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Comparison of Various Periodograms for Sinusoid Detection and Frequency Estimation H. C. SO, Member, IEEE City University of Hong Kong Y. T. CHAN, Senior Member, IEEE Q. MA Royal Military College of Canada P. C. CHING, Senior Member, IEEE The Chinese University of Hong Kong With the advent of the fast Fourier transform (FFT) algorithm, the periodogram and its variants such as the Bartlett’s procedure and Welch method, have become very popular for spectral analysis. However, there has not been a thorough comparison of the detection and estimation performances of these methods. Different forms of the periodogram are studied here for single real tone detection and frequency estimation in the presence of white Gaussian noise. The threshold effect in frequency estimation, that is, when the estimation errors become several orders of magnitude greater than the Cram ´ er-Rao lower bound (CRLB), is also investigated. It is shown that the standard periodogram gives the optimum detection performance for a pure tone while the Welch method is the best detector when there is phase instability in the sinusoid. As expected, since the conventional periodogram is a maximum likelihood estimator of frequency, it generally provides the minimum mean square frequency estimation errors. Manuscript received January 19, 1998; revised November 3, 1998. IEEE Log No. T-AES/35/3/06405. Authors’ current addresses: H. C. So, Dept. of Electronic Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong; Y. T. Chan, Dept. of Electrical and Computer Engineering, Royal Military College of Canada, Kingston, Ontario, Canada, K7K 5L0; Q. Ma, Wireless Networks, Nortel Networks, Ottawa, Canada; P. C. Ching, Dept. of Electronic Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong. 0018-9251/99/$10.00 c ° 1999 IEEE I. INTRODUCTION Detection and frequency estimation of sinusoidal signals from a finite number of noisy discrete-time measurements have applications in many fields. It has been widely used in sonar and radar for moving target detection. Estimation of Doppler shift in this case is usually required in order to find its position and speed. A recent and interesting application is in the search for extraterrestrial intelligence (SETI) [1]. Other well-known examples include demodulation of frequency-shift keying signals, wind profiling [2], geolocation, and identification and tracking of an emergency location transmitter [3]. A direct method for tone detection and frequency estimation is the standard periodogram. The periodogram of an N -point sequence, x(0), x(1), ::: , x(N ¡ 1), is defined as S x (f )= 1 N ¯ ¯ ¯ ¯ ¯ N¡1 X n=0 x(n) exp(¡j 2¼nf ) ¯ ¯ ¯ ¯ ¯ 2 , k = 0, 1, ::: , N ¡ 1: (1) Samples of the periodogram at f k = k=N , k = 0,1, ::: , N ¡ 1 can be computed efficiently with the use of the fast Fourier transform (FFT) algorithm while values at the other frequencies are evaluated by either zero padding or interpolation. In the Bartlett method which is a variant of the periodogram, the sequence x(n) is divided into K nonoverlapping segments, where each segment has length M. For each segment, the periodogram is computed and the Bartlett power spectral estimate is obtained by averaging the periodograms for the K segments. By so doing, the variance in the periodogram estimate is reduced by a factor K but at the expense of reducing the frequency resolution by K. Welch had modified Bartlett’s procedure by allowing the data segments to overlap and at the same time to be multiplied by a window function prior to computing the periodogram. The overlapping is used for further reducing the periodogram variance while the windowing is applied to reduce the spectral leakage [4] associated with finite observation intervals. Although the many forms of periodogram have been derived for more than two decades, there is not a comprehensive study of these approaches in sinusoid detection and frequency estimation. In this paper, the standard periodogram with and without windowing, together with Bartlett’s procedure and the Welch method are compared in order to determine the best detector for a sinusoid in the presence of white Gaussian noise. We would also like to examine if these modifications can delay the threshold effect of the conventional periodogram in frequency estimation of a noisy sinusoid, even though the latter provides the maximum likelihood estimate. It is important to know the theoretical signal-to-noise ratio (SNR) IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 35, NO. 3 JULY 1999 945
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Page 1: Comparison of Various Periodograms for Sinusoid Detection ...hcso/aes_7_99.pdfComparison of Various Periodograms for Sinusoid Detection and Frequency Estimation H. C. SO, Member, IEEE

Comparison of VariousPeriodograms for SinusoidDetection and FrequencyEstimation

H. C. SO, Member, IEEECity University of Hong Kong

Y. T. CHAN, Senior Member, IEEE

Q. MARoyal Military College of Canada

P. C. CHING, Senior Member, IEEEThe Chinese University of Hong Kong

With the advent of the fast Fourier transform (FFT)

algorithm, the periodogram and its variants such as the Bartlett’s

procedure and Welch method, have become very popular for

spectral analysis. However, there has not been a thorough

comparison of the detection and estimation performances of

these methods. Different forms of the periodogram are studied

here for single real tone detection and frequency estimation in

the presence of white Gaussian noise. The threshold effect in

frequency estimation, that is, when the estimation errors become

several orders of magnitude greater than the Cramer-Rao lower

bound (CRLB), is also investigated. It is shown that the standard

periodogram gives the optimum detection performance for a

pure tone while the Welch method is the best detector when

there is phase instability in the sinusoid. As expected, since the

conventional periodogram is a maximum likelihood estimator

of frequency, it generally provides the minimum mean square

frequency estimation errors.

Manuscript received January 19, 1998; revised November 3, 1998.

IEEE Log No. T-AES/35/3/06405.

Authors’ current addresses: H. C. So, Dept. of ElectronicEngineering, City University of Hong Kong, Tat Chee Avenue,Kowloon, Hong Kong; Y. T. Chan, Dept. of Electrical andComputer Engineering, Royal Military College of Canada,Kingston, Ontario, Canada, K7K 5L0; Q. Ma, Wireless Networks,Nortel Networks, Ottawa, Canada; P. C. Ching, Dept. of ElectronicEngineering, The Chinese University of Hong Kong, Shatin, N.T.,Hong Kong.

0018-9251/99/$10.00 c° 1999 IEEE

I. INTRODUCTION

Detection and frequency estimation of sinusoidalsignals from a finite number of noisy discrete-timemeasurements have applications in many fields. Ithas been widely used in sonar and radar for movingtarget detection. Estimation of Doppler shift in thiscase is usually required in order to find its positionand speed. A recent and interesting application is inthe search for extraterrestrial intelligence (SETI) [1].Other well-known examples include demodulation offrequency-shift keying signals, wind profiling [2],geolocation, and identification and tracking of anemergency location transmitter [3].A direct method for tone detection and

frequency estimation is the standard periodogram.The periodogram of an N-point sequence,x(0),x(1), : : : ,x(N ¡ 1), is defined as

Sx(f) =1N

¯¯N¡1Xn=0

x(n)exp(¡j2¼nf)¯¯2

,

k = 0,1, : : : ,N ¡ 1: (1)

Samples of the periodogram at fk = k=N, k =0,1, : : : ,N ¡ 1 can be computed efficiently with theuse of the fast Fourier transform (FFT) algorithmwhile values at the other frequencies are evaluated byeither zero padding or interpolation. In the Bartlettmethod which is a variant of the periodogram, thesequence x(n) is divided into K nonoverlappingsegments, where each segment has length M. Foreach segment, the periodogram is computed andthe Bartlett power spectral estimate is obtained byaveraging the periodograms for the K segments. Byso doing, the variance in the periodogram estimate isreduced by a factor K but at the expense of reducingthe frequency resolution by K. Welch had modifiedBartlett’s procedure by allowing the data segmentsto overlap and at the same time to be multiplied by awindow function prior to computing the periodogram.The overlapping is used for further reducing theperiodogram variance while the windowing is appliedto reduce the spectral leakage [4] associated withfinite observation intervals.Although the many forms of periodogram have

been derived for more than two decades, there isnot a comprehensive study of these approaches insinusoid detection and frequency estimation. In thispaper, the standard periodogram with and withoutwindowing, together with Bartlett’s procedure and theWelch method are compared in order to determine thebest detector for a sinusoid in the presence of whiteGaussian noise. We would also like to examine ifthese modifications can delay the threshold effect ofthe conventional periodogram in frequency estimationof a noisy sinusoid, even though the latter providesthe maximum likelihood estimate. It is importantto know the theoretical signal-to-noise ratio (SNR)

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 35, NO. 3 JULY 1999 945

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below which the estimates become unreliable. Thiswill provide a guide to system designers to decide onthe necessary data length, for a given SNR, to avoidlarge estimation errors. Theoretical developments forsingle tone detection and frequency estimation aregiven in Section II and Section III, respectively. Inparticular, the detection probability, the false alarmprobability, and the mean square frequency error ofthe periodogram and the Bartlett method are derived.Simulation results are presented in Section IV tocorroborate the analytical derivations and to evaluatethe detection and estimation performance of differenttypes of periodograms.

II. SINGLE TONE DETECTION

The problem of detecting a single sinusoid in thepresence of noise is formulated as follows. Given thereceived sequence x(n), a decision has to be madebetween the hypotheses:

H0 : x(n) = q(n), n= 0,1, : : : ,N ¡ 1H1 : x(n) = ®sin(2¼f0n+Á(n)) + q(n)

(2)

where q(n) is a white Gaussian process while ®, f0,and Á(n) are unknown parameters which represent thetone amplitude, frequency, and time-dependent phase,respectively. The first hypothesis H0 assumes thatx(n) consists only of noise while in H1, the sinusoidalsignal is presumed to be present. The aim is to findthe receiver operating characteristic (ROC) whichis determined by the false alarm probability PFA =P(D1 jH0) and detection probability PD = P(D1 jH1),where D1 represents the decision of hypothesis H1.When the standard periodogram is used for

tone detection, three steps are involved. They are1) compute the N samples of the periodogram usingFFT, 2) find the peak value by interpolation, and3) compare it with a threshold TH. If the peakcoefficient is larger than TH, H1 is accepted, otherwiseH0 is chosen. Notice that in the first step, we onlyneed to compute the spectral coefficients for f =1=N,2=N, : : : ,1=2¡ 1=N, because the power spectrumis symmetric for real signals. To derive the PFA and PDin this method, we first obtain the probability densityfunctions (PDFs) of the discrete Fourier transform(DFT) spectral coefficients. For ease of analysis,it is assumed that Á(n) = Á is a constant uniformlydistributed between 0 and 2¼ while f0 2 (0,0:5) suchthat f0 = k0=N where k0 2 f1,2, : : : ,N=2¡ 1g, implyingthat the tone frequency corresponds exactly to one ofthe FFT bins. In this case, only the largest coefficientis selected among the DFT power spectrum and nointerpolation is necessary. However, the simulationresults will also contain experiments with sinusoids ofoff-bin frequencies. It is noteworthy that the standardperiodogram is a digital realization of the quadraturereceiver which is optimum in the sense that it attains

the minimum probability of error. Denote the spectralcoefficients at f = k=N for the noise only case and thesignal present case by Sx0 (k) and Sx1 (k), respectively.The value of Sx1 (k0) is computed as

Sx1 (k0) = A2 +B2 (3)

where

A¢=

1pN

N¡1Xn=0

(®sin(2¼f0n+Á)+ q(n))cosµ2¼nk0N

¶and

B¢=

1pN

N¡1Xn=0

(®sin(2¼f0n+Á)+ q(n))sinµ2¼nk0N

¶are Gaussian random variables. It can be proved[5] that A and B are independent and of identicalvariances equal to ¾2 = ¾2q=2 where ¾

2q represents the

power of q(n). We then follow [6] to derive the PDFof Sx1 (k0), denoted by ps(u), as

ps(u) =12¾2

expµ¡s

2 + u2¾2

¶I0

³pus

¾2

´(4)

which is a noncentral chi-square random variable. Thequantity s equals the square root of the sum of meansquare of A and B, that is, s= ®

pN=2, and Ir is the

modified Bessel function of the first kind of order r.Similarly, it can be shown that the remaining Sx1 (k)

and all Sx0 (k) are of central chi-square distribution andhave the same PDF function pq(u) which is given by

pq(u) =12¾2

exp³¡ u

2¾2

´: (5)

Note that expressions similar to (4) and (5) can befound in [7] which derives the PDFs of the Fouriertransform coefficients of a complex tone signal undernoisy environment.Assuming that the DFT spectral coefficients are

independent which is valid when the time-bandwidthproduct of the signal is sufficiently large, then thefalse alarm rate PFA is calculated as

PFA = 1¡·Z VT

0pq(u)du

¸(N=2)¡1= 1¡

µ1¡ exp

µ¡ VT2¾2

¶¶(N=2)¡1(6)

and the probability of detection PD is derived as

PD = 1¡·Z VT

0

pq(u)du

¸(N=2)¡2Z VT

0

ps(u)du

= 1¡µ1¡ exp

µ¡ VT2¾2

¶¶(N=2)¡2Ã1¡Q1

Ãs

¾,

pVT¾

!!(7)

946 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 35, NO. 3 JULY 1999

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where Q denotes the Marcum Q function [8].The ROC is then acquired by plotting the PDagainst PFA.On the other hand, the DFT spectrum of the

Bartlett method is defined as

S0x(k) =1N

K¡1Xi=0

¯¯M¡1Xn=0

xi(n)expµ¡j2¼nk

M

¶¯¯2

,

k = 0,1, : : : ,M ¡ 1 (8)

where

xi(n) = x(i ¢M + n), i= 0,1, : : : ,K ¡ 1,n= 0,1, : : : ,M ¡ 1:

The derivation for the PDFs is similar to the standardperiodogram case but now the number of the powercoefficients are reduced to M because of the averagingof K segments. In this case, the spectral coefficientsare expressed as

S0x(k) =K¡1Xi=0

(A2i (k)+B2i (k)) (9)

where

Ai(k)¢=

1pN

M¡1Xn=0

xi(n)cosµ2¼nkM

¶and

Bi(k)¢=

1pN

M¡1Xn=0

xi(n)sinµ2¼nkM

¶are independent Gaussian random variables. Byfurther assuming f0 = k

00=M where k00 2 [1,M=2¡ 1]

is an integer and following the previous derivationthat gives rise to (4), it can be shown that when thesignal is present, the PDF of the spectral coefficientsat k = k00, denoted by psa(u), is of the form

psa(u) =12¾2a

µu

s2a

¶(K¡1)=2exp

µ¡s

2a + u2¾2a

¶IK¡1

µpusa¾2a

¶(10)

where ¾2a = ¾2q=(2K) and sa = ®

pM=2. The PDFs of

all other coefficients for the hypotheses H0 and H1 areequal to

pqa(u) =1

¾2Ka 2K¡ (K)uK¡1 exp

µ¡ u

2¾2a

¶(11)

where ¡ (K) is the gamma function. As a result, thefalse alarm and detection probability of single tonedetection using the Bartlett method, PFAa and PDa , aregiven by

PFAa = 1¡·Z VT

0pqa(u)du

¸(M=2)¡1(12)

and

PDa = 1¡·Z VT

0pqa(u)du

¸(M=2)¡2Z VT

0psa(u)du

(13)

respectively.

III. FREQUENCY ESTIMATION

When the hypothesis H1 is chosen, often thereis a need to estimate the tone frequency from thereceived sequence. It is well known that the maximumlikelihood frequency estimate of a pure sinusoid isgiven by the location of the peak of the periodogram[7]. This estimator will attain the Cramer—Rao lowerbound (CRLB) for frequency [9],

varf(N) =3

¼2N(N2¡ 1)SNR (14)

when SNR= ®2=(2¾2q) is greater than a threshold T.When SNR< T, however, the mean square frequencyerror (MSFE) rises very rapidly above the CRLB.This is because the frequency estimation problem isnonlinear and hence will suffer from this thresholdphenomenon, sometimes also known as occurrenceof outliers. Since the modified periodograms doprovide smaller variances, one might expect themto give a lower SNR threshold than the standardperiodogram. In this section, we derive expressionsfor the overall MSFEs of the periodogram and theBartlett’s procedure.When f0 = k0=N, the probability of occurrence of

an anomaly in the standard periodogram is derived as

q= P(Sx1 (k0)· at least one of Sx1 (1),Sx1 (2), : : : ,Sx1 (k0¡ 1),Sx1 (k0 +1), : : : ,Sx1 (N=2¡ 1))

=Z 1

0P(Sx1 (k0) = u) ¢

241¡ (N=2)¡1Yk=1,k 6=k0

P(Sx1 (k)< u)

35du=Z 1

0ps(u)

241¡ (N=2)¡1Yk=1,k 6=k0

Z u

0pq(v)dv

35du (15)

which must be computed numerically. Assuming thatthe anomaly estimate is uniformly distributed between0 and 0.5, the overall MSFE is given by

MSFE = (1¡q)varf(N) +qZ 0:5

02(u¡f0)2du:

(16)

Similarly, the mean square error in the frequencyestimate using the Bartlett method, denoted byMSFEa, is given by

MSFEa =1¡ qaK

varf(M)+ qa

Z 0:5

02(u¡f0)2 du

(17)

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Fig. 1. ROCs for detection of single sinusoid with frequencyf0 = 0:25 at different SNRs. (a) Using periodogram. (b) Using

Bartlett method with four segments.

where qa represents the probability of the occurrenceof an outlier in the averaged periodogram, which canbe calculated using psa(u) and pqa(u). Note that thereis a factor of K in (17) because each segment willgive an independent frequency estimate.

IV. SIMULATION RESULTS

Computer experiments were conducted to verifythe theoretical calculations derived in Sections II andIII. The performances of different variants of theperiodogram, including the windowed periodogram,the Bartlett method and the Welch method in singletone detection and frequency estimation were alsoevaluated. The variance of the white noise was fixedto unity while different SNRs were produced byproperly scaling the signal power. The total samplesize N was 256 and unless stated otherwise, thesegment length M had a value of 64. The results fordetection were averages of 100 000 independent runswhile those for frequency estimation were based on25 000 independent trials.Fig. 1(a) shows the experimental and theoretical

ROCs in detecting a pure sinusoid in the presence ofwhite Gaussian noise using the standard periodogram.The simulation results were obtained by using themethod suggested in [10]. Four values of SNR,

Fig. 2. Comparison of periodogram and Bartlett method withdifferent segment lengths for detecting single sinusoid with

frequency f0 = 0:25 at SNR=¡9 dB.

namely, ¡6 dB, ¡9 dB, ¡12 dB, and ¡15 dB weretried. The frequency f0 was chosen as 0.25 whichcorresponded to one of the FFT bins. It can be seenthat in all cases the simulation results agreed verywell with theory. The test was repeated for the Bartlettmethod and the results are shown in Fig. 1(b). Again,we observe that the experimental and theoreticalresults were very similar. By comparing Figs. 1(a)and 1(b), it can be seen that the periodogram alwaysprovides a better detection performance than theBartlett method. Fig. 2 plots the theoretical ROCs ofthe periodogram and the Bartlett method for differentsegment length M at SNR=¡9 dB. It is observedthat the detection accuracy decreased as the number ofsegment K increased.Fig. 3(a) compares the detection performance

of different forms of periodograms for f0 = 0:25and SNR=¡9 dB. In the windowed periodogram,the Hann window function was used while in theWelch method, rectangular and Hann windows weretried. Again, the standard periodogram gave the bestperformance. It is because it provided the greatestSNR of ®2N=(2¾2q) at f0 = 0:25, although othermodified periodograms had smaller spectral variances.The Welch method with 50% overlap using Hannwindow was superior to the one without overlapand the windowed periodogram, but was inferiorto the Bartlett’s procedure and the Welch methodwith 50% overlap using rectangular window. Inorder to investigate their performance in the presenceof spectral leakage, another similar experimentwas conducted for tone frequency which wasuniformly distributed within one bin and the resultsare illustrated in Fig. 3(b). In this test, f0 was auniform random variable whose value is betweenf0¡ 1=(2N) and f0 +1=(2N) in the standard/windowedperiodogram and f0 2 (f0¡ 1=(2M),f0 +1=(2M))in other methods. It can be seen that the relativedetectabilities of the methods were similar to thosein Fig. 3(a), except that the Welch method with 50%overlap using Hann window outperformed the Bartlett

948 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 35, NO. 3 JULY 1999

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Fig. 3. Comparison of ROCs for detecting single sinusoid usingperiodogram, windowed periodogram, Bartlett’s procedure, Welch

methods at SNR=¡9 dB. (a) f0 = 0:25. (b) f0 uniformlydistributed within one bin.

method at approximately PFA > 0:015. ComparingFigs. 3(a) and 3(b), we observe that the detectionperformance of the periodogram, the 50% overlappedWelch methods and the Bartlett’s procedure degradedwhen the frequency was not exactly on the bin, whilethe remaining two methods had almost identicaldetectablities in both cases.Figs. 4(a) and 4(b) compare the ROCs for a

sinusoid that exhibited phase instability with exact-binfrequency and frequency uniformly distributed withinone bin, respectively, at SNR=¡9 dB. This sinusoidwas modeled by a narrowband random processwith normalized autocorrelation function r(m) =0:95jmj cos(2¼f0m). The results of Figs. 4(a) and 4(b)were almost identical and the detection performance indescending order was as follows, the Welch methodwith 50% overlap using Hann window, the 50%overlapped Welch method with rectangular window,the Bartlett method, the 0% overlapped Welch method,the periodogram and the windowed periodogram.Fig. 5 plots the mean square frequency errors of

the periodogram and the Bartlett method togetherwith the CRLB, when the source signal is a puresinusoid with f0 = 0:25. It is observed that thesimulation results agreed with the theoretical valuesof (16) and (17), particularly at the threshold SNRs.

Fig. 4. Comparison of ROCs for detecting non-stationary-phasesingle sinusoid with autocorrelation function

r(m) = 0:95jmj cos(2¼f0m) using periodogram, windowedperiodogram, Bartlett’s procedure, Welch methods at

SNR =¡9 dB. (a) f0 = 0:25. (b) f0 uniformly distributed withinone bin.

Fig. 5. Theoretical and experimental frequency variances ofperiodogram and Bartlett method with four segments for single

sinusoid with f0 = 0:25 at different SNRs.

Moreover, at SNR¸¡5 dB, the periodogram met theCRLB and had an MSFE which was approximatelyone-tenth of the Bartlett method. Fig. 6(a) showsthe simulation results of all six methods in the sametrial. Although the periodogram did not possessthe smallest threshold SNR, it gave the optimumMSFEs for all SNRs. The estimation performance in

SO ET AL.: COMPARISON OF VARIOUS PERIODOGRAMS 949

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Fig. 6. MSFEs for single sinusoid using periodogram, windowedperiodogram, Bartlett’s procedure, Welch method at different

SNRs. (a) f0 = 0:25. (b) f0 = 0:25+1=(2N) in periodograms andf0 = 0:25+1=(2M) in Bartlett and Welch methods.

descending order is as follows: the basic periodogram,the windowed periodogram, the two 50% overlappedWelch methods, the Bartlett’s procedure, and the0% overlapped Welch method. Comparison ofthe methods for frequency located in the middleof 2 bins is depicted in Fig. 6(b). In general, thestandard periodogram still provided the minimummean square error, and the second best was thewindowed periodogram, then the two 50% overlappedWelch methods, while Bartlett method and the 0%overlapped Welch method were the poorest frequencyestimators.

V. CONCLUSIONS

The theoretical performance of detecting areal tone under noisy environment and estimating

the corresponding frequency using conventionalperiodogram and the Bartlett’s procedure are derivedand confirmed by computer simulations. The standardperiodogram provides the best ROC for a pure tonewhile the Welch method is the optimum detector fora sinusoid with nonstationary phase. Furthermore,the periodogram is the best frequency estimator fora noisy sinusoid among its variants.

ACKNOWLEDGMENT

The authors thank Mr. W. K. Ma for performingsome of the simulations.

REFERENCES

[1] Zimmerman, G. A., and Gulkis, S. (1991)Polyphase-discrete Fourier transform spectrum analysisfor the search for extraterrestrial intelligence sky survey.TDA progress report 42-107, Jet Propulsion Laboratory,Pasadena, CA, July—Sept. 1991, 141—154.

[2] Keeler, R. J., and Griffiths, L. J. (1977)Acoustic Doppler extraction by adaptive linear-predictionfiltering.Journal of the Acoustical Society of America, 61, 5 (May1977), 1218—1227.

[3] Report KMV76-2 (1976)Fourier analysis and identification of ELT signals.Prepared for NASA GSFC, Sept. 1976.

[4] Harris, F. J. (1978)On the use of windows for harmonic analysis with thediscrete Fourier transform.Proceedings of the IEEE, 66, 1 (Jan. 1978), 51—83.

[5] Papoulis, A. (1965)Probability, Random Variables, and Stochastic Processes.New York: McGraw-Hill, 1965.

[6] Proakis, J. G. (1989)Digital Communications.New York: McGraw-Hill, 1989.

[7] Rife, D. C., and Boorstyn, R. R. (1974)Single-tone parameter estimation from discrete-timeobservations.IEEE Transactions on Information Theory, 20, 5 (Sept.1974), 591—598.

[8] Helstrom, C. W. (1992)Computing the generalized Marcum Q-function.IEEE Transactions Information Theory, 38, 4 (July 1992),1422—1428.

[9] Kay, S. M. (1993)Fundamental of Statistical Signal Processing: EstimationTheory.Englewood Cliffs, NJ: Prentice-Hall, 1993.

[10] Hung, E. K. L., and Herring, R. W. (1981)Simulation experiments to compare the signal detectionproperties of DFT and MEM spectra.IEEE Transactions on Acoustics, Speech, Signal Processing,ASSP-29, 5 (Oct. 1981), 1084—1089.

950 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 35, NO. 3 JULY 1999

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H. C. So (M’96) was born in Hong Kong, on June 27, 1968. He received theB.Eng. degree in electronic engineering from City Polytechnic of Hong Kong in1990. In 1995, he received the Ph.D. degree in electronic engineering from theChinese University of Hong Kong.From 1990 to 1991, he was an Electronic Engineer at the Research and

Development Division of Everex Systems Engineering Ltd. During 1995—1996,he worked as a post-doctoral Fellow at the Chinese University of Hong Kong.He is currently a Research Assistant Professor in the Department of ElectronicEngineering of City University of Hong Kong. His research interests includeadaptive signal processing, detection and estimation, source localization, andwavelet transform.

Y. T. Chan (SM’80) was born in Hong Kong. He received the B.Sc. and M.Sc.degrees from Queen’s University, Kingston, Ontario, Canada in 1963 and 1967,and the Ph.D. degree from the University of New Brunswick, Fredericton, Canadain 1973, all in electrical engineering.He has worked with Northern Telecom Ltd. and Bell-Northern Research.

Since 1973, he has been at the Royal Military College of Canada, wherehe is presently professor and Head, Department of Electrical and ComputerEngineering. He has also spent two sabbatical terms at the Chinese Universityof Hong Kong, during 1985—1986 and in 1993. His research interests are inwavelet transform, sonar signal processing and passive localization and trackingtechniques and he has served as a consultant on sonar systems. He is author ofthe book “Wavelet Basics”.He was an Associate Editor (1980—1982) of the IEEE Transactions on Signal

Processing and was the Technical Program Chairman of the 1984 InternationalConference on Acoustics, Speech and Signal Processing (ICASSP 84). Hedirected a NATO Advanced Study Institute on Underwater Acoustic DataProcessing in 1988 and was the General Chairman of ICASSP 91 held inToronto, Canada.

Qiang Ma was born in China, on December 29, 1963. He received the Ph.D.degree in 1995 from Loughborough University, UK.From December 1995 to May 1997, he was a Research Associate at Royal

Military College of Canada. He is currently a DSP Designer in WirelessNetworks, Nortel Networks in Ottawa, where he has been engaged in researchand development of digital radio communication system. His research interestsinclude adaptive filtering, detection, and channel estimation.

SO ET AL.: COMPARISON OF VARIOUS PERIODOGRAMS 951

Page 8: Comparison of Various Periodograms for Sinusoid Detection ...hcso/aes_7_99.pdfComparison of Various Periodograms for Sinusoid Detection and Frequency Estimation H. C. SO, Member, IEEE

P. C. Ching (M’80–SM’90) received the B.Eng. (Hons.) and Ph.D. degrees inelectrical engineering and electronics from the University of Liverpool, UK, in1977 and 1981, respectively.From 1981 to 1982 he worked at the School of Electrical Engineering

of the University of Bath, UK, as a research officer. During 1982—1984, hewas a Lecturer in the Department of Electronic Engineering of Hong KongPolytechnic. Since 1984 he has been with the Chinese University of Hong Kong,where he is presently Dean of Engineering and a professor in the Departmentof Electronic Engineering. He has taught courses in digital signal processing,stochastic processes, speech processing, and communication systems. His researchinterests include adaptive filtering, time delay estimation, signal processing forcommunication, speech coding, synthesis and recognition.Dr. Ching was the Chairman of the IEEE Hong Kong Section in 1993—1994,

and is currently a member of the Signal Processing Theory and MethodsTechnical Committee of the IEEE Signal Processing Society and the IEEHong Kong Centre Executive Committee. Since 1997, he has served as anAssociate Editor for the IEEE Transactions on Signal Processing. He has alsobeen involved in organizing many international conferences including the 1997IEEE International Symposium on Circuits and Systems where he was theVice-Chairman. Dr. Ching is a Fellow of IEE and HKIE.

952 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 35, NO. 3 JULY 1999


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