 120576 ge 1 minus
1
1205762 (22)
where 120576 gt 0 is a certain real number As 119880 approximatelyfollows a standard Gaussian law the probability of thesamples located in the range of threefold the standard error is
4 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7000
025
050
075
100
p(u)
u
minus10 minus9
minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
= minus3719 120572 = 0001
120575 = 01
120575 = 025
120575 = 04
120574
Figure 2 Distribution of the statistic 119880 for 120575 = 01 025 04 atSNR = 0 dB The carrier frequency is set to 19081MHz the lengthof received samples is 1024 the initial phase is 1205876 and the numberof simulations is 1000
greater than 09973 and we can adjust 120576 to obtain the desiredprobability Hence we decide1198671015840
1if
minus120576 le 119880 le 120576 (23)
Consequently the hypothesis test of (2) can be rewritten as
1198670 120583119880ge 0 119867
1 120583119880lt 0 (24)
Therefore we decide1198671if
119880 ge 120574 = minus120576 (25)
where 120574 is the thresholdThe probability of false alarm can bedefined as
119875119903119880 lt 120574119867
0 asymp Φ (120574) = 120572 (26)
where Φ(119909) = (1radic2120587) int
119909
minusinfin119890minus11990522119889119905 is the distribution
function of a standard Gaussian variable and 120572 is a givenfalse alarm Taking the inverse function of (26) leads to thethreshold
120574 = Φminus1(120572) (27)
whereΦminus1(119909) denotes the inverse of the distribution functionof a standard Gaussian probability density function (pdf)
Figure 2 shows the histogram and the fitted pdfs of thestatistic 119880 for different frequency estimator factors It can beseen that under a certain SNR the proposed statistic basedonCI approximated to the standardGaussian and the selectedthreshold can be used to distinguish119867
0and119867
1assumptions
effectively
4 Algorithm Summary
The proposed credibility test algorithm is composed of thefollowing steps
(1) Estimation of the frequency of the signal by using acertain estimation method
(2) Construction of the reference signal 119910(119899) with (3) byusing the estimated frequency
1198910
(3) Computation of the correlation between the receivedsignal and the reference signal and contraction of thestatistic 119880 with (20)
(4) Computation of the test threshold 120574 from the proba-bility of false alarm with (27)
(5) Evaluation of the credibility of frequency estimationby comparing the statistic 119880 to the threshold 120574 with(25)The hypothesis119867
0is chosen if119880 lt 120574 Otherwise
1198671is accepted
5 Performance Analysis
Under 11986710158400assumption with 120582 = 120582
119905 the statistic defined in
(20) is expressed as
119880120582119905
1198671015840
0
=
(119881119888119905)13
minus [1 minus (29) (1 + 119887119905) 119888119905]
radic(29) (1 + 119887119905) 119888119905
sim 119873 (0 1) (28)
where 119888119905= 2 + 120582
119905and 119887 = 120582
119905119888119905 respectively
For 120582 ge 120582119905 the statistic is
119880120582
1198670=
(119881119888)13
minus [1 minus (29) (1 + 119887) 119888]
radic(29) (1 + 119887) 119888
sim 119873 (0 1) (29)
We note that for 120582 gt 120582119905 it can be rewritten as
119880120582119905
1198670=
(119881119888119905)13
minus [1 minus (29) (1 + 119887119905) 119888119905]
radic(29) (1 + 119887119905) 119888119905
=
(119881119888)13
minus 12058913
[1 minus (29) (1 + 119887119905) 119888119905]
12058913radic(29) (1 + 119887
119905) 119888119905
= ((
119881
119888
)
13
minus [1 minus
(29) (1 + 119887)
119888
]
+ [1 minus
(29) (1 + 119887)
119888
]
minus 12058913
[1 minus
(29) (1 + 119887119905)
119888119905
])
times (12058913radic(29) (1 + 119887
119905)
119888119905
)
minus1
(30)
where 120589 = 119888119905119888 and 120589 ge 1 According to the property of the
Gaussian distribution the statistic is given by
119880120582119905
1198670sim 119873(120583
120582119905
1198670 120590120582119905
1198670) (31)
Mathematical Problems in Engineering 5
where
120583120582119905
1198670= 119864 (119880
120582119905
1198670)
=
1 minus (29) (1 + 119887) 119888 minus 12058913
[1 minus (29) (1 + 119887119905) 119888119905]
12058913radic(29) (1 + 119887
119905) 119888119905
(32)
and the variance is
120590120582119905
1198670= radic119863(119880
120582119905
1198670) = 120589minus13
radic
(1 + 119887) 119888
(1 + 119887119905) 119888119905
(33)
Hence we obtain the probability of false alarm as
119875FA = 119875 119880120582119905
0le 120574 | 119867
0
= int
120574
minusinfin
1
radic2120587120590120582119905
1198670
exp[
[
minus
(119909 minus 120583120582119905
1198670)
2
2 (120590120582119905
1198670)
2
]
]
119889119909
asymp Φ(
120574 minus 120583120582119905
1198670
120590120582119905
1198670
)
(34)
Similarly under1198671assumption with 120582 lt 120582
119905 the statistic
119880120582119905
1198671=
(119881119888119905)13
minus [1 minus (29) (1 + 119887119905) 119888119905]
radic(29) (1 + 119887119905) 119888119905
= ((
119881
119888
)
13
minus [1 minus
(29) (1 + 119887)
119888
]
+ [1 minus
(29) (1 + 119887)
119888
]
minus 12058513
[1 minus
(29) (1 + 119887119905)
119888119905
])
times (12058513radic(29) (1 + 119887
119905)
119888119905
)
minus1
(35)
where 120585 = 119888119905119888 and 0 le 120585 le 1 From (35) 119880120582119905
1198671sim 119873(120583
120582119905
1198671 120590120582119905
1198671)
with the mean and variance are respectively given by
Figure 3 Effect of the false alarm probability on the detectionprobability obtained by the credibility test of a frequency estimate
Thus the detection probability can be expressed by
119875119863= Pr 119880120582119905
1198671le 120574 | 119867
1
= int
120574
minusinfin
1
radic2120587120590120582119905
1198671
exp[
[
minus
(119909 minus 120583120582119905
1198671)
2
2 (120590120582119905
1198671)
2
]
]
119889119909
asymp Φ(
120574 minus 120583120582119905
1198671
120590120582119905
1198671
)
(37)
From (34) and (37) we observe that the detection perfor-mance depends largely on the parameter 120582 which is thefunction of the number of samples received the SNR and thefrequency estimation error factor
6 Simulation Results
Monte Carlo simulations were carried out to study the behav-ior of the proposed algorithm in different environments Thesimulations were aimed at detecting whether the absoluteerror of a certain estimated frequency is greater than a quarterof the discrete sampling frequency Ten thousand MonteCarlo trials were performed for each condition Monte Carlotrials were run in each of the following conditions (1) thesinusoid signal contaminated by additive white Gaussiannoises is received (2) SNR = 10 log
1011986021205902 (3) the sampling
frequency is set to 100MHz except in 66
61 Influence of False Alarm Figure 3 shows the detectionbehavior for the credibility test by using the CI method withrespect to the false alarm and the comparisonswith the valuestheoretically calculated using (37) The carrier frequency ofthe sinusoid is 19801MHz while the sample size is 1024 The
Figure 4 Effect of the frequency estimation error factor on thedetection probability obtained by the credibility test of a frequencyestimate
initial phase is 1205876 and the frequency estimation error factoris 05 The SNR is varied from minus15 dB to 6 dB in steps of 3 dBThe values of probability of false alarm are fixed as 01 001and 0001 respectively
In the three cases the probability of detection (119875119863) is close
to 1 at an SNR greater than 0 dB The detection probability isincreased by a higher value of probability of false alarmWith120572 = 01 the probability of detection is close to 1 for an SNRof minus6 dB Moreover the derived theoretical 119875
119863is close to the
simulated value
62 Influence of the Frequency Estimation Error Figure 4shows the probability of detection with respect to the carrierfrequency estimation error and the comparisonswith the the-oretically calculated values The carrier frequency estimationerror is respectively equal to 03 04 and 05 for a sinusoidwith 1024 samples and the probability of false alarm equals001 The rest of the parameter settings of the signal in thesimulation remain the same as that stated in Section 61 Thedetection probability is increased by increasing the carrierfrequency estimation error factor For 120575 = 05 and an SNRclose to minus9 dB the probability of detection is approximatelyequal to 1
In fact a larger frequency estimation error factorimproves the mean of the proposed statistic 119880 and the dis-crimination between nonnull and null is easier In additionit can be observed from Figure 3 that the derived theoretical119875119863is very close to the simulated values in the three cases
63 Influence of the Number of Received Samples N Figure 5shows the detection performance of the credibility test versusthe received samples of the signal and the comparisons withthe theoretical calculated values The number of samples isequal to 512 1024 and 2048 respectively The carrier fre-quency estimation error is set to 04The rest of the parameter
00
01
02
03
04
05
06
07
08
09
10
SNR (dB)0 3 6
PD
minus15 minus12 minus9 minus6 minus3
N = 512 simulatedN = 512 theoreticalN = 1024 simulated
N = 1024 theoreticalN = 2048 simulatedN = 2048 theoretical
Figure 5 Effect of the number of received samples on the detectionprobability obtained by the credibility test of a frequency estimate
settings are the same as that stated in Section 62 In thethree cases with different sample numbers the detectionprobability is close to 1 for an SNR of 0 dBThe credibility testperformance is enhanced by the greater number of receivedsamples at the same SNR When the number of samplesis 2048 the detection probability is close to 1 for an SNRof minus6 dB We can observe from Figure 5 that the derivedtheoretical119875
119863is very close to the simulated values in the three
cases
64 Influence of Frequency Discretization We evaluated theperformance of our method in the case of different carrierfrequencies Figure 6 shows the performance of detectionversus the carrier frequency selected in the simulations Wedefined the discretization factor as 119889 = (119891
0minus 1198960Δ119865) where
1198960
= 199 and five frequencies were generated by setting119889 = 0 01 03 05 07 Clearly if 119889 = 0 the frequencyequals 1953125MHz which is just located on the discretefrequency bin When 119889 = 0 it indicates that the selectedfrequencies are not located on the discrete frequency binThe parameters are 119873 = 1024 120575 = 04 120579 = 1205876 and120572 = 001 Figure 6 shows that the detection probability doesnot depend on the frequencies we selected by using differentdiscretization factors and the same levels of performance areachieved in the five cases and are approximately identical tothe theoretically calculated performance in all the cases
65 Influence of the Initial Phase We investigated the impactof the initial phase on the performance of our methodFigure 7 shows the performance of detection versus the initialfrequency used in the simulations We set three initial phasesas 12058712 1205876 and 1205873 and the sample size as 1024 The restof the parameters set in each simulation are the same asthat mentioned in Section 63 Figure 7 shows that the initialphase of the sinusoid does not affect the detection probability
Figure 7 Effect of the initial phase of the signal on the detectionprobability obtained by the credibility test of a frequency estimate
and the same levels of performance are achieved in the twocases and are identical to the theoretical performances as thestatistic defined in (13) is not relevant to the phase because ofthe use of the modulus operation
66 Influence of the Sampling Frequency We now clarifythe impact of the sampling frequency on the performanceof the proposed method Figure 8 shows the performanceof detection versus the sampling frequency selected in thesimulations The other parameters are 119891
0= 19081MHz
119873 = 1024 120575 = 04 120579 = 1205876 and 120572 = 001 From the figurewe observe that the performance of detection is not affected
0 3 600
01
02
03
04
05
06
07
08
09
10
SNR (dB)
PD
minus15 minus12 minus9 minus6 minus3
Theoreticalfs = 120MHzfs = 100MHz
fs = 80MHz
Figure 8 Effect of the sampling frequency of the signal on thedetection probability obtained by the credibility test of a frequencyestimate
0 3 6
00
01
02
03
04
05
06
07
08
09
10
SNR (dB)minus12 minus9 minus6 minus3
PD (OR rule)PD (AND rule)
PD (CI only)PD (AM only)
PD
Figure 9 Probability of detection and probability of false alarmcomparison between the proposed scheme and others method
by the sampling frequency and is identical to the theoreticalperformance
67 Comparison to Other Methods Comparative experi-ments were also carried out for the proposed statisticAboutanios-Mulgrew (AM) method-based [21] detector anda combination of them by OR and AND rules respectivelyWe have described the AM-based detector that can alsobe used to test the credibility of the frequency estimatein the appendix The parameters are 119891
0= 19081MHz
119873 = 1024 120575 = 04 120579 = 1205876 and 120572 = 001 Thecurves of probabilities of detection for different SNRs areshown in Figure 9 We can observe that under an SNR
8 Mathematical Problems in Engineering
less than minus3 dB the probabilities of detection of the AM-basedmethod are greater than those of the CI-basedmethodFor the fusion method the probabilities of detection usingthe ldquoANDrdquo rule are slightly greater than those of the AM-based method and vice versa for the ldquoORrdquo rule that isthe probability of detection is the smallest when the SNR isless than minus3 dB When the SNRs are greater than minus3 dB theprobabilities of detection of the four detectors all approach1 Therefore under lower SNRs the more efficient fusionmethods combining the CI and AM methods need to befurther developed
It should be noted that calculating the threshold isdifficult because the exact probability distribution function ofthe AM-based statistic 119879AM defined in the appendix is hardto be expressed analytically Therefore here we introducethe bootstrap-based method [22 23] to obtain the thresh-old without depending on the analytical probability of thestatistic based on AM but the computational load increasesmany times because of repeated resampling operations in thebootstrap-based hypothesis test procedures
68 Case Study In this section we use the CI-based credibil-ity test statistic to evaluate three commonly used frequencyestimators the Rife algorithm [4] the maximum likelihoodestimator [6] by Newton iteration and the Aboutanious-Mulgrew estimator [21]The carrier frequency of the sinusoidis 19081MHz the initial phase is 1205876 and the sample size is1024 The range of SNR is from minus18 dB to minus6 dB The valueof probability of false alarm is set to 005 If we let 119863
119894(119894 =
0 1) be the event associated with the decision of choosing119867119894(119894 = 0 1) 119899
119894119895stands for the number of times that deciding
119863119895when hypothesis 119867
119894is correct Therefore Type I error
probability and Type II error probability can be calculatedusing 119899
01(11989900
+ 11989901) and 119899
10(11989910
+ 11989911) respectively The
error probability of the test can be expressed by 119875119864= (11989901+
11989910)10000 while 119875
119863= 11989911(11989911+ 11989910) indicates the ability
of the proposed method to detect the unreliable frequencyestimate which means that the maximum absolute bias of acertain frequency estimate (|Δ119891|) is less than a quarter of thediscrete sampling frequency interval (Δ119865)
Table 1 illustrates the performance of the credibility testfor the ML frequency estimator of the sinusoid by using theproposed CI-based statistic It should be remarked that thecredibility test performance is enhanced by increasing theSNR If the SNR is minus6 dB all the 10000 frequency estimatesbased on ML are reliable and the error probability of theproposed test approximates to 0 When the SNR decreases tominus12 dB 61 estimates are unreliable among 10000 simulationsBy using the proposed statistic all the 61 unreliable estimatescan be detected indicating a detection rate of about 100Among the 9939 reliable estimates there are 21 times whenestimates are mistakenly decided as unreliable in otherwords the error probability is about 021 When the SNR isdecreased to minus15 dB the error probability is about 081 andthe detection rate is greater than 95
Table 2 depicts the performance of the credibility test forthe Rife-based sinusoid frequency estimator The credibilitytest performance is enhanced by a larger SNR For an SNR ofminus6 dB all the 10000 times the frequency estimates obtained
Table 1 Credibility test performance of ML frequency estimator
by using the Rife algorithm are reliable (1198670) and the error
probability is approximately equal to 0 When the SNRdeclines to minus12 dB there are 71 unreliable estimates among10000 simulations By using the CI-based credibility teststatistic these 70 unreliable estimates can be detected andthe detection rate is about 986 On the other hand amongthe 9929 reliable estimates there are 25 times when theestimates are mistakenly decided as unreliable that is theerror probability is about 026 With the SNR decreased tominus15 dB the error probability is about 085 and the detectionrate is greater than 90
Table 3 provides the performance of the credibility test forthe AM-based sinusoid frequency estimator The credibilitytest performance is enhanced by a larger SNR Similar to thatdata in Tables 1 and 2 the test performance is improved bya larger SNR With the SNR decreased to minus15 dB the errorprobability is about 069 and the detection rate is greaterthan 80
Mathematical Problems in Engineering 9
Overall from Tables 2 and 3 we can observe that theproposed credibility test statistic can be used for effectivelyidentifying unreliable frequency estimates by using ML AMand Rife algorithms with a high detection probability and alow error probability when the SNR is greater than minus15 dBCompared to the estimator performances summarized inTables 1 and 2 the estimator performance summarized inTable 3 is the lowest because AM-based estimation has abetter performance than that of bothML and Rife algorithmsand the frequency estimation factors are small
In practice for the detected unreliable estimate we canabandon the results or estimate the frequency of the signalagain thereby enhancing the reliability of the entire signalprocessing unit For example if the FFT-based coarse fre-quency estimator is used in practice andwe set the parametersas 119891119904= 100MHz 119891
0= 1955MHz 119873 = 1000 and the
number of DFT119872 = 1000 there are two cases
(1) Case 1 When the SNRs (eg SNR = minus23 dB) arebelow the SNR threshold the maximum spectrumline cannot be detected correctly In this case theestimate error is larger than 025Δ119865 and the estimatewill be judged as incredible by the proposed CI-basedstatistic If we perform resampling over the frequencygrid by setting 119872 = 2000 the maximum spectrumline still cannot be detected As the absolute estima-tion errors are the same under the two scenariosthe resampling action cannot enhance the credibilityof the estimate and this kind of incredible estimateshould be abandoned
(2) Case 2When an SNR is greater than the SNR thresh-old an incredible frequency estimate is detectedbecause of the displacement of the maximum spec-trum line by the discretization In this case ifwe enlarge 119872 to 2000 by resampling in the fre-quency domain 119891
0can be just moved to the dis-
crete frequency bin Hence the estimation error canbe decreased and the credibility can be enhancedHowever in practice the true frequency cannot beknown by the user hence it is difficult to adjustthe resampling parameter to match the estimationexactly In this case we need to select the moreaccurate estimating method or abandon the resultobtained by original estimation
7 Conclusion
The paper presented a CI-based credibility test algorithm forfrequency estimation of a sinusoid A credibility assessmenttesting model is defined to analyze the mathematical charac-teristics of the correlations between the observed signal andthe reference signal under different hypotheses The test isperformed with the proposed threshold that is based on CIExperimental results revealed that a good performance wasachieved even at low SNRs Credibility tests were performedwith the ML frequency estimator using the proposed thresh-old based on CI as well with the Rife frequency estimator andthe Aboutanious-Mulgrew estimator Experimental results
revealed that a good performance was achieved by theproposed method even at low SNRs
Appendix
Thefrequency estimation credibility testmethod based on theAM estimator includes
(1) estimation of the frequency of the signal by using acertain estimation method
(2) construction of the reference signal 119910(119899) with (3) byusing the estimated frequency
1198910
(3) computation of the correlation between the receivedsignal and the reference signal and estimation ofthe frequency displacement Δ
119891 by the Aboutanios-Mulgrew frequency estimator [21]
(4) evaluation of the credibility of frequency estimationby comparing the statistic 119879AM = ||Δ
119891| minus 025Δ119865| to
the threshold th that can be calculated by a given falsealarm 120572 The hypothesis 119867
0is chosen if 119879AM lt th
Otherwise1198671is accepted
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This study is financially supported by the Natural ScienceFoundation of Jiangsu Province (Project no BK2011837) theNational Natural Science Foundation of China 2012 (Projectno 61201208) and the 333 Research Project of JiangsuProvince (Project no BRA2013171)
References
[1] D C Rife and R R Boorstyn ldquoSingle-tone parameter esti-mation from discrete-time observationrdquo IEEE Transactions onInformation Theory vol 20 no 5 pp 591ndash598 1974
[2] S Peleg and B Porat ldquoLinear FM signal parameter estima-tion from discrete-time observationsrdquo IEEE Transactions onAerospace and Electronic Systems vol 27 no 4 pp 607ndash6161991
[3] M Ghogho A Swami and T Durrani ldquoBlind estimation offrequency offset in the presence of unknown multipathrdquo inProceedings of the IEEE International Conference on PersonalWireless Communications pp 104ndash108 December 2000
[4] D C Rife and G A Vincent ldquoUse of the discrete Fouriertransform in the measurement of frequencies and levels oftonesrdquoThe Bell System Technical Journal vol 49 no 2 pp 197ndash228 1970
[5] J-R Liao andC-M Chen ldquoPhase correction of discrete Fouriertransform coefficients to reduce frequency estimation bias ofsingle tone complex sinusoidrdquo Signal Processing vol 94 no 1pp 108ndash117 2014
[6] T Abatzoglou ldquoA fast maximum likelihood algorithm forfrequency estimation of a sinusoid based on Newtonrsquos methodrdquo
10 Mathematical Problems in Engineering
IEEE Transactions on Acoustics Speech and Signal Processingvol 33 no 1 pp 77ndash89 1985
[7] Y Liu Z Nie Z Zhao and Q H Liu ldquoGeneralization ofiterative Fourier interpolation algorithms for single frequencyestimationrdquo Digital Signal Processing vol 21 no 1 pp 141ndash1492011
[8] G Wei C Yang and F-J Chen ldquoClosed-form frequencyestimator based on narrow-band approximation under noisyenvironmentrdquo Signal Processing vol 91 no 4 pp 841ndash851 2011
[9] C Yang and G Wei ldquoA noniterative frequency estimator withrational combination of three spectrum linesrdquo IEEE Transac-tions on Signal Processing vol 59 no 10 pp 5065ndash5070 2011
[10] C Candan and S Koc ldquoBeamspace approach for detection ofthe number of coherent sourcesrdquo in Proceedings of the IEEERadar Conference Ubiquitous Radar (RADAR rsquo12) pp 0913ndash0918 May 2012
[11] Y Cao G Wei and F-J Chen ldquoA closed-form expandedautocorrelationmethod for frequency estimation of a sinusoidrdquoSignal Processing vol 92 no 4 pp 885ndash892 2012
[12] F Qian S Leung Y Zhu W Wong D Pao and W LauldquoDamped sinusoidal signals parameter estimation in frequencydomainrdquo Signal Processing vol 92 no 2 pp 381ndash391 2012
[13] G Richard ELINT The Interception and Analysis of RadarSignals John Wiley amp Sons Artech House Dedham MassUSA 2nd edition 2006
[14] W Su J A Kosinski and M Yu ldquoDual-use of modulationrecognition techniques for digital communication signalsrdquo inProceedings of the IEEE Long Island Systems Applications andTechnology Conference (LISAT rsquo06) pp 1ndash6 Long Island NYUSA May 2006
[15] L Pucker ldquoReview of contemporaryspectrum sensingtechnologiesrdquo IEEE-SA P19006 Standards Group 2009 httpgrouperieeeorggroupsdyspan6documentswhite papersP19006 Sensor Surveypdf
[16] A Fehske J Gaeddert and JH Reed ldquoAnew approach to signalclassification using spectral correlation and neural networksrdquoin Proceedings of the 1st IEEE International Symposium on NewFrontiers in Dynamic Spectrum Access Networks (DySPAN rsquo05)pp 144ndash150 Baltimore Md USA November 2005
[17] W S Lin andK J R Liu ldquoModulation forensics for wireless dig-ital communicationsrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo08) pp 1789ndash1792 April 2008
[18] G-B Hu L-Z Xu and M Jin ldquoReliability testing for blindprocessing results of LFM signals based on NP criterionrdquo ActaElectronica Sinica vol 41 no 4 pp 739ndash743 2013
[19] A D Whalen Detection of Signals in Noise Academic PressNew York NY USA 2nd edition 1995
[20] M Sankaran ldquoOn the non-central chi-square distributionrdquoBiometrika vol 46 no 1-2 pp 235ndash237 1959
[21] E Aboutanios and B Mulgrew ldquoIterative frequency estimationby interpolation on Fourier coefficientsrdquo IEEE Transactions onSignal Processing vol 53 no 4 pp 1237ndash1242 2005
[22] AM Zoubir andD R Iskander Bootstrap Techniques for SignalProcessing Cambridge University Press Cambridge UK 2004
[23] A M Zoubir and D R Iskandler ldquoBootstrap methods andapplicationsrdquo IEEE Signal Processing Magazine vol 24 no 4pp 10ndash19 2007
Under hypothesis 1198671 because of the low SNR or other
possible reasons the maximum magnitude of the observedsignalsrsquo spectrums might not be located in the correct pointor the discrete frequency interval and the estimation errorΔ119891is larger than Δ1198654 that is 120575 ge 025 Thus
1003816100381610038161003816119864 (1198851)1003816100381610038161003816= 119873119860 sinc (120575) le 09003119873119860 (11)
In order to explain the relationship between the frequencyestimation error and the magnitude of the mean of 119885 wedefine the ratio as
119903 =
|119864(119885)|120575
|119864(119885)|0
asymp sinc (120575) (12)
where |119864(119885)|120575is the magnitude of the mean of 119885 at a
given 120575 According to the sinc function the ratio decreasesmonotonically with the frequency estimating error factor 120575supposing 0 le 120575 le 1 As Δ119891 increases 120575 increases andthe ratio 119903 decreases This relationship is shown in Figure 1For 120575 le 025 the ratio 119903 gt 09003 and it decreases slowlydeparting from 1 with the increase of 120575 For 120575 gt 025 theratio decreases dramatically Hence we can infer the absoluteerror of certain frequency estimation depending on the ratio119903 and obtain the credibility metric of the estimation
In order to facilitate convenient handling of the formuladerivation the statistic used to decide 119867
0and 119867
1can be
defined as follows
119881 =
|119885|2
120590119911
(13)
Clearly the random variable 119881 under 119867119894 119894 = 0 1
follows a standard noncentral law and its contribution can berepresented as [19]
119901119881|119867119894
(V) =
1
2
exp [minus12
(V2 + 1205822
119894)] 1198680(radic120582119894V) V ge 0
0 V lt 0
(14)
where 1198680(119909) is the zero-order modified Bessel function and
120582119894= (1205832
119911119877119894+ 1205832
119911119868119894)1205902
119911denotes the noncentrality
Under the assumption relating to1198670 that is 120575 le 025 we
obtain
1205820ge 120582119905=
2 [119873119860 sinc (025)]2
1198731205902
asymp 16211119873 SNR (15)
where 120582119905is the noncentrality parameter when 120575 = 025
Similarly under the1198671assumption we obtain
1205821= 2 [sinc(120575)]2119873 SNR lt 120582
119905 (16)
From the above discussion we find that under each assump-tion 119867
0or 1198671 the statistic 119881 follows a standard noncentral
law with different parameters Therefore the hypothesistest defined by (2) can be transformed to a parameter testinvolving 120582 that is given by
1198670 120582 ge 120582
119905 119867
1 120582 lt 120582
119905 (17)
32 Decision Rule and the Threshold We first consider thedegradation form of (17) given as
1198671015840
0 120582 = 120582
119905 119867
1015840
1 120582 = 120582
119905 (18)
Consequently letting 119888 = 2 + 120582 119887 = 120582119888 the statistic 119881 canbe approximated to a Gaussian distribution by119882 = (119881119888)
13
[20] whose mean and variance are given by
119864 (119882) = 1 minus
2 (1 + 119887)
9119888
119863 (119882) =
2 (1 + 119887)
9119888
(19)
respectively Thus the standard variable
119880 =
119882 minus 119864 (119882)
radic119863 (119882)
sim 119873 (0 1) (20)
and the hypothesis test expressed by (17) can be simplified by
1198671015840
0 120583119880= 0 119867
1015840
1 120583119880
= 0 (21)
where 120583119880denotes the mean of 119880
As the means of 119880 between the two assumptions aredifferent CI can be used as the decision rule to perform thehypothesis test Hence we write CI as
119875 |119880| le 120576 ge 1 minus
1
1205762 (22)
where 120576 gt 0 is a certain real number As 119880 approximatelyfollows a standard Gaussian law the probability of thesamples located in the range of threefold the standard error is
4 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7000
025
050
075
100
p(u)
u
minus10 minus9
minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
= minus3719 120572 = 0001
120575 = 01
120575 = 025
120575 = 04
120574
Figure 2 Distribution of the statistic 119880 for 120575 = 01 025 04 atSNR = 0 dB The carrier frequency is set to 19081MHz the lengthof received samples is 1024 the initial phase is 1205876 and the numberof simulations is 1000
greater than 09973 and we can adjust 120576 to obtain the desiredprobability Hence we decide1198671015840
1if
minus120576 le 119880 le 120576 (23)
Consequently the hypothesis test of (2) can be rewritten as
1198670 120583119880ge 0 119867
1 120583119880lt 0 (24)
Therefore we decide1198671if
119880 ge 120574 = minus120576 (25)
where 120574 is the thresholdThe probability of false alarm can bedefined as
119875119903119880 lt 120574119867
0 asymp Φ (120574) = 120572 (26)
where Φ(119909) = (1radic2120587) int
119909
minusinfin119890minus11990522119889119905 is the distribution
function of a standard Gaussian variable and 120572 is a givenfalse alarm Taking the inverse function of (26) leads to thethreshold
120574 = Φminus1(120572) (27)
whereΦminus1(119909) denotes the inverse of the distribution functionof a standard Gaussian probability density function (pdf)
Figure 2 shows the histogram and the fitted pdfs of thestatistic 119880 for different frequency estimator factors It can beseen that under a certain SNR the proposed statistic basedonCI approximated to the standardGaussian and the selectedthreshold can be used to distinguish119867
0and119867
1assumptions
effectively
4 Algorithm Summary
The proposed credibility test algorithm is composed of thefollowing steps
(1) Estimation of the frequency of the signal by using acertain estimation method
(2) Construction of the reference signal 119910(119899) with (3) byusing the estimated frequency
1198910
(3) Computation of the correlation between the receivedsignal and the reference signal and contraction of thestatistic 119880 with (20)
(4) Computation of the test threshold 120574 from the proba-bility of false alarm with (27)
(5) Evaluation of the credibility of frequency estimationby comparing the statistic 119880 to the threshold 120574 with(25)The hypothesis119867
0is chosen if119880 lt 120574 Otherwise
1198671is accepted
5 Performance Analysis
Under 11986710158400assumption with 120582 = 120582
119905 the statistic defined in
(20) is expressed as
119880120582119905
1198671015840
0
=
(119881119888119905)13
minus [1 minus (29) (1 + 119887119905) 119888119905]
radic(29) (1 + 119887119905) 119888119905
sim 119873 (0 1) (28)
where 119888119905= 2 + 120582
119905and 119887 = 120582
119905119888119905 respectively
For 120582 ge 120582119905 the statistic is
119880120582
1198670=
(119881119888)13
minus [1 minus (29) (1 + 119887) 119888]
radic(29) (1 + 119887) 119888
sim 119873 (0 1) (29)
We note that for 120582 gt 120582119905 it can be rewritten as
119880120582119905
1198670=
(119881119888119905)13
minus [1 minus (29) (1 + 119887119905) 119888119905]
radic(29) (1 + 119887119905) 119888119905
=
(119881119888)13
minus 12058913
[1 minus (29) (1 + 119887119905) 119888119905]
12058913radic(29) (1 + 119887
119905) 119888119905
= ((
119881
119888
)
13
minus [1 minus
(29) (1 + 119887)
119888
]
+ [1 minus
(29) (1 + 119887)
119888
]
minus 12058913
[1 minus
(29) (1 + 119887119905)
119888119905
])
times (12058913radic(29) (1 + 119887
119905)
119888119905
)
minus1
(30)
where 120589 = 119888119905119888 and 120589 ge 1 According to the property of the
Gaussian distribution the statistic is given by
119880120582119905
1198670sim 119873(120583
120582119905
1198670 120590120582119905
1198670) (31)
Mathematical Problems in Engineering 5
where
120583120582119905
1198670= 119864 (119880
120582119905
1198670)
=
1 minus (29) (1 + 119887) 119888 minus 12058913
[1 minus (29) (1 + 119887119905) 119888119905]
12058913radic(29) (1 + 119887
119905) 119888119905
(32)
and the variance is
120590120582119905
1198670= radic119863(119880
120582119905
1198670) = 120589minus13
radic
(1 + 119887) 119888
(1 + 119887119905) 119888119905
(33)
Hence we obtain the probability of false alarm as
119875FA = 119875 119880120582119905
0le 120574 | 119867
0
= int
120574
minusinfin
1
radic2120587120590120582119905
1198670
exp[
[
minus
(119909 minus 120583120582119905
1198670)
2
2 (120590120582119905
1198670)
2
]
]
119889119909
asymp Φ(
120574 minus 120583120582119905
1198670
120590120582119905
1198670
)
(34)
Similarly under1198671assumption with 120582 lt 120582
119905 the statistic
119880120582119905
1198671=
(119881119888119905)13
minus [1 minus (29) (1 + 119887119905) 119888119905]
radic(29) (1 + 119887119905) 119888119905
= ((
119881
119888
)
13
minus [1 minus
(29) (1 + 119887)
119888
]
+ [1 minus
(29) (1 + 119887)
119888
]
minus 12058513
[1 minus
(29) (1 + 119887119905)
119888119905
])
times (12058513radic(29) (1 + 119887
119905)
119888119905
)
minus1
(35)
where 120585 = 119888119905119888 and 0 le 120585 le 1 From (35) 119880120582119905
1198671sim 119873(120583
120582119905
1198671 120590120582119905
1198671)
with the mean and variance are respectively given by
Figure 3 Effect of the false alarm probability on the detectionprobability obtained by the credibility test of a frequency estimate
Thus the detection probability can be expressed by
119875119863= Pr 119880120582119905
1198671le 120574 | 119867
1
= int
120574
minusinfin
1
radic2120587120590120582119905
1198671
exp[
[
minus
(119909 minus 120583120582119905
1198671)
2
2 (120590120582119905
1198671)
2
]
]
119889119909
asymp Φ(
120574 minus 120583120582119905
1198671
120590120582119905
1198671
)
(37)
From (34) and (37) we observe that the detection perfor-mance depends largely on the parameter 120582 which is thefunction of the number of samples received the SNR and thefrequency estimation error factor
6 Simulation Results
Monte Carlo simulations were carried out to study the behav-ior of the proposed algorithm in different environments Thesimulations were aimed at detecting whether the absoluteerror of a certain estimated frequency is greater than a quarterof the discrete sampling frequency Ten thousand MonteCarlo trials were performed for each condition Monte Carlotrials were run in each of the following conditions (1) thesinusoid signal contaminated by additive white Gaussiannoises is received (2) SNR = 10 log
1011986021205902 (3) the sampling
frequency is set to 100MHz except in 66
61 Influence of False Alarm Figure 3 shows the detectionbehavior for the credibility test by using the CI method withrespect to the false alarm and the comparisonswith the valuestheoretically calculated using (37) The carrier frequency ofthe sinusoid is 19801MHz while the sample size is 1024 The
Figure 4 Effect of the frequency estimation error factor on thedetection probability obtained by the credibility test of a frequencyestimate
initial phase is 1205876 and the frequency estimation error factoris 05 The SNR is varied from minus15 dB to 6 dB in steps of 3 dBThe values of probability of false alarm are fixed as 01 001and 0001 respectively
In the three cases the probability of detection (119875119863) is close
to 1 at an SNR greater than 0 dB The detection probability isincreased by a higher value of probability of false alarmWith120572 = 01 the probability of detection is close to 1 for an SNRof minus6 dB Moreover the derived theoretical 119875
119863is close to the
simulated value
62 Influence of the Frequency Estimation Error Figure 4shows the probability of detection with respect to the carrierfrequency estimation error and the comparisonswith the the-oretically calculated values The carrier frequency estimationerror is respectively equal to 03 04 and 05 for a sinusoidwith 1024 samples and the probability of false alarm equals001 The rest of the parameter settings of the signal in thesimulation remain the same as that stated in Section 61 Thedetection probability is increased by increasing the carrierfrequency estimation error factor For 120575 = 05 and an SNRclose to minus9 dB the probability of detection is approximatelyequal to 1
In fact a larger frequency estimation error factorimproves the mean of the proposed statistic 119880 and the dis-crimination between nonnull and null is easier In additionit can be observed from Figure 3 that the derived theoretical119875119863is very close to the simulated values in the three cases
63 Influence of the Number of Received Samples N Figure 5shows the detection performance of the credibility test versusthe received samples of the signal and the comparisons withthe theoretical calculated values The number of samples isequal to 512 1024 and 2048 respectively The carrier fre-quency estimation error is set to 04The rest of the parameter
00
01
02
03
04
05
06
07
08
09
10
SNR (dB)0 3 6
PD
minus15 minus12 minus9 minus6 minus3
N = 512 simulatedN = 512 theoreticalN = 1024 simulated
N = 1024 theoreticalN = 2048 simulatedN = 2048 theoretical
Figure 5 Effect of the number of received samples on the detectionprobability obtained by the credibility test of a frequency estimate
settings are the same as that stated in Section 62 In thethree cases with different sample numbers the detectionprobability is close to 1 for an SNR of 0 dBThe credibility testperformance is enhanced by the greater number of receivedsamples at the same SNR When the number of samplesis 2048 the detection probability is close to 1 for an SNRof minus6 dB We can observe from Figure 5 that the derivedtheoretical119875
119863is very close to the simulated values in the three
cases
64 Influence of Frequency Discretization We evaluated theperformance of our method in the case of different carrierfrequencies Figure 6 shows the performance of detectionversus the carrier frequency selected in the simulations Wedefined the discretization factor as 119889 = (119891
0minus 1198960Δ119865) where
1198960
= 199 and five frequencies were generated by setting119889 = 0 01 03 05 07 Clearly if 119889 = 0 the frequencyequals 1953125MHz which is just located on the discretefrequency bin When 119889 = 0 it indicates that the selectedfrequencies are not located on the discrete frequency binThe parameters are 119873 = 1024 120575 = 04 120579 = 1205876 and120572 = 001 Figure 6 shows that the detection probability doesnot depend on the frequencies we selected by using differentdiscretization factors and the same levels of performance areachieved in the five cases and are approximately identical tothe theoretically calculated performance in all the cases
65 Influence of the Initial Phase We investigated the impactof the initial phase on the performance of our methodFigure 7 shows the performance of detection versus the initialfrequency used in the simulations We set three initial phasesas 12058712 1205876 and 1205873 and the sample size as 1024 The restof the parameters set in each simulation are the same asthat mentioned in Section 63 Figure 7 shows that the initialphase of the sinusoid does not affect the detection probability
Figure 7 Effect of the initial phase of the signal on the detectionprobability obtained by the credibility test of a frequency estimate
and the same levels of performance are achieved in the twocases and are identical to the theoretical performances as thestatistic defined in (13) is not relevant to the phase because ofthe use of the modulus operation
66 Influence of the Sampling Frequency We now clarifythe impact of the sampling frequency on the performanceof the proposed method Figure 8 shows the performanceof detection versus the sampling frequency selected in thesimulations The other parameters are 119891
0= 19081MHz
119873 = 1024 120575 = 04 120579 = 1205876 and 120572 = 001 From the figurewe observe that the performance of detection is not affected
0 3 600
01
02
03
04
05
06
07
08
09
10
SNR (dB)
PD
minus15 minus12 minus9 minus6 minus3
Theoreticalfs = 120MHzfs = 100MHz
fs = 80MHz
Figure 8 Effect of the sampling frequency of the signal on thedetection probability obtained by the credibility test of a frequencyestimate
0 3 6
00
01
02
03
04
05
06
07
08
09
10
SNR (dB)minus12 minus9 minus6 minus3
PD (OR rule)PD (AND rule)
PD (CI only)PD (AM only)
PD
Figure 9 Probability of detection and probability of false alarmcomparison between the proposed scheme and others method
by the sampling frequency and is identical to the theoreticalperformance
67 Comparison to Other Methods Comparative experi-ments were also carried out for the proposed statisticAboutanios-Mulgrew (AM) method-based [21] detector anda combination of them by OR and AND rules respectivelyWe have described the AM-based detector that can alsobe used to test the credibility of the frequency estimatein the appendix The parameters are 119891
0= 19081MHz
119873 = 1024 120575 = 04 120579 = 1205876 and 120572 = 001 Thecurves of probabilities of detection for different SNRs areshown in Figure 9 We can observe that under an SNR
8 Mathematical Problems in Engineering
less than minus3 dB the probabilities of detection of the AM-basedmethod are greater than those of the CI-basedmethodFor the fusion method the probabilities of detection usingthe ldquoANDrdquo rule are slightly greater than those of the AM-based method and vice versa for the ldquoORrdquo rule that isthe probability of detection is the smallest when the SNR isless than minus3 dB When the SNRs are greater than minus3 dB theprobabilities of detection of the four detectors all approach1 Therefore under lower SNRs the more efficient fusionmethods combining the CI and AM methods need to befurther developed
It should be noted that calculating the threshold isdifficult because the exact probability distribution function ofthe AM-based statistic 119879AM defined in the appendix is hardto be expressed analytically Therefore here we introducethe bootstrap-based method [22 23] to obtain the thresh-old without depending on the analytical probability of thestatistic based on AM but the computational load increasesmany times because of repeated resampling operations in thebootstrap-based hypothesis test procedures
68 Case Study In this section we use the CI-based credibil-ity test statistic to evaluate three commonly used frequencyestimators the Rife algorithm [4] the maximum likelihoodestimator [6] by Newton iteration and the Aboutanious-Mulgrew estimator [21]The carrier frequency of the sinusoidis 19081MHz the initial phase is 1205876 and the sample size is1024 The range of SNR is from minus18 dB to minus6 dB The valueof probability of false alarm is set to 005 If we let 119863
119894(119894 =
0 1) be the event associated with the decision of choosing119867119894(119894 = 0 1) 119899
119894119895stands for the number of times that deciding
119863119895when hypothesis 119867
119894is correct Therefore Type I error
probability and Type II error probability can be calculatedusing 119899
01(11989900
+ 11989901) and 119899
10(11989910
+ 11989911) respectively The
error probability of the test can be expressed by 119875119864= (11989901+
11989910)10000 while 119875
119863= 11989911(11989911+ 11989910) indicates the ability
of the proposed method to detect the unreliable frequencyestimate which means that the maximum absolute bias of acertain frequency estimate (|Δ119891|) is less than a quarter of thediscrete sampling frequency interval (Δ119865)
Table 1 illustrates the performance of the credibility testfor the ML frequency estimator of the sinusoid by using theproposed CI-based statistic It should be remarked that thecredibility test performance is enhanced by increasing theSNR If the SNR is minus6 dB all the 10000 frequency estimatesbased on ML are reliable and the error probability of theproposed test approximates to 0 When the SNR decreases tominus12 dB 61 estimates are unreliable among 10000 simulationsBy using the proposed statistic all the 61 unreliable estimatescan be detected indicating a detection rate of about 100Among the 9939 reliable estimates there are 21 times whenestimates are mistakenly decided as unreliable in otherwords the error probability is about 021 When the SNR isdecreased to minus15 dB the error probability is about 081 andthe detection rate is greater than 95
Table 2 depicts the performance of the credibility test forthe Rife-based sinusoid frequency estimator The credibilitytest performance is enhanced by a larger SNR For an SNR ofminus6 dB all the 10000 times the frequency estimates obtained
Table 1 Credibility test performance of ML frequency estimator
by using the Rife algorithm are reliable (1198670) and the error
probability is approximately equal to 0 When the SNRdeclines to minus12 dB there are 71 unreliable estimates among10000 simulations By using the CI-based credibility teststatistic these 70 unreliable estimates can be detected andthe detection rate is about 986 On the other hand amongthe 9929 reliable estimates there are 25 times when theestimates are mistakenly decided as unreliable that is theerror probability is about 026 With the SNR decreased tominus15 dB the error probability is about 085 and the detectionrate is greater than 90
Table 3 provides the performance of the credibility test forthe AM-based sinusoid frequency estimator The credibilitytest performance is enhanced by a larger SNR Similar to thatdata in Tables 1 and 2 the test performance is improved bya larger SNR With the SNR decreased to minus15 dB the errorprobability is about 069 and the detection rate is greaterthan 80
Mathematical Problems in Engineering 9
Overall from Tables 2 and 3 we can observe that theproposed credibility test statistic can be used for effectivelyidentifying unreliable frequency estimates by using ML AMand Rife algorithms with a high detection probability and alow error probability when the SNR is greater than minus15 dBCompared to the estimator performances summarized inTables 1 and 2 the estimator performance summarized inTable 3 is the lowest because AM-based estimation has abetter performance than that of bothML and Rife algorithmsand the frequency estimation factors are small
In practice for the detected unreliable estimate we canabandon the results or estimate the frequency of the signalagain thereby enhancing the reliability of the entire signalprocessing unit For example if the FFT-based coarse fre-quency estimator is used in practice andwe set the parametersas 119891119904= 100MHz 119891
0= 1955MHz 119873 = 1000 and the
number of DFT119872 = 1000 there are two cases
(1) Case 1 When the SNRs (eg SNR = minus23 dB) arebelow the SNR threshold the maximum spectrumline cannot be detected correctly In this case theestimate error is larger than 025Δ119865 and the estimatewill be judged as incredible by the proposed CI-basedstatistic If we perform resampling over the frequencygrid by setting 119872 = 2000 the maximum spectrumline still cannot be detected As the absolute estima-tion errors are the same under the two scenariosthe resampling action cannot enhance the credibilityof the estimate and this kind of incredible estimateshould be abandoned
(2) Case 2When an SNR is greater than the SNR thresh-old an incredible frequency estimate is detectedbecause of the displacement of the maximum spec-trum line by the discretization In this case ifwe enlarge 119872 to 2000 by resampling in the fre-quency domain 119891
0can be just moved to the dis-
crete frequency bin Hence the estimation error canbe decreased and the credibility can be enhancedHowever in practice the true frequency cannot beknown by the user hence it is difficult to adjustthe resampling parameter to match the estimationexactly In this case we need to select the moreaccurate estimating method or abandon the resultobtained by original estimation
7 Conclusion
The paper presented a CI-based credibility test algorithm forfrequency estimation of a sinusoid A credibility assessmenttesting model is defined to analyze the mathematical charac-teristics of the correlations between the observed signal andthe reference signal under different hypotheses The test isperformed with the proposed threshold that is based on CIExperimental results revealed that a good performance wasachieved even at low SNRs Credibility tests were performedwith the ML frequency estimator using the proposed thresh-old based on CI as well with the Rife frequency estimator andthe Aboutanious-Mulgrew estimator Experimental results
revealed that a good performance was achieved by theproposed method even at low SNRs
Appendix
Thefrequency estimation credibility testmethod based on theAM estimator includes
(1) estimation of the frequency of the signal by using acertain estimation method
(2) construction of the reference signal 119910(119899) with (3) byusing the estimated frequency
1198910
(3) computation of the correlation between the receivedsignal and the reference signal and estimation ofthe frequency displacement Δ
119891 by the Aboutanios-Mulgrew frequency estimator [21]
(4) evaluation of the credibility of frequency estimationby comparing the statistic 119879AM = ||Δ
119891| minus 025Δ119865| to
the threshold th that can be calculated by a given falsealarm 120572 The hypothesis 119867
0is chosen if 119879AM lt th
Otherwise1198671is accepted
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This study is financially supported by the Natural ScienceFoundation of Jiangsu Province (Project no BK2011837) theNational Natural Science Foundation of China 2012 (Projectno 61201208) and the 333 Research Project of JiangsuProvince (Project no BRA2013171)
References
[1] D C Rife and R R Boorstyn ldquoSingle-tone parameter esti-mation from discrete-time observationrdquo IEEE Transactions onInformation Theory vol 20 no 5 pp 591ndash598 1974
[2] S Peleg and B Porat ldquoLinear FM signal parameter estima-tion from discrete-time observationsrdquo IEEE Transactions onAerospace and Electronic Systems vol 27 no 4 pp 607ndash6161991
[3] M Ghogho A Swami and T Durrani ldquoBlind estimation offrequency offset in the presence of unknown multipathrdquo inProceedings of the IEEE International Conference on PersonalWireless Communications pp 104ndash108 December 2000
[4] D C Rife and G A Vincent ldquoUse of the discrete Fouriertransform in the measurement of frequencies and levels oftonesrdquoThe Bell System Technical Journal vol 49 no 2 pp 197ndash228 1970
[5] J-R Liao andC-M Chen ldquoPhase correction of discrete Fouriertransform coefficients to reduce frequency estimation bias ofsingle tone complex sinusoidrdquo Signal Processing vol 94 no 1pp 108ndash117 2014
[6] T Abatzoglou ldquoA fast maximum likelihood algorithm forfrequency estimation of a sinusoid based on Newtonrsquos methodrdquo
10 Mathematical Problems in Engineering
IEEE Transactions on Acoustics Speech and Signal Processingvol 33 no 1 pp 77ndash89 1985
[7] Y Liu Z Nie Z Zhao and Q H Liu ldquoGeneralization ofiterative Fourier interpolation algorithms for single frequencyestimationrdquo Digital Signal Processing vol 21 no 1 pp 141ndash1492011
[8] G Wei C Yang and F-J Chen ldquoClosed-form frequencyestimator based on narrow-band approximation under noisyenvironmentrdquo Signal Processing vol 91 no 4 pp 841ndash851 2011
[9] C Yang and G Wei ldquoA noniterative frequency estimator withrational combination of three spectrum linesrdquo IEEE Transac-tions on Signal Processing vol 59 no 10 pp 5065ndash5070 2011
[10] C Candan and S Koc ldquoBeamspace approach for detection ofthe number of coherent sourcesrdquo in Proceedings of the IEEERadar Conference Ubiquitous Radar (RADAR rsquo12) pp 0913ndash0918 May 2012
[11] Y Cao G Wei and F-J Chen ldquoA closed-form expandedautocorrelationmethod for frequency estimation of a sinusoidrdquoSignal Processing vol 92 no 4 pp 885ndash892 2012
[12] F Qian S Leung Y Zhu W Wong D Pao and W LauldquoDamped sinusoidal signals parameter estimation in frequencydomainrdquo Signal Processing vol 92 no 2 pp 381ndash391 2012
[13] G Richard ELINT The Interception and Analysis of RadarSignals John Wiley amp Sons Artech House Dedham MassUSA 2nd edition 2006
[14] W Su J A Kosinski and M Yu ldquoDual-use of modulationrecognition techniques for digital communication signalsrdquo inProceedings of the IEEE Long Island Systems Applications andTechnology Conference (LISAT rsquo06) pp 1ndash6 Long Island NYUSA May 2006
[15] L Pucker ldquoReview of contemporaryspectrum sensingtechnologiesrdquo IEEE-SA P19006 Standards Group 2009 httpgrouperieeeorggroupsdyspan6documentswhite papersP19006 Sensor Surveypdf
[16] A Fehske J Gaeddert and JH Reed ldquoAnew approach to signalclassification using spectral correlation and neural networksrdquoin Proceedings of the 1st IEEE International Symposium on NewFrontiers in Dynamic Spectrum Access Networks (DySPAN rsquo05)pp 144ndash150 Baltimore Md USA November 2005
[17] W S Lin andK J R Liu ldquoModulation forensics for wireless dig-ital communicationsrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo08) pp 1789ndash1792 April 2008
[18] G-B Hu L-Z Xu and M Jin ldquoReliability testing for blindprocessing results of LFM signals based on NP criterionrdquo ActaElectronica Sinica vol 41 no 4 pp 739ndash743 2013
[19] A D Whalen Detection of Signals in Noise Academic PressNew York NY USA 2nd edition 1995
[20] M Sankaran ldquoOn the non-central chi-square distributionrdquoBiometrika vol 46 no 1-2 pp 235ndash237 1959
[21] E Aboutanios and B Mulgrew ldquoIterative frequency estimationby interpolation on Fourier coefficientsrdquo IEEE Transactions onSignal Processing vol 53 no 4 pp 1237ndash1242 2005
[22] AM Zoubir andD R Iskander Bootstrap Techniques for SignalProcessing Cambridge University Press Cambridge UK 2004
[23] A M Zoubir and D R Iskandler ldquoBootstrap methods andapplicationsrdquo IEEE Signal Processing Magazine vol 24 no 4pp 10ndash19 2007
Figure 2 Distribution of the statistic 119880 for 120575 = 01 025 04 atSNR = 0 dB The carrier frequency is set to 19081MHz the lengthof received samples is 1024 the initial phase is 1205876 and the numberof simulations is 1000
greater than 09973 and we can adjust 120576 to obtain the desiredprobability Hence we decide1198671015840
1if
minus120576 le 119880 le 120576 (23)
Consequently the hypothesis test of (2) can be rewritten as
1198670 120583119880ge 0 119867
1 120583119880lt 0 (24)
Therefore we decide1198671if
119880 ge 120574 = minus120576 (25)
where 120574 is the thresholdThe probability of false alarm can bedefined as
119875119903119880 lt 120574119867
0 asymp Φ (120574) = 120572 (26)
where Φ(119909) = (1radic2120587) int
119909
minusinfin119890minus11990522119889119905 is the distribution
function of a standard Gaussian variable and 120572 is a givenfalse alarm Taking the inverse function of (26) leads to thethreshold
120574 = Φminus1(120572) (27)
whereΦminus1(119909) denotes the inverse of the distribution functionof a standard Gaussian probability density function (pdf)
Figure 2 shows the histogram and the fitted pdfs of thestatistic 119880 for different frequency estimator factors It can beseen that under a certain SNR the proposed statistic basedonCI approximated to the standardGaussian and the selectedthreshold can be used to distinguish119867
0and119867
1assumptions
effectively
4 Algorithm Summary
The proposed credibility test algorithm is composed of thefollowing steps
(1) Estimation of the frequency of the signal by using acertain estimation method
(2) Construction of the reference signal 119910(119899) with (3) byusing the estimated frequency
1198910
(3) Computation of the correlation between the receivedsignal and the reference signal and contraction of thestatistic 119880 with (20)
(4) Computation of the test threshold 120574 from the proba-bility of false alarm with (27)
(5) Evaluation of the credibility of frequency estimationby comparing the statistic 119880 to the threshold 120574 with(25)The hypothesis119867
0is chosen if119880 lt 120574 Otherwise
1198671is accepted
5 Performance Analysis
Under 11986710158400assumption with 120582 = 120582
119905 the statistic defined in
(20) is expressed as
119880120582119905
1198671015840
0
=
(119881119888119905)13
minus [1 minus (29) (1 + 119887119905) 119888119905]
radic(29) (1 + 119887119905) 119888119905
sim 119873 (0 1) (28)
where 119888119905= 2 + 120582
119905and 119887 = 120582
119905119888119905 respectively
For 120582 ge 120582119905 the statistic is
119880120582
1198670=
(119881119888)13
minus [1 minus (29) (1 + 119887) 119888]
radic(29) (1 + 119887) 119888
sim 119873 (0 1) (29)
We note that for 120582 gt 120582119905 it can be rewritten as
119880120582119905
1198670=
(119881119888119905)13
minus [1 minus (29) (1 + 119887119905) 119888119905]
radic(29) (1 + 119887119905) 119888119905
=
(119881119888)13
minus 12058913
[1 minus (29) (1 + 119887119905) 119888119905]
12058913radic(29) (1 + 119887
119905) 119888119905
= ((
119881
119888
)
13
minus [1 minus
(29) (1 + 119887)
119888
]
+ [1 minus
(29) (1 + 119887)
119888
]
minus 12058913
[1 minus
(29) (1 + 119887119905)
119888119905
])
times (12058913radic(29) (1 + 119887
119905)
119888119905
)
minus1
(30)
where 120589 = 119888119905119888 and 120589 ge 1 According to the property of the
Gaussian distribution the statistic is given by
119880120582119905
1198670sim 119873(120583
120582119905
1198670 120590120582119905
1198670) (31)
Mathematical Problems in Engineering 5
where
120583120582119905
1198670= 119864 (119880
120582119905
1198670)
=
1 minus (29) (1 + 119887) 119888 minus 12058913
[1 minus (29) (1 + 119887119905) 119888119905]
12058913radic(29) (1 + 119887
119905) 119888119905
(32)
and the variance is
120590120582119905
1198670= radic119863(119880
120582119905
1198670) = 120589minus13
radic
(1 + 119887) 119888
(1 + 119887119905) 119888119905
(33)
Hence we obtain the probability of false alarm as
119875FA = 119875 119880120582119905
0le 120574 | 119867
0
= int
120574
minusinfin
1
radic2120587120590120582119905
1198670
exp[
[
minus
(119909 minus 120583120582119905
1198670)
2
2 (120590120582119905
1198670)
2
]
]
119889119909
asymp Φ(
120574 minus 120583120582119905
1198670
120590120582119905
1198670
)
(34)
Similarly under1198671assumption with 120582 lt 120582
119905 the statistic
119880120582119905
1198671=
(119881119888119905)13
minus [1 minus (29) (1 + 119887119905) 119888119905]
radic(29) (1 + 119887119905) 119888119905
= ((
119881
119888
)
13
minus [1 minus
(29) (1 + 119887)
119888
]
+ [1 minus
(29) (1 + 119887)
119888
]
minus 12058513
[1 minus
(29) (1 + 119887119905)
119888119905
])
times (12058513radic(29) (1 + 119887
119905)
119888119905
)
minus1
(35)
where 120585 = 119888119905119888 and 0 le 120585 le 1 From (35) 119880120582119905
1198671sim 119873(120583
120582119905
1198671 120590120582119905
1198671)
with the mean and variance are respectively given by
Figure 3 Effect of the false alarm probability on the detectionprobability obtained by the credibility test of a frequency estimate
Thus the detection probability can be expressed by
119875119863= Pr 119880120582119905
1198671le 120574 | 119867
1
= int
120574
minusinfin
1
radic2120587120590120582119905
1198671
exp[
[
minus
(119909 minus 120583120582119905
1198671)
2
2 (120590120582119905
1198671)
2
]
]
119889119909
asymp Φ(
120574 minus 120583120582119905
1198671
120590120582119905
1198671
)
(37)
From (34) and (37) we observe that the detection perfor-mance depends largely on the parameter 120582 which is thefunction of the number of samples received the SNR and thefrequency estimation error factor
6 Simulation Results
Monte Carlo simulations were carried out to study the behav-ior of the proposed algorithm in different environments Thesimulations were aimed at detecting whether the absoluteerror of a certain estimated frequency is greater than a quarterof the discrete sampling frequency Ten thousand MonteCarlo trials were performed for each condition Monte Carlotrials were run in each of the following conditions (1) thesinusoid signal contaminated by additive white Gaussiannoises is received (2) SNR = 10 log
1011986021205902 (3) the sampling
frequency is set to 100MHz except in 66
61 Influence of False Alarm Figure 3 shows the detectionbehavior for the credibility test by using the CI method withrespect to the false alarm and the comparisonswith the valuestheoretically calculated using (37) The carrier frequency ofthe sinusoid is 19801MHz while the sample size is 1024 The
Figure 4 Effect of the frequency estimation error factor on thedetection probability obtained by the credibility test of a frequencyestimate
initial phase is 1205876 and the frequency estimation error factoris 05 The SNR is varied from minus15 dB to 6 dB in steps of 3 dBThe values of probability of false alarm are fixed as 01 001and 0001 respectively
In the three cases the probability of detection (119875119863) is close
to 1 at an SNR greater than 0 dB The detection probability isincreased by a higher value of probability of false alarmWith120572 = 01 the probability of detection is close to 1 for an SNRof minus6 dB Moreover the derived theoretical 119875
119863is close to the
simulated value
62 Influence of the Frequency Estimation Error Figure 4shows the probability of detection with respect to the carrierfrequency estimation error and the comparisonswith the the-oretically calculated values The carrier frequency estimationerror is respectively equal to 03 04 and 05 for a sinusoidwith 1024 samples and the probability of false alarm equals001 The rest of the parameter settings of the signal in thesimulation remain the same as that stated in Section 61 Thedetection probability is increased by increasing the carrierfrequency estimation error factor For 120575 = 05 and an SNRclose to minus9 dB the probability of detection is approximatelyequal to 1
In fact a larger frequency estimation error factorimproves the mean of the proposed statistic 119880 and the dis-crimination between nonnull and null is easier In additionit can be observed from Figure 3 that the derived theoretical119875119863is very close to the simulated values in the three cases
63 Influence of the Number of Received Samples N Figure 5shows the detection performance of the credibility test versusthe received samples of the signal and the comparisons withthe theoretical calculated values The number of samples isequal to 512 1024 and 2048 respectively The carrier fre-quency estimation error is set to 04The rest of the parameter
00
01
02
03
04
05
06
07
08
09
10
SNR (dB)0 3 6
PD
minus15 minus12 minus9 minus6 minus3
N = 512 simulatedN = 512 theoreticalN = 1024 simulated
N = 1024 theoreticalN = 2048 simulatedN = 2048 theoretical
Figure 5 Effect of the number of received samples on the detectionprobability obtained by the credibility test of a frequency estimate
settings are the same as that stated in Section 62 In thethree cases with different sample numbers the detectionprobability is close to 1 for an SNR of 0 dBThe credibility testperformance is enhanced by the greater number of receivedsamples at the same SNR When the number of samplesis 2048 the detection probability is close to 1 for an SNRof minus6 dB We can observe from Figure 5 that the derivedtheoretical119875
119863is very close to the simulated values in the three
cases
64 Influence of Frequency Discretization We evaluated theperformance of our method in the case of different carrierfrequencies Figure 6 shows the performance of detectionversus the carrier frequency selected in the simulations Wedefined the discretization factor as 119889 = (119891
0minus 1198960Δ119865) where
1198960
= 199 and five frequencies were generated by setting119889 = 0 01 03 05 07 Clearly if 119889 = 0 the frequencyequals 1953125MHz which is just located on the discretefrequency bin When 119889 = 0 it indicates that the selectedfrequencies are not located on the discrete frequency binThe parameters are 119873 = 1024 120575 = 04 120579 = 1205876 and120572 = 001 Figure 6 shows that the detection probability doesnot depend on the frequencies we selected by using differentdiscretization factors and the same levels of performance areachieved in the five cases and are approximately identical tothe theoretically calculated performance in all the cases
65 Influence of the Initial Phase We investigated the impactof the initial phase on the performance of our methodFigure 7 shows the performance of detection versus the initialfrequency used in the simulations We set three initial phasesas 12058712 1205876 and 1205873 and the sample size as 1024 The restof the parameters set in each simulation are the same asthat mentioned in Section 63 Figure 7 shows that the initialphase of the sinusoid does not affect the detection probability
Figure 7 Effect of the initial phase of the signal on the detectionprobability obtained by the credibility test of a frequency estimate
and the same levels of performance are achieved in the twocases and are identical to the theoretical performances as thestatistic defined in (13) is not relevant to the phase because ofthe use of the modulus operation
66 Influence of the Sampling Frequency We now clarifythe impact of the sampling frequency on the performanceof the proposed method Figure 8 shows the performanceof detection versus the sampling frequency selected in thesimulations The other parameters are 119891
0= 19081MHz
119873 = 1024 120575 = 04 120579 = 1205876 and 120572 = 001 From the figurewe observe that the performance of detection is not affected
0 3 600
01
02
03
04
05
06
07
08
09
10
SNR (dB)
PD
minus15 minus12 minus9 minus6 minus3
Theoreticalfs = 120MHzfs = 100MHz
fs = 80MHz
Figure 8 Effect of the sampling frequency of the signal on thedetection probability obtained by the credibility test of a frequencyestimate
0 3 6
00
01
02
03
04
05
06
07
08
09
10
SNR (dB)minus12 minus9 minus6 minus3
PD (OR rule)PD (AND rule)
PD (CI only)PD (AM only)
PD
Figure 9 Probability of detection and probability of false alarmcomparison between the proposed scheme and others method
by the sampling frequency and is identical to the theoreticalperformance
67 Comparison to Other Methods Comparative experi-ments were also carried out for the proposed statisticAboutanios-Mulgrew (AM) method-based [21] detector anda combination of them by OR and AND rules respectivelyWe have described the AM-based detector that can alsobe used to test the credibility of the frequency estimatein the appendix The parameters are 119891
0= 19081MHz
119873 = 1024 120575 = 04 120579 = 1205876 and 120572 = 001 Thecurves of probabilities of detection for different SNRs areshown in Figure 9 We can observe that under an SNR
8 Mathematical Problems in Engineering
less than minus3 dB the probabilities of detection of the AM-basedmethod are greater than those of the CI-basedmethodFor the fusion method the probabilities of detection usingthe ldquoANDrdquo rule are slightly greater than those of the AM-based method and vice versa for the ldquoORrdquo rule that isthe probability of detection is the smallest when the SNR isless than minus3 dB When the SNRs are greater than minus3 dB theprobabilities of detection of the four detectors all approach1 Therefore under lower SNRs the more efficient fusionmethods combining the CI and AM methods need to befurther developed
It should be noted that calculating the threshold isdifficult because the exact probability distribution function ofthe AM-based statistic 119879AM defined in the appendix is hardto be expressed analytically Therefore here we introducethe bootstrap-based method [22 23] to obtain the thresh-old without depending on the analytical probability of thestatistic based on AM but the computational load increasesmany times because of repeated resampling operations in thebootstrap-based hypothesis test procedures
68 Case Study In this section we use the CI-based credibil-ity test statistic to evaluate three commonly used frequencyestimators the Rife algorithm [4] the maximum likelihoodestimator [6] by Newton iteration and the Aboutanious-Mulgrew estimator [21]The carrier frequency of the sinusoidis 19081MHz the initial phase is 1205876 and the sample size is1024 The range of SNR is from minus18 dB to minus6 dB The valueof probability of false alarm is set to 005 If we let 119863
119894(119894 =
0 1) be the event associated with the decision of choosing119867119894(119894 = 0 1) 119899
119894119895stands for the number of times that deciding
119863119895when hypothesis 119867
119894is correct Therefore Type I error
probability and Type II error probability can be calculatedusing 119899
01(11989900
+ 11989901) and 119899
10(11989910
+ 11989911) respectively The
error probability of the test can be expressed by 119875119864= (11989901+
11989910)10000 while 119875
119863= 11989911(11989911+ 11989910) indicates the ability
of the proposed method to detect the unreliable frequencyestimate which means that the maximum absolute bias of acertain frequency estimate (|Δ119891|) is less than a quarter of thediscrete sampling frequency interval (Δ119865)
Table 1 illustrates the performance of the credibility testfor the ML frequency estimator of the sinusoid by using theproposed CI-based statistic It should be remarked that thecredibility test performance is enhanced by increasing theSNR If the SNR is minus6 dB all the 10000 frequency estimatesbased on ML are reliable and the error probability of theproposed test approximates to 0 When the SNR decreases tominus12 dB 61 estimates are unreliable among 10000 simulationsBy using the proposed statistic all the 61 unreliable estimatescan be detected indicating a detection rate of about 100Among the 9939 reliable estimates there are 21 times whenestimates are mistakenly decided as unreliable in otherwords the error probability is about 021 When the SNR isdecreased to minus15 dB the error probability is about 081 andthe detection rate is greater than 95
Table 2 depicts the performance of the credibility test forthe Rife-based sinusoid frequency estimator The credibilitytest performance is enhanced by a larger SNR For an SNR ofminus6 dB all the 10000 times the frequency estimates obtained
Table 1 Credibility test performance of ML frequency estimator
by using the Rife algorithm are reliable (1198670) and the error
probability is approximately equal to 0 When the SNRdeclines to minus12 dB there are 71 unreliable estimates among10000 simulations By using the CI-based credibility teststatistic these 70 unreliable estimates can be detected andthe detection rate is about 986 On the other hand amongthe 9929 reliable estimates there are 25 times when theestimates are mistakenly decided as unreliable that is theerror probability is about 026 With the SNR decreased tominus15 dB the error probability is about 085 and the detectionrate is greater than 90
Table 3 provides the performance of the credibility test forthe AM-based sinusoid frequency estimator The credibilitytest performance is enhanced by a larger SNR Similar to thatdata in Tables 1 and 2 the test performance is improved bya larger SNR With the SNR decreased to minus15 dB the errorprobability is about 069 and the detection rate is greaterthan 80
Mathematical Problems in Engineering 9
Overall from Tables 2 and 3 we can observe that theproposed credibility test statistic can be used for effectivelyidentifying unreliable frequency estimates by using ML AMand Rife algorithms with a high detection probability and alow error probability when the SNR is greater than minus15 dBCompared to the estimator performances summarized inTables 1 and 2 the estimator performance summarized inTable 3 is the lowest because AM-based estimation has abetter performance than that of bothML and Rife algorithmsand the frequency estimation factors are small
In practice for the detected unreliable estimate we canabandon the results or estimate the frequency of the signalagain thereby enhancing the reliability of the entire signalprocessing unit For example if the FFT-based coarse fre-quency estimator is used in practice andwe set the parametersas 119891119904= 100MHz 119891
0= 1955MHz 119873 = 1000 and the
number of DFT119872 = 1000 there are two cases
(1) Case 1 When the SNRs (eg SNR = minus23 dB) arebelow the SNR threshold the maximum spectrumline cannot be detected correctly In this case theestimate error is larger than 025Δ119865 and the estimatewill be judged as incredible by the proposed CI-basedstatistic If we perform resampling over the frequencygrid by setting 119872 = 2000 the maximum spectrumline still cannot be detected As the absolute estima-tion errors are the same under the two scenariosthe resampling action cannot enhance the credibilityof the estimate and this kind of incredible estimateshould be abandoned
(2) Case 2When an SNR is greater than the SNR thresh-old an incredible frequency estimate is detectedbecause of the displacement of the maximum spec-trum line by the discretization In this case ifwe enlarge 119872 to 2000 by resampling in the fre-quency domain 119891
0can be just moved to the dis-
crete frequency bin Hence the estimation error canbe decreased and the credibility can be enhancedHowever in practice the true frequency cannot beknown by the user hence it is difficult to adjustthe resampling parameter to match the estimationexactly In this case we need to select the moreaccurate estimating method or abandon the resultobtained by original estimation
7 Conclusion
The paper presented a CI-based credibility test algorithm forfrequency estimation of a sinusoid A credibility assessmenttesting model is defined to analyze the mathematical charac-teristics of the correlations between the observed signal andthe reference signal under different hypotheses The test isperformed with the proposed threshold that is based on CIExperimental results revealed that a good performance wasachieved even at low SNRs Credibility tests were performedwith the ML frequency estimator using the proposed thresh-old based on CI as well with the Rife frequency estimator andthe Aboutanious-Mulgrew estimator Experimental results
revealed that a good performance was achieved by theproposed method even at low SNRs
Appendix
Thefrequency estimation credibility testmethod based on theAM estimator includes
(1) estimation of the frequency of the signal by using acertain estimation method
(2) construction of the reference signal 119910(119899) with (3) byusing the estimated frequency
1198910
(3) computation of the correlation between the receivedsignal and the reference signal and estimation ofthe frequency displacement Δ
119891 by the Aboutanios-Mulgrew frequency estimator [21]
(4) evaluation of the credibility of frequency estimationby comparing the statistic 119879AM = ||Δ
119891| minus 025Δ119865| to
the threshold th that can be calculated by a given falsealarm 120572 The hypothesis 119867
0is chosen if 119879AM lt th
Otherwise1198671is accepted
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This study is financially supported by the Natural ScienceFoundation of Jiangsu Province (Project no BK2011837) theNational Natural Science Foundation of China 2012 (Projectno 61201208) and the 333 Research Project of JiangsuProvince (Project no BRA2013171)
References
[1] D C Rife and R R Boorstyn ldquoSingle-tone parameter esti-mation from discrete-time observationrdquo IEEE Transactions onInformation Theory vol 20 no 5 pp 591ndash598 1974
[2] S Peleg and B Porat ldquoLinear FM signal parameter estima-tion from discrete-time observationsrdquo IEEE Transactions onAerospace and Electronic Systems vol 27 no 4 pp 607ndash6161991
[3] M Ghogho A Swami and T Durrani ldquoBlind estimation offrequency offset in the presence of unknown multipathrdquo inProceedings of the IEEE International Conference on PersonalWireless Communications pp 104ndash108 December 2000
[4] D C Rife and G A Vincent ldquoUse of the discrete Fouriertransform in the measurement of frequencies and levels oftonesrdquoThe Bell System Technical Journal vol 49 no 2 pp 197ndash228 1970
[5] J-R Liao andC-M Chen ldquoPhase correction of discrete Fouriertransform coefficients to reduce frequency estimation bias ofsingle tone complex sinusoidrdquo Signal Processing vol 94 no 1pp 108ndash117 2014
[6] T Abatzoglou ldquoA fast maximum likelihood algorithm forfrequency estimation of a sinusoid based on Newtonrsquos methodrdquo
10 Mathematical Problems in Engineering
IEEE Transactions on Acoustics Speech and Signal Processingvol 33 no 1 pp 77ndash89 1985
[7] Y Liu Z Nie Z Zhao and Q H Liu ldquoGeneralization ofiterative Fourier interpolation algorithms for single frequencyestimationrdquo Digital Signal Processing vol 21 no 1 pp 141ndash1492011
[8] G Wei C Yang and F-J Chen ldquoClosed-form frequencyestimator based on narrow-band approximation under noisyenvironmentrdquo Signal Processing vol 91 no 4 pp 841ndash851 2011
[9] C Yang and G Wei ldquoA noniterative frequency estimator withrational combination of three spectrum linesrdquo IEEE Transac-tions on Signal Processing vol 59 no 10 pp 5065ndash5070 2011
[10] C Candan and S Koc ldquoBeamspace approach for detection ofthe number of coherent sourcesrdquo in Proceedings of the IEEERadar Conference Ubiquitous Radar (RADAR rsquo12) pp 0913ndash0918 May 2012
[11] Y Cao G Wei and F-J Chen ldquoA closed-form expandedautocorrelationmethod for frequency estimation of a sinusoidrdquoSignal Processing vol 92 no 4 pp 885ndash892 2012
[12] F Qian S Leung Y Zhu W Wong D Pao and W LauldquoDamped sinusoidal signals parameter estimation in frequencydomainrdquo Signal Processing vol 92 no 2 pp 381ndash391 2012
[13] G Richard ELINT The Interception and Analysis of RadarSignals John Wiley amp Sons Artech House Dedham MassUSA 2nd edition 2006
[14] W Su J A Kosinski and M Yu ldquoDual-use of modulationrecognition techniques for digital communication signalsrdquo inProceedings of the IEEE Long Island Systems Applications andTechnology Conference (LISAT rsquo06) pp 1ndash6 Long Island NYUSA May 2006
[15] L Pucker ldquoReview of contemporaryspectrum sensingtechnologiesrdquo IEEE-SA P19006 Standards Group 2009 httpgrouperieeeorggroupsdyspan6documentswhite papersP19006 Sensor Surveypdf
[16] A Fehske J Gaeddert and JH Reed ldquoAnew approach to signalclassification using spectral correlation and neural networksrdquoin Proceedings of the 1st IEEE International Symposium on NewFrontiers in Dynamic Spectrum Access Networks (DySPAN rsquo05)pp 144ndash150 Baltimore Md USA November 2005
[17] W S Lin andK J R Liu ldquoModulation forensics for wireless dig-ital communicationsrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo08) pp 1789ndash1792 April 2008
[18] G-B Hu L-Z Xu and M Jin ldquoReliability testing for blindprocessing results of LFM signals based on NP criterionrdquo ActaElectronica Sinica vol 41 no 4 pp 739ndash743 2013
[19] A D Whalen Detection of Signals in Noise Academic PressNew York NY USA 2nd edition 1995
[20] M Sankaran ldquoOn the non-central chi-square distributionrdquoBiometrika vol 46 no 1-2 pp 235ndash237 1959
[21] E Aboutanios and B Mulgrew ldquoIterative frequency estimationby interpolation on Fourier coefficientsrdquo IEEE Transactions onSignal Processing vol 53 no 4 pp 1237ndash1242 2005
[22] AM Zoubir andD R Iskander Bootstrap Techniques for SignalProcessing Cambridge University Press Cambridge UK 2004
[23] A M Zoubir and D R Iskandler ldquoBootstrap methods andapplicationsrdquo IEEE Signal Processing Magazine vol 24 no 4pp 10ndash19 2007
Figure 3 Effect of the false alarm probability on the detectionprobability obtained by the credibility test of a frequency estimate
Thus the detection probability can be expressed by
119875119863= Pr 119880120582119905
1198671le 120574 | 119867
1
= int
120574
minusinfin
1
radic2120587120590120582119905
1198671
exp[
[
minus
(119909 minus 120583120582119905
1198671)
2
2 (120590120582119905
1198671)
2
]
]
119889119909
asymp Φ(
120574 minus 120583120582119905
1198671
120590120582119905
1198671
)
(37)
From (34) and (37) we observe that the detection perfor-mance depends largely on the parameter 120582 which is thefunction of the number of samples received the SNR and thefrequency estimation error factor
6 Simulation Results
Monte Carlo simulations were carried out to study the behav-ior of the proposed algorithm in different environments Thesimulations were aimed at detecting whether the absoluteerror of a certain estimated frequency is greater than a quarterof the discrete sampling frequency Ten thousand MonteCarlo trials were performed for each condition Monte Carlotrials were run in each of the following conditions (1) thesinusoid signal contaminated by additive white Gaussiannoises is received (2) SNR = 10 log
1011986021205902 (3) the sampling
frequency is set to 100MHz except in 66
61 Influence of False Alarm Figure 3 shows the detectionbehavior for the credibility test by using the CI method withrespect to the false alarm and the comparisonswith the valuestheoretically calculated using (37) The carrier frequency ofthe sinusoid is 19801MHz while the sample size is 1024 The
Figure 4 Effect of the frequency estimation error factor on thedetection probability obtained by the credibility test of a frequencyestimate
initial phase is 1205876 and the frequency estimation error factoris 05 The SNR is varied from minus15 dB to 6 dB in steps of 3 dBThe values of probability of false alarm are fixed as 01 001and 0001 respectively
In the three cases the probability of detection (119875119863) is close
to 1 at an SNR greater than 0 dB The detection probability isincreased by a higher value of probability of false alarmWith120572 = 01 the probability of detection is close to 1 for an SNRof minus6 dB Moreover the derived theoretical 119875
119863is close to the
simulated value
62 Influence of the Frequency Estimation Error Figure 4shows the probability of detection with respect to the carrierfrequency estimation error and the comparisonswith the the-oretically calculated values The carrier frequency estimationerror is respectively equal to 03 04 and 05 for a sinusoidwith 1024 samples and the probability of false alarm equals001 The rest of the parameter settings of the signal in thesimulation remain the same as that stated in Section 61 Thedetection probability is increased by increasing the carrierfrequency estimation error factor For 120575 = 05 and an SNRclose to minus9 dB the probability of detection is approximatelyequal to 1
In fact a larger frequency estimation error factorimproves the mean of the proposed statistic 119880 and the dis-crimination between nonnull and null is easier In additionit can be observed from Figure 3 that the derived theoretical119875119863is very close to the simulated values in the three cases
63 Influence of the Number of Received Samples N Figure 5shows the detection performance of the credibility test versusthe received samples of the signal and the comparisons withthe theoretical calculated values The number of samples isequal to 512 1024 and 2048 respectively The carrier fre-quency estimation error is set to 04The rest of the parameter
00
01
02
03
04
05
06
07
08
09
10
SNR (dB)0 3 6
PD
minus15 minus12 minus9 minus6 minus3
N = 512 simulatedN = 512 theoreticalN = 1024 simulated
N = 1024 theoreticalN = 2048 simulatedN = 2048 theoretical
Figure 5 Effect of the number of received samples on the detectionprobability obtained by the credibility test of a frequency estimate
settings are the same as that stated in Section 62 In thethree cases with different sample numbers the detectionprobability is close to 1 for an SNR of 0 dBThe credibility testperformance is enhanced by the greater number of receivedsamples at the same SNR When the number of samplesis 2048 the detection probability is close to 1 for an SNRof minus6 dB We can observe from Figure 5 that the derivedtheoretical119875
119863is very close to the simulated values in the three
cases
64 Influence of Frequency Discretization We evaluated theperformance of our method in the case of different carrierfrequencies Figure 6 shows the performance of detectionversus the carrier frequency selected in the simulations Wedefined the discretization factor as 119889 = (119891
0minus 1198960Δ119865) where
1198960
= 199 and five frequencies were generated by setting119889 = 0 01 03 05 07 Clearly if 119889 = 0 the frequencyequals 1953125MHz which is just located on the discretefrequency bin When 119889 = 0 it indicates that the selectedfrequencies are not located on the discrete frequency binThe parameters are 119873 = 1024 120575 = 04 120579 = 1205876 and120572 = 001 Figure 6 shows that the detection probability doesnot depend on the frequencies we selected by using differentdiscretization factors and the same levels of performance areachieved in the five cases and are approximately identical tothe theoretically calculated performance in all the cases
65 Influence of the Initial Phase We investigated the impactof the initial phase on the performance of our methodFigure 7 shows the performance of detection versus the initialfrequency used in the simulations We set three initial phasesas 12058712 1205876 and 1205873 and the sample size as 1024 The restof the parameters set in each simulation are the same asthat mentioned in Section 63 Figure 7 shows that the initialphase of the sinusoid does not affect the detection probability
Figure 7 Effect of the initial phase of the signal on the detectionprobability obtained by the credibility test of a frequency estimate
and the same levels of performance are achieved in the twocases and are identical to the theoretical performances as thestatistic defined in (13) is not relevant to the phase because ofthe use of the modulus operation
66 Influence of the Sampling Frequency We now clarifythe impact of the sampling frequency on the performanceof the proposed method Figure 8 shows the performanceof detection versus the sampling frequency selected in thesimulations The other parameters are 119891
0= 19081MHz
119873 = 1024 120575 = 04 120579 = 1205876 and 120572 = 001 From the figurewe observe that the performance of detection is not affected
0 3 600
01
02
03
04
05
06
07
08
09
10
SNR (dB)
PD
minus15 minus12 minus9 minus6 minus3
Theoreticalfs = 120MHzfs = 100MHz
fs = 80MHz
Figure 8 Effect of the sampling frequency of the signal on thedetection probability obtained by the credibility test of a frequencyestimate
0 3 6
00
01
02
03
04
05
06
07
08
09
10
SNR (dB)minus12 minus9 minus6 minus3
PD (OR rule)PD (AND rule)
PD (CI only)PD (AM only)
PD
Figure 9 Probability of detection and probability of false alarmcomparison between the proposed scheme and others method
by the sampling frequency and is identical to the theoreticalperformance
67 Comparison to Other Methods Comparative experi-ments were also carried out for the proposed statisticAboutanios-Mulgrew (AM) method-based [21] detector anda combination of them by OR and AND rules respectivelyWe have described the AM-based detector that can alsobe used to test the credibility of the frequency estimatein the appendix The parameters are 119891
0= 19081MHz
119873 = 1024 120575 = 04 120579 = 1205876 and 120572 = 001 Thecurves of probabilities of detection for different SNRs areshown in Figure 9 We can observe that under an SNR
8 Mathematical Problems in Engineering
less than minus3 dB the probabilities of detection of the AM-basedmethod are greater than those of the CI-basedmethodFor the fusion method the probabilities of detection usingthe ldquoANDrdquo rule are slightly greater than those of the AM-based method and vice versa for the ldquoORrdquo rule that isthe probability of detection is the smallest when the SNR isless than minus3 dB When the SNRs are greater than minus3 dB theprobabilities of detection of the four detectors all approach1 Therefore under lower SNRs the more efficient fusionmethods combining the CI and AM methods need to befurther developed
It should be noted that calculating the threshold isdifficult because the exact probability distribution function ofthe AM-based statistic 119879AM defined in the appendix is hardto be expressed analytically Therefore here we introducethe bootstrap-based method [22 23] to obtain the thresh-old without depending on the analytical probability of thestatistic based on AM but the computational load increasesmany times because of repeated resampling operations in thebootstrap-based hypothesis test procedures
68 Case Study In this section we use the CI-based credibil-ity test statistic to evaluate three commonly used frequencyestimators the Rife algorithm [4] the maximum likelihoodestimator [6] by Newton iteration and the Aboutanious-Mulgrew estimator [21]The carrier frequency of the sinusoidis 19081MHz the initial phase is 1205876 and the sample size is1024 The range of SNR is from minus18 dB to minus6 dB The valueof probability of false alarm is set to 005 If we let 119863
119894(119894 =
0 1) be the event associated with the decision of choosing119867119894(119894 = 0 1) 119899
119894119895stands for the number of times that deciding
119863119895when hypothesis 119867
119894is correct Therefore Type I error
probability and Type II error probability can be calculatedusing 119899
01(11989900
+ 11989901) and 119899
10(11989910
+ 11989911) respectively The
error probability of the test can be expressed by 119875119864= (11989901+
11989910)10000 while 119875
119863= 11989911(11989911+ 11989910) indicates the ability
of the proposed method to detect the unreliable frequencyestimate which means that the maximum absolute bias of acertain frequency estimate (|Δ119891|) is less than a quarter of thediscrete sampling frequency interval (Δ119865)
Table 1 illustrates the performance of the credibility testfor the ML frequency estimator of the sinusoid by using theproposed CI-based statistic It should be remarked that thecredibility test performance is enhanced by increasing theSNR If the SNR is minus6 dB all the 10000 frequency estimatesbased on ML are reliable and the error probability of theproposed test approximates to 0 When the SNR decreases tominus12 dB 61 estimates are unreliable among 10000 simulationsBy using the proposed statistic all the 61 unreliable estimatescan be detected indicating a detection rate of about 100Among the 9939 reliable estimates there are 21 times whenestimates are mistakenly decided as unreliable in otherwords the error probability is about 021 When the SNR isdecreased to minus15 dB the error probability is about 081 andthe detection rate is greater than 95
Table 2 depicts the performance of the credibility test forthe Rife-based sinusoid frequency estimator The credibilitytest performance is enhanced by a larger SNR For an SNR ofminus6 dB all the 10000 times the frequency estimates obtained
Table 1 Credibility test performance of ML frequency estimator
by using the Rife algorithm are reliable (1198670) and the error
probability is approximately equal to 0 When the SNRdeclines to minus12 dB there are 71 unreliable estimates among10000 simulations By using the CI-based credibility teststatistic these 70 unreliable estimates can be detected andthe detection rate is about 986 On the other hand amongthe 9929 reliable estimates there are 25 times when theestimates are mistakenly decided as unreliable that is theerror probability is about 026 With the SNR decreased tominus15 dB the error probability is about 085 and the detectionrate is greater than 90
Table 3 provides the performance of the credibility test forthe AM-based sinusoid frequency estimator The credibilitytest performance is enhanced by a larger SNR Similar to thatdata in Tables 1 and 2 the test performance is improved bya larger SNR With the SNR decreased to minus15 dB the errorprobability is about 069 and the detection rate is greaterthan 80
Mathematical Problems in Engineering 9
Overall from Tables 2 and 3 we can observe that theproposed credibility test statistic can be used for effectivelyidentifying unreliable frequency estimates by using ML AMand Rife algorithms with a high detection probability and alow error probability when the SNR is greater than minus15 dBCompared to the estimator performances summarized inTables 1 and 2 the estimator performance summarized inTable 3 is the lowest because AM-based estimation has abetter performance than that of bothML and Rife algorithmsand the frequency estimation factors are small
In practice for the detected unreliable estimate we canabandon the results or estimate the frequency of the signalagain thereby enhancing the reliability of the entire signalprocessing unit For example if the FFT-based coarse fre-quency estimator is used in practice andwe set the parametersas 119891119904= 100MHz 119891
0= 1955MHz 119873 = 1000 and the
number of DFT119872 = 1000 there are two cases
(1) Case 1 When the SNRs (eg SNR = minus23 dB) arebelow the SNR threshold the maximum spectrumline cannot be detected correctly In this case theestimate error is larger than 025Δ119865 and the estimatewill be judged as incredible by the proposed CI-basedstatistic If we perform resampling over the frequencygrid by setting 119872 = 2000 the maximum spectrumline still cannot be detected As the absolute estima-tion errors are the same under the two scenariosthe resampling action cannot enhance the credibilityof the estimate and this kind of incredible estimateshould be abandoned
(2) Case 2When an SNR is greater than the SNR thresh-old an incredible frequency estimate is detectedbecause of the displacement of the maximum spec-trum line by the discretization In this case ifwe enlarge 119872 to 2000 by resampling in the fre-quency domain 119891
0can be just moved to the dis-
crete frequency bin Hence the estimation error canbe decreased and the credibility can be enhancedHowever in practice the true frequency cannot beknown by the user hence it is difficult to adjustthe resampling parameter to match the estimationexactly In this case we need to select the moreaccurate estimating method or abandon the resultobtained by original estimation
7 Conclusion
The paper presented a CI-based credibility test algorithm forfrequency estimation of a sinusoid A credibility assessmenttesting model is defined to analyze the mathematical charac-teristics of the correlations between the observed signal andthe reference signal under different hypotheses The test isperformed with the proposed threshold that is based on CIExperimental results revealed that a good performance wasachieved even at low SNRs Credibility tests were performedwith the ML frequency estimator using the proposed thresh-old based on CI as well with the Rife frequency estimator andthe Aboutanious-Mulgrew estimator Experimental results
revealed that a good performance was achieved by theproposed method even at low SNRs
Appendix
Thefrequency estimation credibility testmethod based on theAM estimator includes
(1) estimation of the frequency of the signal by using acertain estimation method
(2) construction of the reference signal 119910(119899) with (3) byusing the estimated frequency
1198910
(3) computation of the correlation between the receivedsignal and the reference signal and estimation ofthe frequency displacement Δ
119891 by the Aboutanios-Mulgrew frequency estimator [21]
(4) evaluation of the credibility of frequency estimationby comparing the statistic 119879AM = ||Δ
119891| minus 025Δ119865| to
the threshold th that can be calculated by a given falsealarm 120572 The hypothesis 119867
0is chosen if 119879AM lt th
Otherwise1198671is accepted
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This study is financially supported by the Natural ScienceFoundation of Jiangsu Province (Project no BK2011837) theNational Natural Science Foundation of China 2012 (Projectno 61201208) and the 333 Research Project of JiangsuProvince (Project no BRA2013171)
References
[1] D C Rife and R R Boorstyn ldquoSingle-tone parameter esti-mation from discrete-time observationrdquo IEEE Transactions onInformation Theory vol 20 no 5 pp 591ndash598 1974
[2] S Peleg and B Porat ldquoLinear FM signal parameter estima-tion from discrete-time observationsrdquo IEEE Transactions onAerospace and Electronic Systems vol 27 no 4 pp 607ndash6161991
[3] M Ghogho A Swami and T Durrani ldquoBlind estimation offrequency offset in the presence of unknown multipathrdquo inProceedings of the IEEE International Conference on PersonalWireless Communications pp 104ndash108 December 2000
[4] D C Rife and G A Vincent ldquoUse of the discrete Fouriertransform in the measurement of frequencies and levels oftonesrdquoThe Bell System Technical Journal vol 49 no 2 pp 197ndash228 1970
[5] J-R Liao andC-M Chen ldquoPhase correction of discrete Fouriertransform coefficients to reduce frequency estimation bias ofsingle tone complex sinusoidrdquo Signal Processing vol 94 no 1pp 108ndash117 2014
[6] T Abatzoglou ldquoA fast maximum likelihood algorithm forfrequency estimation of a sinusoid based on Newtonrsquos methodrdquo
10 Mathematical Problems in Engineering
IEEE Transactions on Acoustics Speech and Signal Processingvol 33 no 1 pp 77ndash89 1985
[7] Y Liu Z Nie Z Zhao and Q H Liu ldquoGeneralization ofiterative Fourier interpolation algorithms for single frequencyestimationrdquo Digital Signal Processing vol 21 no 1 pp 141ndash1492011
[8] G Wei C Yang and F-J Chen ldquoClosed-form frequencyestimator based on narrow-band approximation under noisyenvironmentrdquo Signal Processing vol 91 no 4 pp 841ndash851 2011
[9] C Yang and G Wei ldquoA noniterative frequency estimator withrational combination of three spectrum linesrdquo IEEE Transac-tions on Signal Processing vol 59 no 10 pp 5065ndash5070 2011
[10] C Candan and S Koc ldquoBeamspace approach for detection ofthe number of coherent sourcesrdquo in Proceedings of the IEEERadar Conference Ubiquitous Radar (RADAR rsquo12) pp 0913ndash0918 May 2012
[11] Y Cao G Wei and F-J Chen ldquoA closed-form expandedautocorrelationmethod for frequency estimation of a sinusoidrdquoSignal Processing vol 92 no 4 pp 885ndash892 2012
[12] F Qian S Leung Y Zhu W Wong D Pao and W LauldquoDamped sinusoidal signals parameter estimation in frequencydomainrdquo Signal Processing vol 92 no 2 pp 381ndash391 2012
[13] G Richard ELINT The Interception and Analysis of RadarSignals John Wiley amp Sons Artech House Dedham MassUSA 2nd edition 2006
[14] W Su J A Kosinski and M Yu ldquoDual-use of modulationrecognition techniques for digital communication signalsrdquo inProceedings of the IEEE Long Island Systems Applications andTechnology Conference (LISAT rsquo06) pp 1ndash6 Long Island NYUSA May 2006
[15] L Pucker ldquoReview of contemporaryspectrum sensingtechnologiesrdquo IEEE-SA P19006 Standards Group 2009 httpgrouperieeeorggroupsdyspan6documentswhite papersP19006 Sensor Surveypdf
[16] A Fehske J Gaeddert and JH Reed ldquoAnew approach to signalclassification using spectral correlation and neural networksrdquoin Proceedings of the 1st IEEE International Symposium on NewFrontiers in Dynamic Spectrum Access Networks (DySPAN rsquo05)pp 144ndash150 Baltimore Md USA November 2005
[17] W S Lin andK J R Liu ldquoModulation forensics for wireless dig-ital communicationsrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo08) pp 1789ndash1792 April 2008
[18] G-B Hu L-Z Xu and M Jin ldquoReliability testing for blindprocessing results of LFM signals based on NP criterionrdquo ActaElectronica Sinica vol 41 no 4 pp 739ndash743 2013
[19] A D Whalen Detection of Signals in Noise Academic PressNew York NY USA 2nd edition 1995
[20] M Sankaran ldquoOn the non-central chi-square distributionrdquoBiometrika vol 46 no 1-2 pp 235ndash237 1959
[21] E Aboutanios and B Mulgrew ldquoIterative frequency estimationby interpolation on Fourier coefficientsrdquo IEEE Transactions onSignal Processing vol 53 no 4 pp 1237ndash1242 2005
[22] AM Zoubir andD R Iskander Bootstrap Techniques for SignalProcessing Cambridge University Press Cambridge UK 2004
[23] A M Zoubir and D R Iskandler ldquoBootstrap methods andapplicationsrdquo IEEE Signal Processing Magazine vol 24 no 4pp 10ndash19 2007
Figure 4 Effect of the frequency estimation error factor on thedetection probability obtained by the credibility test of a frequencyestimate
initial phase is 1205876 and the frequency estimation error factoris 05 The SNR is varied from minus15 dB to 6 dB in steps of 3 dBThe values of probability of false alarm are fixed as 01 001and 0001 respectively
In the three cases the probability of detection (119875119863) is close
to 1 at an SNR greater than 0 dB The detection probability isincreased by a higher value of probability of false alarmWith120572 = 01 the probability of detection is close to 1 for an SNRof minus6 dB Moreover the derived theoretical 119875
119863is close to the
simulated value
62 Influence of the Frequency Estimation Error Figure 4shows the probability of detection with respect to the carrierfrequency estimation error and the comparisonswith the the-oretically calculated values The carrier frequency estimationerror is respectively equal to 03 04 and 05 for a sinusoidwith 1024 samples and the probability of false alarm equals001 The rest of the parameter settings of the signal in thesimulation remain the same as that stated in Section 61 Thedetection probability is increased by increasing the carrierfrequency estimation error factor For 120575 = 05 and an SNRclose to minus9 dB the probability of detection is approximatelyequal to 1
In fact a larger frequency estimation error factorimproves the mean of the proposed statistic 119880 and the dis-crimination between nonnull and null is easier In additionit can be observed from Figure 3 that the derived theoretical119875119863is very close to the simulated values in the three cases
63 Influence of the Number of Received Samples N Figure 5shows the detection performance of the credibility test versusthe received samples of the signal and the comparisons withthe theoretical calculated values The number of samples isequal to 512 1024 and 2048 respectively The carrier fre-quency estimation error is set to 04The rest of the parameter
00
01
02
03
04
05
06
07
08
09
10
SNR (dB)0 3 6
PD
minus15 minus12 minus9 minus6 minus3
N = 512 simulatedN = 512 theoreticalN = 1024 simulated
N = 1024 theoreticalN = 2048 simulatedN = 2048 theoretical
Figure 5 Effect of the number of received samples on the detectionprobability obtained by the credibility test of a frequency estimate
settings are the same as that stated in Section 62 In thethree cases with different sample numbers the detectionprobability is close to 1 for an SNR of 0 dBThe credibility testperformance is enhanced by the greater number of receivedsamples at the same SNR When the number of samplesis 2048 the detection probability is close to 1 for an SNRof minus6 dB We can observe from Figure 5 that the derivedtheoretical119875
119863is very close to the simulated values in the three
cases
64 Influence of Frequency Discretization We evaluated theperformance of our method in the case of different carrierfrequencies Figure 6 shows the performance of detectionversus the carrier frequency selected in the simulations Wedefined the discretization factor as 119889 = (119891
0minus 1198960Δ119865) where
1198960
= 199 and five frequencies were generated by setting119889 = 0 01 03 05 07 Clearly if 119889 = 0 the frequencyequals 1953125MHz which is just located on the discretefrequency bin When 119889 = 0 it indicates that the selectedfrequencies are not located on the discrete frequency binThe parameters are 119873 = 1024 120575 = 04 120579 = 1205876 and120572 = 001 Figure 6 shows that the detection probability doesnot depend on the frequencies we selected by using differentdiscretization factors and the same levels of performance areachieved in the five cases and are approximately identical tothe theoretically calculated performance in all the cases
65 Influence of the Initial Phase We investigated the impactof the initial phase on the performance of our methodFigure 7 shows the performance of detection versus the initialfrequency used in the simulations We set three initial phasesas 12058712 1205876 and 1205873 and the sample size as 1024 The restof the parameters set in each simulation are the same asthat mentioned in Section 63 Figure 7 shows that the initialphase of the sinusoid does not affect the detection probability
Figure 7 Effect of the initial phase of the signal on the detectionprobability obtained by the credibility test of a frequency estimate
and the same levels of performance are achieved in the twocases and are identical to the theoretical performances as thestatistic defined in (13) is not relevant to the phase because ofthe use of the modulus operation
66 Influence of the Sampling Frequency We now clarifythe impact of the sampling frequency on the performanceof the proposed method Figure 8 shows the performanceof detection versus the sampling frequency selected in thesimulations The other parameters are 119891
0= 19081MHz
119873 = 1024 120575 = 04 120579 = 1205876 and 120572 = 001 From the figurewe observe that the performance of detection is not affected
0 3 600
01
02
03
04
05
06
07
08
09
10
SNR (dB)
PD
minus15 minus12 minus9 minus6 minus3
Theoreticalfs = 120MHzfs = 100MHz
fs = 80MHz
Figure 8 Effect of the sampling frequency of the signal on thedetection probability obtained by the credibility test of a frequencyestimate
0 3 6
00
01
02
03
04
05
06
07
08
09
10
SNR (dB)minus12 minus9 minus6 minus3
PD (OR rule)PD (AND rule)
PD (CI only)PD (AM only)
PD
Figure 9 Probability of detection and probability of false alarmcomparison between the proposed scheme and others method
by the sampling frequency and is identical to the theoreticalperformance
67 Comparison to Other Methods Comparative experi-ments were also carried out for the proposed statisticAboutanios-Mulgrew (AM) method-based [21] detector anda combination of them by OR and AND rules respectivelyWe have described the AM-based detector that can alsobe used to test the credibility of the frequency estimatein the appendix The parameters are 119891
0= 19081MHz
119873 = 1024 120575 = 04 120579 = 1205876 and 120572 = 001 Thecurves of probabilities of detection for different SNRs areshown in Figure 9 We can observe that under an SNR
8 Mathematical Problems in Engineering
less than minus3 dB the probabilities of detection of the AM-basedmethod are greater than those of the CI-basedmethodFor the fusion method the probabilities of detection usingthe ldquoANDrdquo rule are slightly greater than those of the AM-based method and vice versa for the ldquoORrdquo rule that isthe probability of detection is the smallest when the SNR isless than minus3 dB When the SNRs are greater than minus3 dB theprobabilities of detection of the four detectors all approach1 Therefore under lower SNRs the more efficient fusionmethods combining the CI and AM methods need to befurther developed
It should be noted that calculating the threshold isdifficult because the exact probability distribution function ofthe AM-based statistic 119879AM defined in the appendix is hardto be expressed analytically Therefore here we introducethe bootstrap-based method [22 23] to obtain the thresh-old without depending on the analytical probability of thestatistic based on AM but the computational load increasesmany times because of repeated resampling operations in thebootstrap-based hypothesis test procedures
68 Case Study In this section we use the CI-based credibil-ity test statistic to evaluate three commonly used frequencyestimators the Rife algorithm [4] the maximum likelihoodestimator [6] by Newton iteration and the Aboutanious-Mulgrew estimator [21]The carrier frequency of the sinusoidis 19081MHz the initial phase is 1205876 and the sample size is1024 The range of SNR is from minus18 dB to minus6 dB The valueof probability of false alarm is set to 005 If we let 119863
119894(119894 =
0 1) be the event associated with the decision of choosing119867119894(119894 = 0 1) 119899
119894119895stands for the number of times that deciding
119863119895when hypothesis 119867
119894is correct Therefore Type I error
probability and Type II error probability can be calculatedusing 119899
01(11989900
+ 11989901) and 119899
10(11989910
+ 11989911) respectively The
error probability of the test can be expressed by 119875119864= (11989901+
11989910)10000 while 119875
119863= 11989911(11989911+ 11989910) indicates the ability
of the proposed method to detect the unreliable frequencyestimate which means that the maximum absolute bias of acertain frequency estimate (|Δ119891|) is less than a quarter of thediscrete sampling frequency interval (Δ119865)
Table 1 illustrates the performance of the credibility testfor the ML frequency estimator of the sinusoid by using theproposed CI-based statistic It should be remarked that thecredibility test performance is enhanced by increasing theSNR If the SNR is minus6 dB all the 10000 frequency estimatesbased on ML are reliable and the error probability of theproposed test approximates to 0 When the SNR decreases tominus12 dB 61 estimates are unreliable among 10000 simulationsBy using the proposed statistic all the 61 unreliable estimatescan be detected indicating a detection rate of about 100Among the 9939 reliable estimates there are 21 times whenestimates are mistakenly decided as unreliable in otherwords the error probability is about 021 When the SNR isdecreased to minus15 dB the error probability is about 081 andthe detection rate is greater than 95
Table 2 depicts the performance of the credibility test forthe Rife-based sinusoid frequency estimator The credibilitytest performance is enhanced by a larger SNR For an SNR ofminus6 dB all the 10000 times the frequency estimates obtained
Table 1 Credibility test performance of ML frequency estimator
by using the Rife algorithm are reliable (1198670) and the error
probability is approximately equal to 0 When the SNRdeclines to minus12 dB there are 71 unreliable estimates among10000 simulations By using the CI-based credibility teststatistic these 70 unreliable estimates can be detected andthe detection rate is about 986 On the other hand amongthe 9929 reliable estimates there are 25 times when theestimates are mistakenly decided as unreliable that is theerror probability is about 026 With the SNR decreased tominus15 dB the error probability is about 085 and the detectionrate is greater than 90
Table 3 provides the performance of the credibility test forthe AM-based sinusoid frequency estimator The credibilitytest performance is enhanced by a larger SNR Similar to thatdata in Tables 1 and 2 the test performance is improved bya larger SNR With the SNR decreased to minus15 dB the errorprobability is about 069 and the detection rate is greaterthan 80
Mathematical Problems in Engineering 9
Overall from Tables 2 and 3 we can observe that theproposed credibility test statistic can be used for effectivelyidentifying unreliable frequency estimates by using ML AMand Rife algorithms with a high detection probability and alow error probability when the SNR is greater than minus15 dBCompared to the estimator performances summarized inTables 1 and 2 the estimator performance summarized inTable 3 is the lowest because AM-based estimation has abetter performance than that of bothML and Rife algorithmsand the frequency estimation factors are small
In practice for the detected unreliable estimate we canabandon the results or estimate the frequency of the signalagain thereby enhancing the reliability of the entire signalprocessing unit For example if the FFT-based coarse fre-quency estimator is used in practice andwe set the parametersas 119891119904= 100MHz 119891
0= 1955MHz 119873 = 1000 and the
number of DFT119872 = 1000 there are two cases
(1) Case 1 When the SNRs (eg SNR = minus23 dB) arebelow the SNR threshold the maximum spectrumline cannot be detected correctly In this case theestimate error is larger than 025Δ119865 and the estimatewill be judged as incredible by the proposed CI-basedstatistic If we perform resampling over the frequencygrid by setting 119872 = 2000 the maximum spectrumline still cannot be detected As the absolute estima-tion errors are the same under the two scenariosthe resampling action cannot enhance the credibilityof the estimate and this kind of incredible estimateshould be abandoned
(2) Case 2When an SNR is greater than the SNR thresh-old an incredible frequency estimate is detectedbecause of the displacement of the maximum spec-trum line by the discretization In this case ifwe enlarge 119872 to 2000 by resampling in the fre-quency domain 119891
0can be just moved to the dis-
crete frequency bin Hence the estimation error canbe decreased and the credibility can be enhancedHowever in practice the true frequency cannot beknown by the user hence it is difficult to adjustthe resampling parameter to match the estimationexactly In this case we need to select the moreaccurate estimating method or abandon the resultobtained by original estimation
7 Conclusion
The paper presented a CI-based credibility test algorithm forfrequency estimation of a sinusoid A credibility assessmenttesting model is defined to analyze the mathematical charac-teristics of the correlations between the observed signal andthe reference signal under different hypotheses The test isperformed with the proposed threshold that is based on CIExperimental results revealed that a good performance wasachieved even at low SNRs Credibility tests were performedwith the ML frequency estimator using the proposed thresh-old based on CI as well with the Rife frequency estimator andthe Aboutanious-Mulgrew estimator Experimental results
revealed that a good performance was achieved by theproposed method even at low SNRs
Appendix
Thefrequency estimation credibility testmethod based on theAM estimator includes
(1) estimation of the frequency of the signal by using acertain estimation method
(2) construction of the reference signal 119910(119899) with (3) byusing the estimated frequency
1198910
(3) computation of the correlation between the receivedsignal and the reference signal and estimation ofthe frequency displacement Δ
119891 by the Aboutanios-Mulgrew frequency estimator [21]
(4) evaluation of the credibility of frequency estimationby comparing the statistic 119879AM = ||Δ
119891| minus 025Δ119865| to
the threshold th that can be calculated by a given falsealarm 120572 The hypothesis 119867
0is chosen if 119879AM lt th
Otherwise1198671is accepted
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This study is financially supported by the Natural ScienceFoundation of Jiangsu Province (Project no BK2011837) theNational Natural Science Foundation of China 2012 (Projectno 61201208) and the 333 Research Project of JiangsuProvince (Project no BRA2013171)
References
[1] D C Rife and R R Boorstyn ldquoSingle-tone parameter esti-mation from discrete-time observationrdquo IEEE Transactions onInformation Theory vol 20 no 5 pp 591ndash598 1974
[2] S Peleg and B Porat ldquoLinear FM signal parameter estima-tion from discrete-time observationsrdquo IEEE Transactions onAerospace and Electronic Systems vol 27 no 4 pp 607ndash6161991
[3] M Ghogho A Swami and T Durrani ldquoBlind estimation offrequency offset in the presence of unknown multipathrdquo inProceedings of the IEEE International Conference on PersonalWireless Communications pp 104ndash108 December 2000
[4] D C Rife and G A Vincent ldquoUse of the discrete Fouriertransform in the measurement of frequencies and levels oftonesrdquoThe Bell System Technical Journal vol 49 no 2 pp 197ndash228 1970
[5] J-R Liao andC-M Chen ldquoPhase correction of discrete Fouriertransform coefficients to reduce frequency estimation bias ofsingle tone complex sinusoidrdquo Signal Processing vol 94 no 1pp 108ndash117 2014
[6] T Abatzoglou ldquoA fast maximum likelihood algorithm forfrequency estimation of a sinusoid based on Newtonrsquos methodrdquo
10 Mathematical Problems in Engineering
IEEE Transactions on Acoustics Speech and Signal Processingvol 33 no 1 pp 77ndash89 1985
[7] Y Liu Z Nie Z Zhao and Q H Liu ldquoGeneralization ofiterative Fourier interpolation algorithms for single frequencyestimationrdquo Digital Signal Processing vol 21 no 1 pp 141ndash1492011
[8] G Wei C Yang and F-J Chen ldquoClosed-form frequencyestimator based on narrow-band approximation under noisyenvironmentrdquo Signal Processing vol 91 no 4 pp 841ndash851 2011
[9] C Yang and G Wei ldquoA noniterative frequency estimator withrational combination of three spectrum linesrdquo IEEE Transac-tions on Signal Processing vol 59 no 10 pp 5065ndash5070 2011
[10] C Candan and S Koc ldquoBeamspace approach for detection ofthe number of coherent sourcesrdquo in Proceedings of the IEEERadar Conference Ubiquitous Radar (RADAR rsquo12) pp 0913ndash0918 May 2012
[11] Y Cao G Wei and F-J Chen ldquoA closed-form expandedautocorrelationmethod for frequency estimation of a sinusoidrdquoSignal Processing vol 92 no 4 pp 885ndash892 2012
[12] F Qian S Leung Y Zhu W Wong D Pao and W LauldquoDamped sinusoidal signals parameter estimation in frequencydomainrdquo Signal Processing vol 92 no 2 pp 381ndash391 2012
[13] G Richard ELINT The Interception and Analysis of RadarSignals John Wiley amp Sons Artech House Dedham MassUSA 2nd edition 2006
[14] W Su J A Kosinski and M Yu ldquoDual-use of modulationrecognition techniques for digital communication signalsrdquo inProceedings of the IEEE Long Island Systems Applications andTechnology Conference (LISAT rsquo06) pp 1ndash6 Long Island NYUSA May 2006
[15] L Pucker ldquoReview of contemporaryspectrum sensingtechnologiesrdquo IEEE-SA P19006 Standards Group 2009 httpgrouperieeeorggroupsdyspan6documentswhite papersP19006 Sensor Surveypdf
[16] A Fehske J Gaeddert and JH Reed ldquoAnew approach to signalclassification using spectral correlation and neural networksrdquoin Proceedings of the 1st IEEE International Symposium on NewFrontiers in Dynamic Spectrum Access Networks (DySPAN rsquo05)pp 144ndash150 Baltimore Md USA November 2005
[17] W S Lin andK J R Liu ldquoModulation forensics for wireless dig-ital communicationsrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo08) pp 1789ndash1792 April 2008
[18] G-B Hu L-Z Xu and M Jin ldquoReliability testing for blindprocessing results of LFM signals based on NP criterionrdquo ActaElectronica Sinica vol 41 no 4 pp 739ndash743 2013
[19] A D Whalen Detection of Signals in Noise Academic PressNew York NY USA 2nd edition 1995
[20] M Sankaran ldquoOn the non-central chi-square distributionrdquoBiometrika vol 46 no 1-2 pp 235ndash237 1959
[21] E Aboutanios and B Mulgrew ldquoIterative frequency estimationby interpolation on Fourier coefficientsrdquo IEEE Transactions onSignal Processing vol 53 no 4 pp 1237ndash1242 2005
[22] AM Zoubir andD R Iskander Bootstrap Techniques for SignalProcessing Cambridge University Press Cambridge UK 2004
[23] A M Zoubir and D R Iskandler ldquoBootstrap methods andapplicationsrdquo IEEE Signal Processing Magazine vol 24 no 4pp 10ndash19 2007
Figure 7 Effect of the initial phase of the signal on the detectionprobability obtained by the credibility test of a frequency estimate
and the same levels of performance are achieved in the twocases and are identical to the theoretical performances as thestatistic defined in (13) is not relevant to the phase because ofthe use of the modulus operation
66 Influence of the Sampling Frequency We now clarifythe impact of the sampling frequency on the performanceof the proposed method Figure 8 shows the performanceof detection versus the sampling frequency selected in thesimulations The other parameters are 119891
0= 19081MHz
119873 = 1024 120575 = 04 120579 = 1205876 and 120572 = 001 From the figurewe observe that the performance of detection is not affected
0 3 600
01
02
03
04
05
06
07
08
09
10
SNR (dB)
PD
minus15 minus12 minus9 minus6 minus3
Theoreticalfs = 120MHzfs = 100MHz
fs = 80MHz
Figure 8 Effect of the sampling frequency of the signal on thedetection probability obtained by the credibility test of a frequencyestimate
0 3 6
00
01
02
03
04
05
06
07
08
09
10
SNR (dB)minus12 minus9 minus6 minus3
PD (OR rule)PD (AND rule)
PD (CI only)PD (AM only)
PD
Figure 9 Probability of detection and probability of false alarmcomparison between the proposed scheme and others method
by the sampling frequency and is identical to the theoreticalperformance
67 Comparison to Other Methods Comparative experi-ments were also carried out for the proposed statisticAboutanios-Mulgrew (AM) method-based [21] detector anda combination of them by OR and AND rules respectivelyWe have described the AM-based detector that can alsobe used to test the credibility of the frequency estimatein the appendix The parameters are 119891
0= 19081MHz
119873 = 1024 120575 = 04 120579 = 1205876 and 120572 = 001 Thecurves of probabilities of detection for different SNRs areshown in Figure 9 We can observe that under an SNR
8 Mathematical Problems in Engineering
less than minus3 dB the probabilities of detection of the AM-basedmethod are greater than those of the CI-basedmethodFor the fusion method the probabilities of detection usingthe ldquoANDrdquo rule are slightly greater than those of the AM-based method and vice versa for the ldquoORrdquo rule that isthe probability of detection is the smallest when the SNR isless than minus3 dB When the SNRs are greater than minus3 dB theprobabilities of detection of the four detectors all approach1 Therefore under lower SNRs the more efficient fusionmethods combining the CI and AM methods need to befurther developed
It should be noted that calculating the threshold isdifficult because the exact probability distribution function ofthe AM-based statistic 119879AM defined in the appendix is hardto be expressed analytically Therefore here we introducethe bootstrap-based method [22 23] to obtain the thresh-old without depending on the analytical probability of thestatistic based on AM but the computational load increasesmany times because of repeated resampling operations in thebootstrap-based hypothesis test procedures
68 Case Study In this section we use the CI-based credibil-ity test statistic to evaluate three commonly used frequencyestimators the Rife algorithm [4] the maximum likelihoodestimator [6] by Newton iteration and the Aboutanious-Mulgrew estimator [21]The carrier frequency of the sinusoidis 19081MHz the initial phase is 1205876 and the sample size is1024 The range of SNR is from minus18 dB to minus6 dB The valueof probability of false alarm is set to 005 If we let 119863
119894(119894 =
0 1) be the event associated with the decision of choosing119867119894(119894 = 0 1) 119899
119894119895stands for the number of times that deciding
119863119895when hypothesis 119867
119894is correct Therefore Type I error
probability and Type II error probability can be calculatedusing 119899
01(11989900
+ 11989901) and 119899
10(11989910
+ 11989911) respectively The
error probability of the test can be expressed by 119875119864= (11989901+
11989910)10000 while 119875
119863= 11989911(11989911+ 11989910) indicates the ability
of the proposed method to detect the unreliable frequencyestimate which means that the maximum absolute bias of acertain frequency estimate (|Δ119891|) is less than a quarter of thediscrete sampling frequency interval (Δ119865)
Table 1 illustrates the performance of the credibility testfor the ML frequency estimator of the sinusoid by using theproposed CI-based statistic It should be remarked that thecredibility test performance is enhanced by increasing theSNR If the SNR is minus6 dB all the 10000 frequency estimatesbased on ML are reliable and the error probability of theproposed test approximates to 0 When the SNR decreases tominus12 dB 61 estimates are unreliable among 10000 simulationsBy using the proposed statistic all the 61 unreliable estimatescan be detected indicating a detection rate of about 100Among the 9939 reliable estimates there are 21 times whenestimates are mistakenly decided as unreliable in otherwords the error probability is about 021 When the SNR isdecreased to minus15 dB the error probability is about 081 andthe detection rate is greater than 95
Table 2 depicts the performance of the credibility test forthe Rife-based sinusoid frequency estimator The credibilitytest performance is enhanced by a larger SNR For an SNR ofminus6 dB all the 10000 times the frequency estimates obtained
Table 1 Credibility test performance of ML frequency estimator
by using the Rife algorithm are reliable (1198670) and the error
probability is approximately equal to 0 When the SNRdeclines to minus12 dB there are 71 unreliable estimates among10000 simulations By using the CI-based credibility teststatistic these 70 unreliable estimates can be detected andthe detection rate is about 986 On the other hand amongthe 9929 reliable estimates there are 25 times when theestimates are mistakenly decided as unreliable that is theerror probability is about 026 With the SNR decreased tominus15 dB the error probability is about 085 and the detectionrate is greater than 90
Table 3 provides the performance of the credibility test forthe AM-based sinusoid frequency estimator The credibilitytest performance is enhanced by a larger SNR Similar to thatdata in Tables 1 and 2 the test performance is improved bya larger SNR With the SNR decreased to minus15 dB the errorprobability is about 069 and the detection rate is greaterthan 80
Mathematical Problems in Engineering 9
Overall from Tables 2 and 3 we can observe that theproposed credibility test statistic can be used for effectivelyidentifying unreliable frequency estimates by using ML AMand Rife algorithms with a high detection probability and alow error probability when the SNR is greater than minus15 dBCompared to the estimator performances summarized inTables 1 and 2 the estimator performance summarized inTable 3 is the lowest because AM-based estimation has abetter performance than that of bothML and Rife algorithmsand the frequency estimation factors are small
In practice for the detected unreliable estimate we canabandon the results or estimate the frequency of the signalagain thereby enhancing the reliability of the entire signalprocessing unit For example if the FFT-based coarse fre-quency estimator is used in practice andwe set the parametersas 119891119904= 100MHz 119891
0= 1955MHz 119873 = 1000 and the
number of DFT119872 = 1000 there are two cases
(1) Case 1 When the SNRs (eg SNR = minus23 dB) arebelow the SNR threshold the maximum spectrumline cannot be detected correctly In this case theestimate error is larger than 025Δ119865 and the estimatewill be judged as incredible by the proposed CI-basedstatistic If we perform resampling over the frequencygrid by setting 119872 = 2000 the maximum spectrumline still cannot be detected As the absolute estima-tion errors are the same under the two scenariosthe resampling action cannot enhance the credibilityof the estimate and this kind of incredible estimateshould be abandoned
(2) Case 2When an SNR is greater than the SNR thresh-old an incredible frequency estimate is detectedbecause of the displacement of the maximum spec-trum line by the discretization In this case ifwe enlarge 119872 to 2000 by resampling in the fre-quency domain 119891
0can be just moved to the dis-
crete frequency bin Hence the estimation error canbe decreased and the credibility can be enhancedHowever in practice the true frequency cannot beknown by the user hence it is difficult to adjustthe resampling parameter to match the estimationexactly In this case we need to select the moreaccurate estimating method or abandon the resultobtained by original estimation
7 Conclusion
The paper presented a CI-based credibility test algorithm forfrequency estimation of a sinusoid A credibility assessmenttesting model is defined to analyze the mathematical charac-teristics of the correlations between the observed signal andthe reference signal under different hypotheses The test isperformed with the proposed threshold that is based on CIExperimental results revealed that a good performance wasachieved even at low SNRs Credibility tests were performedwith the ML frequency estimator using the proposed thresh-old based on CI as well with the Rife frequency estimator andthe Aboutanious-Mulgrew estimator Experimental results
revealed that a good performance was achieved by theproposed method even at low SNRs
Appendix
Thefrequency estimation credibility testmethod based on theAM estimator includes
(1) estimation of the frequency of the signal by using acertain estimation method
(2) construction of the reference signal 119910(119899) with (3) byusing the estimated frequency
1198910
(3) computation of the correlation between the receivedsignal and the reference signal and estimation ofthe frequency displacement Δ
119891 by the Aboutanios-Mulgrew frequency estimator [21]
(4) evaluation of the credibility of frequency estimationby comparing the statistic 119879AM = ||Δ
119891| minus 025Δ119865| to
the threshold th that can be calculated by a given falsealarm 120572 The hypothesis 119867
0is chosen if 119879AM lt th
Otherwise1198671is accepted
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This study is financially supported by the Natural ScienceFoundation of Jiangsu Province (Project no BK2011837) theNational Natural Science Foundation of China 2012 (Projectno 61201208) and the 333 Research Project of JiangsuProvince (Project no BRA2013171)
References
[1] D C Rife and R R Boorstyn ldquoSingle-tone parameter esti-mation from discrete-time observationrdquo IEEE Transactions onInformation Theory vol 20 no 5 pp 591ndash598 1974
[2] S Peleg and B Porat ldquoLinear FM signal parameter estima-tion from discrete-time observationsrdquo IEEE Transactions onAerospace and Electronic Systems vol 27 no 4 pp 607ndash6161991
[3] M Ghogho A Swami and T Durrani ldquoBlind estimation offrequency offset in the presence of unknown multipathrdquo inProceedings of the IEEE International Conference on PersonalWireless Communications pp 104ndash108 December 2000
[4] D C Rife and G A Vincent ldquoUse of the discrete Fouriertransform in the measurement of frequencies and levels oftonesrdquoThe Bell System Technical Journal vol 49 no 2 pp 197ndash228 1970
[5] J-R Liao andC-M Chen ldquoPhase correction of discrete Fouriertransform coefficients to reduce frequency estimation bias ofsingle tone complex sinusoidrdquo Signal Processing vol 94 no 1pp 108ndash117 2014
[6] T Abatzoglou ldquoA fast maximum likelihood algorithm forfrequency estimation of a sinusoid based on Newtonrsquos methodrdquo
10 Mathematical Problems in Engineering
IEEE Transactions on Acoustics Speech and Signal Processingvol 33 no 1 pp 77ndash89 1985
[7] Y Liu Z Nie Z Zhao and Q H Liu ldquoGeneralization ofiterative Fourier interpolation algorithms for single frequencyestimationrdquo Digital Signal Processing vol 21 no 1 pp 141ndash1492011
[8] G Wei C Yang and F-J Chen ldquoClosed-form frequencyestimator based on narrow-band approximation under noisyenvironmentrdquo Signal Processing vol 91 no 4 pp 841ndash851 2011
[9] C Yang and G Wei ldquoA noniterative frequency estimator withrational combination of three spectrum linesrdquo IEEE Transac-tions on Signal Processing vol 59 no 10 pp 5065ndash5070 2011
[10] C Candan and S Koc ldquoBeamspace approach for detection ofthe number of coherent sourcesrdquo in Proceedings of the IEEERadar Conference Ubiquitous Radar (RADAR rsquo12) pp 0913ndash0918 May 2012
[11] Y Cao G Wei and F-J Chen ldquoA closed-form expandedautocorrelationmethod for frequency estimation of a sinusoidrdquoSignal Processing vol 92 no 4 pp 885ndash892 2012
[12] F Qian S Leung Y Zhu W Wong D Pao and W LauldquoDamped sinusoidal signals parameter estimation in frequencydomainrdquo Signal Processing vol 92 no 2 pp 381ndash391 2012
[13] G Richard ELINT The Interception and Analysis of RadarSignals John Wiley amp Sons Artech House Dedham MassUSA 2nd edition 2006
[14] W Su J A Kosinski and M Yu ldquoDual-use of modulationrecognition techniques for digital communication signalsrdquo inProceedings of the IEEE Long Island Systems Applications andTechnology Conference (LISAT rsquo06) pp 1ndash6 Long Island NYUSA May 2006
[15] L Pucker ldquoReview of contemporaryspectrum sensingtechnologiesrdquo IEEE-SA P19006 Standards Group 2009 httpgrouperieeeorggroupsdyspan6documentswhite papersP19006 Sensor Surveypdf
[16] A Fehske J Gaeddert and JH Reed ldquoAnew approach to signalclassification using spectral correlation and neural networksrdquoin Proceedings of the 1st IEEE International Symposium on NewFrontiers in Dynamic Spectrum Access Networks (DySPAN rsquo05)pp 144ndash150 Baltimore Md USA November 2005
[17] W S Lin andK J R Liu ldquoModulation forensics for wireless dig-ital communicationsrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo08) pp 1789ndash1792 April 2008
[18] G-B Hu L-Z Xu and M Jin ldquoReliability testing for blindprocessing results of LFM signals based on NP criterionrdquo ActaElectronica Sinica vol 41 no 4 pp 739ndash743 2013
[19] A D Whalen Detection of Signals in Noise Academic PressNew York NY USA 2nd edition 1995
[20] M Sankaran ldquoOn the non-central chi-square distributionrdquoBiometrika vol 46 no 1-2 pp 235ndash237 1959
[21] E Aboutanios and B Mulgrew ldquoIterative frequency estimationby interpolation on Fourier coefficientsrdquo IEEE Transactions onSignal Processing vol 53 no 4 pp 1237ndash1242 2005
[22] AM Zoubir andD R Iskander Bootstrap Techniques for SignalProcessing Cambridge University Press Cambridge UK 2004
[23] A M Zoubir and D R Iskandler ldquoBootstrap methods andapplicationsrdquo IEEE Signal Processing Magazine vol 24 no 4pp 10ndash19 2007
less than minus3 dB the probabilities of detection of the AM-basedmethod are greater than those of the CI-basedmethodFor the fusion method the probabilities of detection usingthe ldquoANDrdquo rule are slightly greater than those of the AM-based method and vice versa for the ldquoORrdquo rule that isthe probability of detection is the smallest when the SNR isless than minus3 dB When the SNRs are greater than minus3 dB theprobabilities of detection of the four detectors all approach1 Therefore under lower SNRs the more efficient fusionmethods combining the CI and AM methods need to befurther developed
It should be noted that calculating the threshold isdifficult because the exact probability distribution function ofthe AM-based statistic 119879AM defined in the appendix is hardto be expressed analytically Therefore here we introducethe bootstrap-based method [22 23] to obtain the thresh-old without depending on the analytical probability of thestatistic based on AM but the computational load increasesmany times because of repeated resampling operations in thebootstrap-based hypothesis test procedures
68 Case Study In this section we use the CI-based credibil-ity test statistic to evaluate three commonly used frequencyestimators the Rife algorithm [4] the maximum likelihoodestimator [6] by Newton iteration and the Aboutanious-Mulgrew estimator [21]The carrier frequency of the sinusoidis 19081MHz the initial phase is 1205876 and the sample size is1024 The range of SNR is from minus18 dB to minus6 dB The valueof probability of false alarm is set to 005 If we let 119863
119894(119894 =
0 1) be the event associated with the decision of choosing119867119894(119894 = 0 1) 119899
119894119895stands for the number of times that deciding
119863119895when hypothesis 119867
119894is correct Therefore Type I error
probability and Type II error probability can be calculatedusing 119899
01(11989900
+ 11989901) and 119899
10(11989910
+ 11989911) respectively The
error probability of the test can be expressed by 119875119864= (11989901+
11989910)10000 while 119875
119863= 11989911(11989911+ 11989910) indicates the ability
of the proposed method to detect the unreliable frequencyestimate which means that the maximum absolute bias of acertain frequency estimate (|Δ119891|) is less than a quarter of thediscrete sampling frequency interval (Δ119865)
Table 1 illustrates the performance of the credibility testfor the ML frequency estimator of the sinusoid by using theproposed CI-based statistic It should be remarked that thecredibility test performance is enhanced by increasing theSNR If the SNR is minus6 dB all the 10000 frequency estimatesbased on ML are reliable and the error probability of theproposed test approximates to 0 When the SNR decreases tominus12 dB 61 estimates are unreliable among 10000 simulationsBy using the proposed statistic all the 61 unreliable estimatescan be detected indicating a detection rate of about 100Among the 9939 reliable estimates there are 21 times whenestimates are mistakenly decided as unreliable in otherwords the error probability is about 021 When the SNR isdecreased to minus15 dB the error probability is about 081 andthe detection rate is greater than 95
Table 2 depicts the performance of the credibility test forthe Rife-based sinusoid frequency estimator The credibilitytest performance is enhanced by a larger SNR For an SNR ofminus6 dB all the 10000 times the frequency estimates obtained
Table 1 Credibility test performance of ML frequency estimator
by using the Rife algorithm are reliable (1198670) and the error
probability is approximately equal to 0 When the SNRdeclines to minus12 dB there are 71 unreliable estimates among10000 simulations By using the CI-based credibility teststatistic these 70 unreliable estimates can be detected andthe detection rate is about 986 On the other hand amongthe 9929 reliable estimates there are 25 times when theestimates are mistakenly decided as unreliable that is theerror probability is about 026 With the SNR decreased tominus15 dB the error probability is about 085 and the detectionrate is greater than 90
Table 3 provides the performance of the credibility test forthe AM-based sinusoid frequency estimator The credibilitytest performance is enhanced by a larger SNR Similar to thatdata in Tables 1 and 2 the test performance is improved bya larger SNR With the SNR decreased to minus15 dB the errorprobability is about 069 and the detection rate is greaterthan 80
Mathematical Problems in Engineering 9
Overall from Tables 2 and 3 we can observe that theproposed credibility test statistic can be used for effectivelyidentifying unreliable frequency estimates by using ML AMand Rife algorithms with a high detection probability and alow error probability when the SNR is greater than minus15 dBCompared to the estimator performances summarized inTables 1 and 2 the estimator performance summarized inTable 3 is the lowest because AM-based estimation has abetter performance than that of bothML and Rife algorithmsand the frequency estimation factors are small
In practice for the detected unreliable estimate we canabandon the results or estimate the frequency of the signalagain thereby enhancing the reliability of the entire signalprocessing unit For example if the FFT-based coarse fre-quency estimator is used in practice andwe set the parametersas 119891119904= 100MHz 119891
0= 1955MHz 119873 = 1000 and the
number of DFT119872 = 1000 there are two cases
(1) Case 1 When the SNRs (eg SNR = minus23 dB) arebelow the SNR threshold the maximum spectrumline cannot be detected correctly In this case theestimate error is larger than 025Δ119865 and the estimatewill be judged as incredible by the proposed CI-basedstatistic If we perform resampling over the frequencygrid by setting 119872 = 2000 the maximum spectrumline still cannot be detected As the absolute estima-tion errors are the same under the two scenariosthe resampling action cannot enhance the credibilityof the estimate and this kind of incredible estimateshould be abandoned
(2) Case 2When an SNR is greater than the SNR thresh-old an incredible frequency estimate is detectedbecause of the displacement of the maximum spec-trum line by the discretization In this case ifwe enlarge 119872 to 2000 by resampling in the fre-quency domain 119891
0can be just moved to the dis-
crete frequency bin Hence the estimation error canbe decreased and the credibility can be enhancedHowever in practice the true frequency cannot beknown by the user hence it is difficult to adjustthe resampling parameter to match the estimationexactly In this case we need to select the moreaccurate estimating method or abandon the resultobtained by original estimation
7 Conclusion
The paper presented a CI-based credibility test algorithm forfrequency estimation of a sinusoid A credibility assessmenttesting model is defined to analyze the mathematical charac-teristics of the correlations between the observed signal andthe reference signal under different hypotheses The test isperformed with the proposed threshold that is based on CIExperimental results revealed that a good performance wasachieved even at low SNRs Credibility tests were performedwith the ML frequency estimator using the proposed thresh-old based on CI as well with the Rife frequency estimator andthe Aboutanious-Mulgrew estimator Experimental results
revealed that a good performance was achieved by theproposed method even at low SNRs
Appendix
Thefrequency estimation credibility testmethod based on theAM estimator includes
(1) estimation of the frequency of the signal by using acertain estimation method
(2) construction of the reference signal 119910(119899) with (3) byusing the estimated frequency
1198910
(3) computation of the correlation between the receivedsignal and the reference signal and estimation ofthe frequency displacement Δ
119891 by the Aboutanios-Mulgrew frequency estimator [21]
(4) evaluation of the credibility of frequency estimationby comparing the statistic 119879AM = ||Δ
119891| minus 025Δ119865| to
the threshold th that can be calculated by a given falsealarm 120572 The hypothesis 119867
0is chosen if 119879AM lt th
Otherwise1198671is accepted
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This study is financially supported by the Natural ScienceFoundation of Jiangsu Province (Project no BK2011837) theNational Natural Science Foundation of China 2012 (Projectno 61201208) and the 333 Research Project of JiangsuProvince (Project no BRA2013171)
References
[1] D C Rife and R R Boorstyn ldquoSingle-tone parameter esti-mation from discrete-time observationrdquo IEEE Transactions onInformation Theory vol 20 no 5 pp 591ndash598 1974
[2] S Peleg and B Porat ldquoLinear FM signal parameter estima-tion from discrete-time observationsrdquo IEEE Transactions onAerospace and Electronic Systems vol 27 no 4 pp 607ndash6161991
[3] M Ghogho A Swami and T Durrani ldquoBlind estimation offrequency offset in the presence of unknown multipathrdquo inProceedings of the IEEE International Conference on PersonalWireless Communications pp 104ndash108 December 2000
[4] D C Rife and G A Vincent ldquoUse of the discrete Fouriertransform in the measurement of frequencies and levels oftonesrdquoThe Bell System Technical Journal vol 49 no 2 pp 197ndash228 1970
[5] J-R Liao andC-M Chen ldquoPhase correction of discrete Fouriertransform coefficients to reduce frequency estimation bias ofsingle tone complex sinusoidrdquo Signal Processing vol 94 no 1pp 108ndash117 2014
[6] T Abatzoglou ldquoA fast maximum likelihood algorithm forfrequency estimation of a sinusoid based on Newtonrsquos methodrdquo
10 Mathematical Problems in Engineering
IEEE Transactions on Acoustics Speech and Signal Processingvol 33 no 1 pp 77ndash89 1985
[7] Y Liu Z Nie Z Zhao and Q H Liu ldquoGeneralization ofiterative Fourier interpolation algorithms for single frequencyestimationrdquo Digital Signal Processing vol 21 no 1 pp 141ndash1492011
[8] G Wei C Yang and F-J Chen ldquoClosed-form frequencyestimator based on narrow-band approximation under noisyenvironmentrdquo Signal Processing vol 91 no 4 pp 841ndash851 2011
[9] C Yang and G Wei ldquoA noniterative frequency estimator withrational combination of three spectrum linesrdquo IEEE Transac-tions on Signal Processing vol 59 no 10 pp 5065ndash5070 2011
[10] C Candan and S Koc ldquoBeamspace approach for detection ofthe number of coherent sourcesrdquo in Proceedings of the IEEERadar Conference Ubiquitous Radar (RADAR rsquo12) pp 0913ndash0918 May 2012
[11] Y Cao G Wei and F-J Chen ldquoA closed-form expandedautocorrelationmethod for frequency estimation of a sinusoidrdquoSignal Processing vol 92 no 4 pp 885ndash892 2012
[12] F Qian S Leung Y Zhu W Wong D Pao and W LauldquoDamped sinusoidal signals parameter estimation in frequencydomainrdquo Signal Processing vol 92 no 2 pp 381ndash391 2012
[13] G Richard ELINT The Interception and Analysis of RadarSignals John Wiley amp Sons Artech House Dedham MassUSA 2nd edition 2006
[14] W Su J A Kosinski and M Yu ldquoDual-use of modulationrecognition techniques for digital communication signalsrdquo inProceedings of the IEEE Long Island Systems Applications andTechnology Conference (LISAT rsquo06) pp 1ndash6 Long Island NYUSA May 2006
[15] L Pucker ldquoReview of contemporaryspectrum sensingtechnologiesrdquo IEEE-SA P19006 Standards Group 2009 httpgrouperieeeorggroupsdyspan6documentswhite papersP19006 Sensor Surveypdf
[16] A Fehske J Gaeddert and JH Reed ldquoAnew approach to signalclassification using spectral correlation and neural networksrdquoin Proceedings of the 1st IEEE International Symposium on NewFrontiers in Dynamic Spectrum Access Networks (DySPAN rsquo05)pp 144ndash150 Baltimore Md USA November 2005
[17] W S Lin andK J R Liu ldquoModulation forensics for wireless dig-ital communicationsrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo08) pp 1789ndash1792 April 2008
[18] G-B Hu L-Z Xu and M Jin ldquoReliability testing for blindprocessing results of LFM signals based on NP criterionrdquo ActaElectronica Sinica vol 41 no 4 pp 739ndash743 2013
[19] A D Whalen Detection of Signals in Noise Academic PressNew York NY USA 2nd edition 1995
[20] M Sankaran ldquoOn the non-central chi-square distributionrdquoBiometrika vol 46 no 1-2 pp 235ndash237 1959
[21] E Aboutanios and B Mulgrew ldquoIterative frequency estimationby interpolation on Fourier coefficientsrdquo IEEE Transactions onSignal Processing vol 53 no 4 pp 1237ndash1242 2005
[22] AM Zoubir andD R Iskander Bootstrap Techniques for SignalProcessing Cambridge University Press Cambridge UK 2004
[23] A M Zoubir and D R Iskandler ldquoBootstrap methods andapplicationsrdquo IEEE Signal Processing Magazine vol 24 no 4pp 10ndash19 2007
Overall from Tables 2 and 3 we can observe that theproposed credibility test statistic can be used for effectivelyidentifying unreliable frequency estimates by using ML AMand Rife algorithms with a high detection probability and alow error probability when the SNR is greater than minus15 dBCompared to the estimator performances summarized inTables 1 and 2 the estimator performance summarized inTable 3 is the lowest because AM-based estimation has abetter performance than that of bothML and Rife algorithmsand the frequency estimation factors are small
In practice for the detected unreliable estimate we canabandon the results or estimate the frequency of the signalagain thereby enhancing the reliability of the entire signalprocessing unit For example if the FFT-based coarse fre-quency estimator is used in practice andwe set the parametersas 119891119904= 100MHz 119891
0= 1955MHz 119873 = 1000 and the
number of DFT119872 = 1000 there are two cases
(1) Case 1 When the SNRs (eg SNR = minus23 dB) arebelow the SNR threshold the maximum spectrumline cannot be detected correctly In this case theestimate error is larger than 025Δ119865 and the estimatewill be judged as incredible by the proposed CI-basedstatistic If we perform resampling over the frequencygrid by setting 119872 = 2000 the maximum spectrumline still cannot be detected As the absolute estima-tion errors are the same under the two scenariosthe resampling action cannot enhance the credibilityof the estimate and this kind of incredible estimateshould be abandoned
(2) Case 2When an SNR is greater than the SNR thresh-old an incredible frequency estimate is detectedbecause of the displacement of the maximum spec-trum line by the discretization In this case ifwe enlarge 119872 to 2000 by resampling in the fre-quency domain 119891
0can be just moved to the dis-
crete frequency bin Hence the estimation error canbe decreased and the credibility can be enhancedHowever in practice the true frequency cannot beknown by the user hence it is difficult to adjustthe resampling parameter to match the estimationexactly In this case we need to select the moreaccurate estimating method or abandon the resultobtained by original estimation
7 Conclusion
The paper presented a CI-based credibility test algorithm forfrequency estimation of a sinusoid A credibility assessmenttesting model is defined to analyze the mathematical charac-teristics of the correlations between the observed signal andthe reference signal under different hypotheses The test isperformed with the proposed threshold that is based on CIExperimental results revealed that a good performance wasachieved even at low SNRs Credibility tests were performedwith the ML frequency estimator using the proposed thresh-old based on CI as well with the Rife frequency estimator andthe Aboutanious-Mulgrew estimator Experimental results
revealed that a good performance was achieved by theproposed method even at low SNRs
Appendix
Thefrequency estimation credibility testmethod based on theAM estimator includes
(1) estimation of the frequency of the signal by using acertain estimation method
(2) construction of the reference signal 119910(119899) with (3) byusing the estimated frequency
1198910
(3) computation of the correlation between the receivedsignal and the reference signal and estimation ofthe frequency displacement Δ
119891 by the Aboutanios-Mulgrew frequency estimator [21]
(4) evaluation of the credibility of frequency estimationby comparing the statistic 119879AM = ||Δ
119891| minus 025Δ119865| to
the threshold th that can be calculated by a given falsealarm 120572 The hypothesis 119867
0is chosen if 119879AM lt th
Otherwise1198671is accepted
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This study is financially supported by the Natural ScienceFoundation of Jiangsu Province (Project no BK2011837) theNational Natural Science Foundation of China 2012 (Projectno 61201208) and the 333 Research Project of JiangsuProvince (Project no BRA2013171)
References
[1] D C Rife and R R Boorstyn ldquoSingle-tone parameter esti-mation from discrete-time observationrdquo IEEE Transactions onInformation Theory vol 20 no 5 pp 591ndash598 1974
[2] S Peleg and B Porat ldquoLinear FM signal parameter estima-tion from discrete-time observationsrdquo IEEE Transactions onAerospace and Electronic Systems vol 27 no 4 pp 607ndash6161991
[3] M Ghogho A Swami and T Durrani ldquoBlind estimation offrequency offset in the presence of unknown multipathrdquo inProceedings of the IEEE International Conference on PersonalWireless Communications pp 104ndash108 December 2000
[4] D C Rife and G A Vincent ldquoUse of the discrete Fouriertransform in the measurement of frequencies and levels oftonesrdquoThe Bell System Technical Journal vol 49 no 2 pp 197ndash228 1970
[5] J-R Liao andC-M Chen ldquoPhase correction of discrete Fouriertransform coefficients to reduce frequency estimation bias ofsingle tone complex sinusoidrdquo Signal Processing vol 94 no 1pp 108ndash117 2014
[6] T Abatzoglou ldquoA fast maximum likelihood algorithm forfrequency estimation of a sinusoid based on Newtonrsquos methodrdquo
10 Mathematical Problems in Engineering
IEEE Transactions on Acoustics Speech and Signal Processingvol 33 no 1 pp 77ndash89 1985
[7] Y Liu Z Nie Z Zhao and Q H Liu ldquoGeneralization ofiterative Fourier interpolation algorithms for single frequencyestimationrdquo Digital Signal Processing vol 21 no 1 pp 141ndash1492011
[8] G Wei C Yang and F-J Chen ldquoClosed-form frequencyestimator based on narrow-band approximation under noisyenvironmentrdquo Signal Processing vol 91 no 4 pp 841ndash851 2011
[9] C Yang and G Wei ldquoA noniterative frequency estimator withrational combination of three spectrum linesrdquo IEEE Transac-tions on Signal Processing vol 59 no 10 pp 5065ndash5070 2011
[10] C Candan and S Koc ldquoBeamspace approach for detection ofthe number of coherent sourcesrdquo in Proceedings of the IEEERadar Conference Ubiquitous Radar (RADAR rsquo12) pp 0913ndash0918 May 2012
[11] Y Cao G Wei and F-J Chen ldquoA closed-form expandedautocorrelationmethod for frequency estimation of a sinusoidrdquoSignal Processing vol 92 no 4 pp 885ndash892 2012
[12] F Qian S Leung Y Zhu W Wong D Pao and W LauldquoDamped sinusoidal signals parameter estimation in frequencydomainrdquo Signal Processing vol 92 no 2 pp 381ndash391 2012
[13] G Richard ELINT The Interception and Analysis of RadarSignals John Wiley amp Sons Artech House Dedham MassUSA 2nd edition 2006
[14] W Su J A Kosinski and M Yu ldquoDual-use of modulationrecognition techniques for digital communication signalsrdquo inProceedings of the IEEE Long Island Systems Applications andTechnology Conference (LISAT rsquo06) pp 1ndash6 Long Island NYUSA May 2006
[15] L Pucker ldquoReview of contemporaryspectrum sensingtechnologiesrdquo IEEE-SA P19006 Standards Group 2009 httpgrouperieeeorggroupsdyspan6documentswhite papersP19006 Sensor Surveypdf
[16] A Fehske J Gaeddert and JH Reed ldquoAnew approach to signalclassification using spectral correlation and neural networksrdquoin Proceedings of the 1st IEEE International Symposium on NewFrontiers in Dynamic Spectrum Access Networks (DySPAN rsquo05)pp 144ndash150 Baltimore Md USA November 2005
[17] W S Lin andK J R Liu ldquoModulation forensics for wireless dig-ital communicationsrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo08) pp 1789ndash1792 April 2008
[18] G-B Hu L-Z Xu and M Jin ldquoReliability testing for blindprocessing results of LFM signals based on NP criterionrdquo ActaElectronica Sinica vol 41 no 4 pp 739ndash743 2013
[19] A D Whalen Detection of Signals in Noise Academic PressNew York NY USA 2nd edition 1995
[20] M Sankaran ldquoOn the non-central chi-square distributionrdquoBiometrika vol 46 no 1-2 pp 235ndash237 1959
[21] E Aboutanios and B Mulgrew ldquoIterative frequency estimationby interpolation on Fourier coefficientsrdquo IEEE Transactions onSignal Processing vol 53 no 4 pp 1237ndash1242 2005
[22] AM Zoubir andD R Iskander Bootstrap Techniques for SignalProcessing Cambridge University Press Cambridge UK 2004
[23] A M Zoubir and D R Iskandler ldquoBootstrap methods andapplicationsrdquo IEEE Signal Processing Magazine vol 24 no 4pp 10ndash19 2007
IEEE Transactions on Acoustics Speech and Signal Processingvol 33 no 1 pp 77ndash89 1985
[7] Y Liu Z Nie Z Zhao and Q H Liu ldquoGeneralization ofiterative Fourier interpolation algorithms for single frequencyestimationrdquo Digital Signal Processing vol 21 no 1 pp 141ndash1492011
[8] G Wei C Yang and F-J Chen ldquoClosed-form frequencyestimator based on narrow-band approximation under noisyenvironmentrdquo Signal Processing vol 91 no 4 pp 841ndash851 2011
[9] C Yang and G Wei ldquoA noniterative frequency estimator withrational combination of three spectrum linesrdquo IEEE Transac-tions on Signal Processing vol 59 no 10 pp 5065ndash5070 2011
[10] C Candan and S Koc ldquoBeamspace approach for detection ofthe number of coherent sourcesrdquo in Proceedings of the IEEERadar Conference Ubiquitous Radar (RADAR rsquo12) pp 0913ndash0918 May 2012
[11] Y Cao G Wei and F-J Chen ldquoA closed-form expandedautocorrelationmethod for frequency estimation of a sinusoidrdquoSignal Processing vol 92 no 4 pp 885ndash892 2012
[12] F Qian S Leung Y Zhu W Wong D Pao and W LauldquoDamped sinusoidal signals parameter estimation in frequencydomainrdquo Signal Processing vol 92 no 2 pp 381ndash391 2012
[13] G Richard ELINT The Interception and Analysis of RadarSignals John Wiley amp Sons Artech House Dedham MassUSA 2nd edition 2006
[14] W Su J A Kosinski and M Yu ldquoDual-use of modulationrecognition techniques for digital communication signalsrdquo inProceedings of the IEEE Long Island Systems Applications andTechnology Conference (LISAT rsquo06) pp 1ndash6 Long Island NYUSA May 2006
[15] L Pucker ldquoReview of contemporaryspectrum sensingtechnologiesrdquo IEEE-SA P19006 Standards Group 2009 httpgrouperieeeorggroupsdyspan6documentswhite papersP19006 Sensor Surveypdf
[16] A Fehske J Gaeddert and JH Reed ldquoAnew approach to signalclassification using spectral correlation and neural networksrdquoin Proceedings of the 1st IEEE International Symposium on NewFrontiers in Dynamic Spectrum Access Networks (DySPAN rsquo05)pp 144ndash150 Baltimore Md USA November 2005
[17] W S Lin andK J R Liu ldquoModulation forensics for wireless dig-ital communicationsrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo08) pp 1789ndash1792 April 2008
[18] G-B Hu L-Z Xu and M Jin ldquoReliability testing for blindprocessing results of LFM signals based on NP criterionrdquo ActaElectronica Sinica vol 41 no 4 pp 739ndash743 2013
[19] A D Whalen Detection of Signals in Noise Academic PressNew York NY USA 2nd edition 1995
[20] M Sankaran ldquoOn the non-central chi-square distributionrdquoBiometrika vol 46 no 1-2 pp 235ndash237 1959
[21] E Aboutanios and B Mulgrew ldquoIterative frequency estimationby interpolation on Fourier coefficientsrdquo IEEE Transactions onSignal Processing vol 53 no 4 pp 1237ndash1242 2005
[22] AM Zoubir andD R Iskander Bootstrap Techniques for SignalProcessing Cambridge University Press Cambridge UK 2004
[23] A M Zoubir and D R Iskandler ldquoBootstrap methods andapplicationsrdquo IEEE Signal Processing Magazine vol 24 no 4pp 10ndash19 2007
