Research ArticleFeature-Based Digital Modulation RecognitionUsing Compressive Sampling
Zhuo Sun Sese Wang and Xuantong Chen
Beijing University of Posts and Telecommunications Beijing 100876 China
Correspondence should be addressed to Zhuo Sun zhuosunbupteducn
Received 2 November 2015 Accepted 8 December 2015
Academic Editor Qilian Liang
Copyright copy 2016 Zhuo Sun et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Compressive sensing theory can be applied to reconstruct the signal with far fewer measurements than what is usually considerednecessary while in many scenarios such as spectrum detection and modulation recognition we only expect to acquire usefulcharacteristics rather than the original signals where selecting the feature with sparsity becomes the main challenge With the aimof digital modulation recognition the paper mainly constructs two features which can be recovered directly from compressivesamples The two features are the spectrum of received data and its nonlinear transformation and the compositional feature ofmultiple high-ordermoments of the received data both of them have desired sparsity required for reconstruction from subsamplesRecognition of multiple frequency shift keying multiple phase shift keying and multiple quadrature amplitude modulation areconsidered in our paper and implemented in a unified procedure Simulation shows that the two identification features can workeffectively in the digital modulation recognition even at a relatively low signal-to-noise ratio
1 Introduction
Constantly increasing volume of data transmitted throughthe mobile communication networks and the needs of usersto increase the data rates lead to rapid development ofmobile communication systems The future fifth generation(5G) wireless communication tends to achieve a remarkablebreakthrough both in data rate and spectral efficiency [1]With demand for large data size and high data rate vastspectrum resources are required urgently However mostspectrum resources below 2G are fixedly occupied by otherindustries although they have not been fully utilized Con-sidering this the purpose of spectrum sensing in mobilecommunication networks is to share spectrum resourceswith other industries without interfering with their normaloperations Moreover spectrum sensing can be also appliedto coordinate the public resources which is a revolutionarychange of the fixed spectrum allocation system [2]
On account of the rearrangement function needed inspectrum sensing and the fact that modulation recognitioncan provide reliable parameters for it digital modulationrecognition is of great importance in the whole system [3]The goal of digital modulation recognition is to identify
the modulation format of an unknown digital communica-tion signal For modulation classification two general classesof classicalmethods exist likelihood-based and feature-basedmethods respectively [4 5] Based on the likelihood functionof the received digital signal the former method makes thedecision by comparing the likelihood ratio with a thresholdIn the feature-based method several features are usuallychosen and the decision is made jointly
However in traditional sensing process two approachesare based on Shannon-Nyquist sampling theorem and thedata scale to deal with can be enormous with a quite wideband especially in the cooperation networks These yearsresearchers have brought compressive sensing (CS) in whichcan solve the problem of high sampling rate caused byShannon-Nyquist sampling theorem It is declared that ifthe signal has a sparse representation in a fixed basis wecan reconstruct the sparse domain of the signal by solvingan optimization algorithm using samplings far fewer thandimensions of the original signal and the original signal canbe obtained by a simple matrix operation [6 7]
In many CS conditions we expect to acquire somesignal characteristics rather than recovering the originalsignal since reconstructing signals allows for lots of extra
Hindawi Publishing CorporationMobile Information SystemsVolume 2016 Article ID 9754162 10 pageshttpdxdoiorg10115520169754162
2 Mobile Information Systems
operations which results in higher complexity in both timeand space Researchers have already carried out much relatedvaluable work in reconstructing signal characteristics basedon CS [8 9] Inspired by these researches we are devoted forfinding identification features which can be used in the digitalmodulation recognition and simultaneously have sparsitymeaning they can be reconstructed directly by compressivesamples
In this paper we propose a feature-based method basedon CS for digital modulation recognition We constructtwo identification features and use compressive samples torecover themdirectly without recovering the original signalsOne identification feature is the spectrum of received dataand its nonlinear transformation which is based on thefeature proposed by [10] and the other is a compositionalfeature of multiple high-order moments of the received dataThese two features can be used to identify various kindsof modulation modes and in this paper we only focuson multiple frequency shift keying (MFSK) multiple phaseshift keying (MPSK) and multiple quadrature amplitudemodulation (MQAM) Simulations would be carried out toindicate that the performance of our method can be effectiveand reliable with lower complexity and better antinoiseproperty than traditional ones [11 12]
The rest of this paper is organized as follows Section 2would present the system model adopted throughout thework both the signal model and compressive sensing modelIn Section 3 we construct two identification features andanalysis sparsity of them In Section 4 we build the linearrelationships between identification features and compressivesamples and give a brief introduction of the recoverymethodThen the whole recognition flowchart will be shown inSection 5 Simulations and analysis are present in Section 6And finally we draw the conclusion in Section 7
2 System Model
21 Signal Model In a spectrum sensing scenario we assumethat a wide band received signal has one of the followingmodulation modes
119910 (119905) =
119899=infin
sum
119899=minusinfin
119860119892 (119905 minus 119899119879) 119903 (119905) + V (119905) (1)
where 119860 represents the amplitude of the received signal 119879represents the symbol period 119892(119905) represents the impulseresponse of pulse shaping low-pass filter in which we chooserectangular pulse in this paper and V(119905) stands for additiveGaussian white noise (AWGN) In the whole process thetiming offset and the carrier offset are both assumed to bezero and the form of 119903(119905) is chosen as follows
where 119891119888and 120573 respectively stand for the carrier frequency
and order of the chosen modulation mode Δ119891 is the carrierspacing and 119886
119894119888
and 119886119894119904
are a set of discrete levels It isworth noting that there is no need of pulse shaping forMFSKwhile in order to unity the form as (1) we regard 119860119892(119905 minus 119899119879)
in MFSK as 1 which would not influence the use of it
22 Compressive Sampling Model According to the theoryof CS the compressive sampling process can be modeledanalytically as
z = Ay (3)
where y is the 119873-length sampling vector of the receivedsignal 119910(119905) at a rate no lower than Nyquist sampling ratez represents the subsampling measurements A is a real-value measurement matrix of size 119872 times 119873 which complieswith the restricted isometry property (RIP) such as Gaus-sian matrix partial Fourier transform matrix or others Inorder to reconstruct the 120574th power of signal in Section 41we adopt a special measurement matrix with the valueof ldquo1rdquo randomly located in each row and other elementsbeing zero Owing to the randomization of row elementsthe matrix satisfies the RIP requirement as well as Gaus-sian matrix Furthermore the two-valued property of thematrix can enormously simplify thematrix operations whichmake it possible to establish linear relationship betweenthe nonlinear reconstruction target with the compressivesamples
To classify the modulated signal based on z we will firstlybuild the linear relationships between z and the identificationfeatures and then reconstruct the features directly withcompressive samples z by solving an optimization algorithmwhich would then be used to do digital modulation recogni-tion
3 Construction of the Identification Features
To achieve the goal of classifyingmodulation types accuratelyidentification features should be chosen with distinguishingdetails for each modulation type firstly Secondly identifi-cation features should have desired sparsity in order to beconstructed by compressive samples based on the theory ofCS According to these two requirements we propose andconstruct the following two identification features
31 Feature 1 Spectrum of the Signalrsquos 120574th Power NonlinearTransformation Referring to [9] we calculate the119873th powerof the received signal 119910(119905) that is
[119910 (119905)]120574
=
119899=infin
sum
119899=minusinfin
119860120574
119892120574
(119905 minus 119899119879) 119903120574
(119905) + V1015840 (119905) (4)
where 120574 = 2119896 (119896 = 0 1 2 ) In the following it will
be shown that the spectrum of specific power order ofsignal level presents the recognizable characters for certainmodulation types and orders V1015840(119905) is the noise caused bynonlinear transformation of V(119905)
Mobile Information Systems 3
Then we calculate the spectrum of [119910(119905)]120574 which is rep-
resented as y120574 with 120574 ranging from 0 to a larger number Wehave the following relationship
y120574 = FS120574 (5)
where F = [119890minus1198952120587119886119887119873
](119886119887)
represents the 119873-point IFFTmatrix
For different kinds of modulation modes the results arequite different which can be used to do the recognition andwe call this feature the spectrum feature below
ForMFSK according to (1) and (2) the spectrum of it canbe calculated as follows
120575 [119891 minus (119891119888+ 119894Δ119891)]
(6)
where 120575(sdot) stands for the impulse function Obviously thereis impulse in the spectrum of MFSK and the number ofthese impulses just corresponds to order 120573 As a contrast thespectrum for MPSK can be presented as follows
SMPSK = int
infin
minusinfin
infin
sum
minusinfin
119860119892 (119905 minus 119899119879) 119890119895(2120587119891119888119905+2120587((119894minus1)120573))
119890minus1198952120587119891119905
119889119905
= sum
119894
119860 sin 119888 (119891) lowast 120575 (119891 minus 119891119888) 1198901198952120587((119894minus1)120573)
= sum
119894
119860 sin 119888 (119891 minus 119891119888) 1198901198952120587((119894minus1)120573)
(7)
where lowast stands for the convolution operation The spectrumof MPSK comes out to be a monotone decreasing sine func-tion with no impulse The calculation process of spectrumof MQAM is similar to MPSK and their consequences arealso similar Figure 1 shows the results of differentmodulationtypes by 120574 = 1 From (a) and (b) we can see that there isapparent impulse forMFSK just aswe analyze in theory quitedifferent from that in (c) and (d) which represent the resultsof MPSK and MQAM respectively That is to say we candistinguish MFSK from others by the spectrum of the signaland the number of pulses indicates the order of MFSK
For MPSK when 120574 lt 120573
S120574MPSK
= int
infin
minusinfin
infin
sum
minusinfin
119860120574
119892120574
(119905 minus 119899119879) 119890119895120574(2120587119891
119888119905+2120587((119894minus1)120573))
119890minus1198952120587119891119905
119889119905
= sum
119894
119860120574 sin 119888 (119891) lowast 120575 (119891 minus 120574119891
From the expression we can see that there is no impulse inthis condition However when 120574 = 120573
S120573MPSK
= int
infin
minusinfin
infin
sum
minusinfin
119860120573
119892120573
(119905 minus 119899119879) 119890119895120573(2120587119891
119888119905+2120587((119894minus1)120573))
119890minus1198952120587119891119905
119889119905
= int
infin
minusinfin
infin
sum
minusinfin
119860120573
119892120573
(119905 minus 119899119879) 119890minus1198952120587119891119905
119889119905
= 119860120573
int
infin
minusinfin
119890minus1198952120587119891119905
119889119905 = 119860120573
120575 (119891)
(9)
For 119892(119905 minus 119899119879) we choose rectangle filter Since119890119895120573(2120587119891
119888119905+2120587((119894minus1)120573))
= 1 suminfin
minusinfin119860120573
119892120573
(119905 minus 119899119879) becomes aconstant and the Fourier transform of it is an impulse Basedon it we compare the results of 120574 = 1 120574 = 2 120574 = 4 and120574 = 8 referring to Figure 2 It can be seen that the impulsefirstly appears when 120574 = 120573 with 120574 varying from small to bigwhich can be used to determine the order of MPSK
As forMQAM owing to similarity of the signal constella-tion the property of MQAM is similar with QPSK shown asFigure 3 It can be easily seen that the impulse firstly appearswhen 120574 = 4 and it has no relationship with the specific orderof it
The sparsity of this feature is in inverse proportion with119873 which represents length of the signal as well as the IFFTsize on condition that there appears the impulse
To sum up the only remaining problem is to distinguishQPSK and MQAM Therefore we construct another featurefor it
32 Feature 2 AComposition ofMultipleHigh-OrderMomentsof the Signal For a digital modulated communication signal119909(119898) = 119909(119905) | (119905 = 119898119879119904 119898 = 1 2 ) the mixed momentsof order 119901 + 119902 are defined as (6) at a zero delay vector [10]
119872119901+119902119902
(119909) = 119864 (119909 (119898)119901
(119909lowast
(119898))119902
) (10)
where the superscript lowast denotes conjugation and 119864(sdot) meanscalculating the mean value
In our system model we intend to acquire 11987221
(119910) and11987240
(119910) as recognition parameters which we call the high-order moment feature
With carrier known symbols in the digital signals can beregarded as points in the signal constellation [13 14] Sincepoints in digital signals of linear modulations are of equalprobabilities when the data size is large enough we can usepoints in the signal constellation to calculate the theoreticalvalues of 119872
21and 119872
40 as Table 1 shows
Referring to Table 111987240
of different modulation formatsare of different theoretical times compared to 119864
We take QPSK and 16QAM as examples According to(11) the theoretical values 120572 of QPSK and 16QAM respec-tively come out to be 1 and 068 If we get the identifica-tion characteristic 120572 of a signal we can then identify themodulation format by comparing 120572 with a suitable decisionthreshold
Since high-order moment is a kind of statistics we needsample several times Then to obtain 119872
Figure 2 Spectrum of 120574th power of signal modulated by MPSK
ℎ = 119896 the value comes out to be zero for the uncorrelationbetween symbols of the signal For R
11991040 119910ℎ119910119896corresponds
to (119873(ℎ minus 1) + 119896)th element of vecyy119879 When ℎ = 119896 therelationship is that 119910
ℎ
2 corresponds to the (119873(ℎ minus 1) + ℎ)thelement of vecyy119879 According to (10) 119864(119910
ℎ
4
) is the desiredvalue 119872
40(119910) so the (119873(ℎ minus 1) + ℎ)th diagonal elements
(ℎ = 1 2 119873) ofR11991040
are equal to11987240
(119910) Other elementsare zero for the uncorrelation between symbols of the signalThe theoretical figures of R
11991021and R
11991040are shown as
Figure 4It is obvious thatR
11991021andR
11991040in Figure 4 are sparse For
R11991021
all diagonal elements are nonzero meaning the sparsity
degree of it is 1119873 For R11991040
the ((ℎ minus 1) times119873+ ℎ)th elementsof vecR
119909119879 are nonzero meaning the sparsity degree of it is
11198733
4 Recovery of the Identification Features withCompressing Samples
In this section we introduce the approaches of recoveringthe two identification features based on CS We firstly buildthe linear relationships between compressive samples and thedefined features and then give a brief introduction of the
Figure 3 Spectrum of 120574th power of signal modulated by MQAM
020
4060
80
020
4060
800
05
1
15
t
Tau
Auto
corr
elat
ion
mat
rix o
f the
sign
al
(a) R11991021
050
100150
050
100150
0
02
04
06
08
The c
onstr
ucte
d m
atrix
Ry40
(b) R11991040
Figure 4 R11991021
and R11991040
of the signal modulated by 16QAM
reconstruction algorithm and the practical selection strategyfor the measurement matrix
41 Linear Relationships between Compressive Samples andthe Identification Features
411 Linear Relationships between Compressive Samples andthe Spectrum Feature It is obviously that the 120574th power of thesignal is a nonlinear transformation To get linear relationshipbetween compressive samples and the spectrumof the signalrsquos120574th power nonlinear transformation we choose the specialmeasurement matrix proposed in Section 2 According tothe nature of this certain-form matrix we can easily get thefollowing relationship based on (3)
z120574 = A1y120574 (15)
and A1is the measurement matrix for feature 1 Then refer-
ring to (5) we obtain
z120574 = A1FS120574= ΘS120574 (16)
whereΘ = A1F is the sensing matrix we needed
412 Linear Relationships between Compressive Samples andthe High-Order Moment Feature For this identification fea-ture the sampling matrix can be chosen as any one as long asit satisfies the restricted isometry property (RIP)
(i) R11991021
according to (3) and the nature of transpose weget the following relationship and A
2stands for the
measurement matrix for R11991021
zz119867 = A2(yy119867)A
2
119867
(17)
Mobile Information Systems 7
Take the average of both sides
119864 (zz119867) = A2sdot 119864 (yy119867) sdot A
2
119867
(18)
We useR11991121
to represent119864(zz119867) simultaneously referto (12) and then get
R11991121
= A2sdot R11991021
sdot A2
119867
(19)
Next we apply the property vecUXV = (V119879 otimes
U)vecX to transform (19) to (20) It is worth notic-ing that A
2
119867
= A2
119879 for A is a real-value matrix
vec R11991121
= A2otimes A2vec R
11991021 = Ψvec R
11991021 (20)
where Ψ = A2otimes A2can be regarded as the sensing
matrix with the scale of 1198722 times 1198732
(ii) R11991040
since the sparsity degree of R11991040
is far fewerthan that of R
11991021 the dimension of signal needed
and scale of measurement can also be very low Werepresent the measurement for R
11991040as A3 while the
only difference of it from A2is the dimension
Similar to (17) there is
zz119879 = A3(yy119879)A
3
119879
(21)
Then according to vecUXV = (V119879 otimes U)vecX wecan transform the two-dimensional relationship intoone-dimensional relationship
vec zz119879 = A3otimes A3vec yy119879 (22)
We can obtain
vec zz119879 vec119879 zz119879
= (A3otimes A3) vec yy119879 vec119879 yy119879 (A
3otimes A3)
(23)
Take the average of both sides
119864 (vec zz119879 vec119879 zz119879)
= (A3otimes A3) 119864 (vec yy119879 vec119879 yy119879) (A
3otimes A3)
(24)
Based on (13) we get the relationship
R11991140
= (A3otimes A3)R11991040
(A3otimes A3) (25)
where R11991140
denotes 119864(veczz119879vec119879zz119879) And thenwe have
vec R11991140
= (A3otimes A3) otimes (A
3otimes A3) vec R
11991040
= Φvec R11991040
(26)
whereΦ = (A3otimesA3)otimes(A3otimesA3) is the sensingmatrix
42 Reconstruction of Identification Features z120574 R11991121
andR11991140
can be calculated by the sampling value z With sensingmatrixes and measurement vectors known the reconstruc-tion of the sparse vectors can be regarded as the signalrecovery problem by solving the NP-hard puzzle as followstaking R
11991021as an example
vec Ry21 = argmin 10038171003817100381710038171003817vec R
11991021100381710038171003817100381710038170
st vec R11991121
= Φ vec R11991121
(27)
This can be transformed into a linear programming problem
minvecRy21
10038171003817100381710038171003817vec R
11991121 minusΦ vec R
1199102110038171003817100381710038171003817
12
2
+ 11989410038171003817100381710038171003817vec R
11991021100381710038171003817100381710038171
(28)
which is called 1198971-norm least square programming problemand is proved to be convex that there exists a unique optimumsolution 119894 gt 0 is a weighting scalar that balances the sparsityof the solution induced by the 1198971-norm term and the datareconstruction error reflected by the 1198972-norm LS term
In Section 41 we havementioned recovering three recog-nition features by using measurement matrixes A
1 A2 and
A3 respectively However practically only using A
1as the
compressive measurement may meet the requirement ofrecovering all of the features The reason is that A
2and
A3differ in the dimension but are both designed with the
constraint of RIP property only From the other aspect theprimary requirement of constructing matrix A
1is also the
RIP condition
5 Modulation Recognition withthe Identification Features
Given a received communication signalmodulated byMFSKMPSK or MQAM we firstly get compressive samples usingmeasurement matrixes present in Section 2 In this processdue to difference of sparsity we have analyzed in Section 3various features may apply various length of the signal andthis can be decided based on actual situations According tothe approaches proposed above the identification featurescan be easily obtained Then we can recognize the modu-lation format effectively referring to the flowchart shown inFigure 5 and specific steps are listed in the following
Step 1 Reconstruct the spectrum feature when 120574 = 1 withcompressive samples If there is impulse in the recoveredspectrum the modulation mode can be identified as MFSKand the number of impulses indicates the order of it How-ever if there is no impulse in the feature the communicationsignal is modulated by MPSK or MQAM and then Step 2should be conducted
Step 2 Reconstruct the spectrum feature when 120574 = 2 4 8
with compressive samples and observe value of 120574 when theimpulse firstly appears If 120574 = 4 when the impulse appearsthe modulation mode can be regarded as QPSK or MQAM
8 Mobile Information Systems
MFSK MPSK and MQAM
Order of MPSK
Order of MFSK
MPSK and MQAM
QPSK and MQAM
QPSK and order ofMQAM
The spectrumNumber of
impulses
No impulse
The spectrum
Impulse appears when
Impulse appears
feature (120574 = 1)
when 120574 ne 4
feature (120574 ne 1)
120574 = 4
(120573 ne 4)
The high-ordermoment feature
Value range of 120572
Figure 5 The process of digital modulation recognition
and then we go to Step 3 However if 120574 = 4when the impulseappears the signal is modulated by MPSK and this value of 120574is the order of it
Step 3 Reconstruct R11991021
and R11991040
of the signal with com-pressive samples get average values of the diagonal as119872
21(119910)
and 11987240
(119910) respectively and then calculate 120572 based on (11)Compare 120572 with the calculated boundary values shown inTable 1 and determine the modulation type
6 Numerical Results
This section presents the simulation results of our feature-based recognition method We firstly generate a stream ofsignals modulated by MPSK MFSK or MQAM All the sig-nals share the same bit rate 1 kbits and the carrier frequency2 kHz and the carrier spacing for MFSK is 025 kHz Forthe two proposed features the observation time is variousbecause data volume needed by the two features are all
120574 = 1
120574 = 2
120574 = 4
120574 = 8
0
01
02
03
04
05
06
07
08
09
1
Cor
rect
det
ectio
n ra
te
84 102 60minus2minus4minus6minus8minus10
SNR
Figure 6 Correct detection rate of impulse in reconstructed feature1
differentThe performance of reconstruction is closely relatedto the signal-to-noise ratio (SNR) which is set as a variable inour simulation and simulations at every SNR are carried outfor 500 times
As mentioned above information we need to capturein feature 1 is whether there are impulses and the numberof them rather than accurate numerical values Thereforewe apply correct detection rate of pulse to evaluate theperformance of reconstruction of spectrum feature whichis shown in Figure 6 We set a decision threshold whichequals two-thirds of the biggest reconstructed value and ifthere is no other value larger than the threshold the biggestvalue would be regarded as the impulse In this scenario thecompressive ratio is set as 03 which means 119872119873 = 03We calculate the detection rate for MFSK signal on 120574 = 1BPSK on 120574 = 2 QPSK and MQAM on 120574 = 4 and 8PSKon 120574 = 8 respectively It is obvious that the detection ratevaries a lot with 120574 The reason is that 120574th power of signal is anonlinear transform meaning that the uniformly distributednoise ismagnified and the degree ofmagnification extends asthe increasing of 120574Therefore detection rate of impulse when120574 = 8 is the worst one
Figure 7 shows the mean square error (MSE) of recon-structed feature 2 with respect to the theoretical ones Thatis
MSE = 119864
1003817100381710038171003817100381710038171003817
vec S120574 minus vec S
120574
1003817100381710038171003817100381710038171003817
2
2
10038171003817100381710038171003817vec S
12057410038171003817100381710038171003817
2
2
(29)
We give the MSE of reconstructed R11991021
and R11991040
respec-tively with the compressive ratio chosen as 03 and 045 FromFigure 7 we can see that the performance of reconstructionof R11991040
is closely related to the compressive ratio while theperformance of reconstruction of R
11991021is relatively perfect
Mobile Information Systems 9
5 6 7 8 9 10 11 12 13 140
005
015
025
03
02
01
035
04
045
SNR (dB)
MSE
Ry21
Ry40
Ry21
Ry40
MN = 03
MN = 03
MN = 045MN = 045
Figure 7 MSE of reconstructed R11991021
and R11991040
with differentcompressive ratio
even at a low compressive ratio Moreover we can easily getthe conclusion that when the compressive ratio is suitable theprecision of feature 2 is high enough as long as the SNR ishigher than 10 dB
Figure 8 shows the correct classification rate of differentmodulation modes at relatively low SNR Difference of thecorrect classification comes from various performance ofreconstruction of features which has been shown in Figures6 and 7 MFSK has high recognition rate larger than 093
even when SNR = minus6 dB For MPSK the correct recognitionrate declines as 120573 increases However for QPSK andMQAMthe performance is quite different and we give the followinganalysis
According to [14] we have the fact that 11987240
of just thesignal and mixture of noise and signal are of the same valueso the main cause of the error comes from 119872
21
As for 11987221 we have the following proof stating the
variation of the value in noisy condition and noiselesscondition To describe this clearly 119872
21(1199100) 11987221
(V) and11987221
(119910) are respectively used to replace 11987221
while beingin the following condition of signal only noise only and themixture of noise and signal
11987221
(1199100) = 119864 (119910
0ℎ1199100ℎ
lowast
)
11987221
(V) = 119864 (VℎVℎ
lowast
)
11987221
(119910) = 119864 ((1199100ℎ
+ Vℎ) (1199100ℎ
+ Vℎ)lowast
)
= 119864 ((1199100ℎ
+ Vℎ) (1199100ℎ
lowast
+ Vℎ
lowast
))
= 119864 (1199100ℎ1199100ℎ
lowast
+ 1199100ℎ
lowastVℎ+ 1199100ℎVℎ
lowast
+ VℎVℎ
lowast
)
= 119864 (1199100ℎ1199100ℎ
lowast
) + 119864 (1199100ℎ
lowastVℎ) + 119864 (119910
0ℎVℎ
lowast
)
+ 119864 (VℎVℎ
lowast
)
(30)
0
01
02
03
04
05
06
07
08
09
1
SNR (dB)C
orre
ct re
cogn
ition
rate
minus10 minus5 0 5 10 15
MFSKBPSK8PSK
QPSK16QAM
Figure 8 Correct classification rate of different modulation modes
V is zero-mean random measure noises with Gaussiandistribution which is independent from 119910 According to thenature of expectation we know that
119864 (1199100ℎ
lowastVℎ) = 119864 (119910
0ℎ1199100ℎ
lowast
) = 0 (31)
Therefore we can obtain the following relationship
11987221
(119910) = 119864 (1199100ℎ1199100ℎ
lowast
) + 119864 (VℎVℎ
lowast
)
= 11987221
(1199100) + 119872
21(V)
(32)
meaning11987221
(119910) is the sum of signal power and noise powerFrom (11) and (27) we can obtain the relationship of the
where 119875V denotes noise power and 1198751199100
denotes signal powerTo sum up 119872
21(119910) is added by the power of noise and
as a consequence the identification parameter 120572 becomes
10 Mobile Information Systems
smaller thus QPSKmay be recognized as 16QAMThereforethe correct recognition rate of 16QAM is much higher thanQPSK when SNR is lower than 10 dB as shown in Figure 8
7 Conclusion
To solve the problem of high sampling rate for digital modu-lation recognition in spectrum sensing we have proposed afeature-based method to identify the modulation formats ofdigital modulated communication signals using compressivesamples and have greatly lowered the sampling rate basedon CS Two features are constructed in our method oneof which is the spectrum of signalrsquos 120574th power nonlineartransformation and the other is a composition of multiplehigh-order moments of the signal both with desired sparsityBy these two features we have applied suitable measurementmatrixes and built linear relationships referring to themThemethod successfully avoids reconstructing original signalsand uses recognition features to classify signals directlydeclining the algorithm complexity effectively Simulationsshow that correct recognition rates are different for differentmodulation types but are all relatively ideal even in noisy sce-narios In actual situations the method can be decomposedaiming at variable demands and for further work we tend toimprove the performance of the whole method continuouslyespecially the noise elimination in the classification of QPSKand MQAM
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by The China National NaturalScience Fund under Grants 61271181 and 61171109 and theJoint Project withChina Southwest Institute of Electronic andTelecommunication Technology
References
[1] H Bogucka P Kryszkiewicz and A Kliks ldquoDynamic spectrumaggregation for future 5G communicationsrdquo IEEE Communica-tions Magazine vol 53 no 5 pp 35ndash43 2015
[2] T Irnich J Kronander and Y Selen ldquoSpectrum sharing sce-narios and resulting technical requirements for 5G systemsrdquoin Proceedings of the IEEE 24th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCWorkshops rsquo13) pp 127ndash132 IEEE London UK September2013
[3] S Fengpan Research on Modulation Classification for Compres-sive Sensing in Cognitive Radio Ningbo University 2013
[4] O A Dobre A Abdi Y Bar-Ness and W Su ldquoSurveyof automatic modulation classification techniques classicalapproaches and new trendsrdquo IET Communications vol 1 no2 pp 137ndash156 2007
[5] F Wang and X Wang ldquoFast and robust modulation classi-fication via Kolmogorov-Smirnov testrdquo IEEE Transactions onCommunications vol 58 no 8 pp 2324ndash2332 2010
[6] E Cands ldquoCompressive samplingrdquo inProceedings of the Interna-tional Congress ofMathematicians vol 3 pp 1433ndash1452MadridSpain 2006
[7] E J Candes and M B Wakin ldquoAn introduction to compressivesamplingrdquo IEEE Signal Processing Magazine vol 25 no 2 pp21ndash30 2008
[8] Z Tian Y Tafesse and B M Sadler ldquoCyclic feature detectionwith sub-nyquist sampling for wideband spectrum sensingrdquoIEEE Journal on Selected Topics in Signal Processing vol 6 no 1pp 58ndash69 2012
[9] L Zhou and H Man ldquoDistributed automatic modulationclassification based on cyclic feature via compressive sensingrdquoin Proceedings of the IEEEMilitary Communications Conference(MILCOM rsquo13) pp 40ndash45 IEEE San Diego Calif USANovember 2013
[10] J Reichert ldquoAutomatic classification of communication signalsusing higher order statisticsrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo92) vol 5 pp 221ndash224 San Francisco Calif USAMarch 1992
[11] V Orlic and M L Dukic ldquoAlgorithm for automatic modula-tion classification in multipath channel based on sixth-ordercumulantsrdquo inProceedings of the 9th International Conference onTelecommunication inModern Satellite Cable and BroadcastingServices (TELSIKS rsquo09) pp 423ndash426 IEEE Nis Serbia October2009
[12] D C Chang and P K Shih ldquoCumulants-based modulationclassification technique in multipath fading channelsrdquo IETCommunications vol 9 no 6 pp 828ndash835 2015
[13] B Wang and L Ge ldquoA novel algorithm for identification ofOFDM signalrdquo in Proceedings of the International Conference onWireless Communications Networking and Mobile Computing(WCNM rsquo05) pp 261ndash264 September 2005
[14] D Grimaldi S Rapuano and G Truglia ldquoAn automatic digitalmodulation classifier for measurement on telecommunicationnetworksrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology ConferencemdashConference Record pp1711ndash1720 Sorrento Italy 2002
operations which results in higher complexity in both timeand space Researchers have already carried out much relatedvaluable work in reconstructing signal characteristics basedon CS [8 9] Inspired by these researches we are devoted forfinding identification features which can be used in the digitalmodulation recognition and simultaneously have sparsitymeaning they can be reconstructed directly by compressivesamples
In this paper we propose a feature-based method basedon CS for digital modulation recognition We constructtwo identification features and use compressive samples torecover themdirectly without recovering the original signalsOne identification feature is the spectrum of received dataand its nonlinear transformation which is based on thefeature proposed by [10] and the other is a compositionalfeature of multiple high-order moments of the received dataThese two features can be used to identify various kindsof modulation modes and in this paper we only focuson multiple frequency shift keying (MFSK) multiple phaseshift keying (MPSK) and multiple quadrature amplitudemodulation (MQAM) Simulations would be carried out toindicate that the performance of our method can be effectiveand reliable with lower complexity and better antinoiseproperty than traditional ones [11 12]
The rest of this paper is organized as follows Section 2would present the system model adopted throughout thework both the signal model and compressive sensing modelIn Section 3 we construct two identification features andanalysis sparsity of them In Section 4 we build the linearrelationships between identification features and compressivesamples and give a brief introduction of the recoverymethodThen the whole recognition flowchart will be shown inSection 5 Simulations and analysis are present in Section 6And finally we draw the conclusion in Section 7
2 System Model
21 Signal Model In a spectrum sensing scenario we assumethat a wide band received signal has one of the followingmodulation modes
119910 (119905) =
119899=infin
sum
119899=minusinfin
119860119892 (119905 minus 119899119879) 119903 (119905) + V (119905) (1)
where 119860 represents the amplitude of the received signal 119879represents the symbol period 119892(119905) represents the impulseresponse of pulse shaping low-pass filter in which we chooserectangular pulse in this paper and V(119905) stands for additiveGaussian white noise (AWGN) In the whole process thetiming offset and the carrier offset are both assumed to bezero and the form of 119903(119905) is chosen as follows
where 119891119888and 120573 respectively stand for the carrier frequency
and order of the chosen modulation mode Δ119891 is the carrierspacing and 119886
119894119888
and 119886119894119904
are a set of discrete levels It isworth noting that there is no need of pulse shaping forMFSKwhile in order to unity the form as (1) we regard 119860119892(119905 minus 119899119879)
in MFSK as 1 which would not influence the use of it
22 Compressive Sampling Model According to the theoryof CS the compressive sampling process can be modeledanalytically as
z = Ay (3)
where y is the 119873-length sampling vector of the receivedsignal 119910(119905) at a rate no lower than Nyquist sampling ratez represents the subsampling measurements A is a real-value measurement matrix of size 119872 times 119873 which complieswith the restricted isometry property (RIP) such as Gaus-sian matrix partial Fourier transform matrix or others Inorder to reconstruct the 120574th power of signal in Section 41we adopt a special measurement matrix with the valueof ldquo1rdquo randomly located in each row and other elementsbeing zero Owing to the randomization of row elementsthe matrix satisfies the RIP requirement as well as Gaus-sian matrix Furthermore the two-valued property of thematrix can enormously simplify thematrix operations whichmake it possible to establish linear relationship betweenthe nonlinear reconstruction target with the compressivesamples
To classify the modulated signal based on z we will firstlybuild the linear relationships between z and the identificationfeatures and then reconstruct the features directly withcompressive samples z by solving an optimization algorithmwhich would then be used to do digital modulation recogni-tion
3 Construction of the Identification Features
To achieve the goal of classifyingmodulation types accuratelyidentification features should be chosen with distinguishingdetails for each modulation type firstly Secondly identifi-cation features should have desired sparsity in order to beconstructed by compressive samples based on the theory ofCS According to these two requirements we propose andconstruct the following two identification features
31 Feature 1 Spectrum of the Signalrsquos 120574th Power NonlinearTransformation Referring to [9] we calculate the119873th powerof the received signal 119910(119905) that is
[119910 (119905)]120574
=
119899=infin
sum
119899=minusinfin
119860120574
119892120574
(119905 minus 119899119879) 119903120574
(119905) + V1015840 (119905) (4)
where 120574 = 2119896 (119896 = 0 1 2 ) In the following it will
be shown that the spectrum of specific power order ofsignal level presents the recognizable characters for certainmodulation types and orders V1015840(119905) is the noise caused bynonlinear transformation of V(119905)
Mobile Information Systems 3
Then we calculate the spectrum of [119910(119905)]120574 which is rep-
resented as y120574 with 120574 ranging from 0 to a larger number Wehave the following relationship
y120574 = FS120574 (5)
where F = [119890minus1198952120587119886119887119873
](119886119887)
represents the 119873-point IFFTmatrix
For different kinds of modulation modes the results arequite different which can be used to do the recognition andwe call this feature the spectrum feature below
ForMFSK according to (1) and (2) the spectrum of it canbe calculated as follows
120575 [119891 minus (119891119888+ 119894Δ119891)]
(6)
where 120575(sdot) stands for the impulse function Obviously thereis impulse in the spectrum of MFSK and the number ofthese impulses just corresponds to order 120573 As a contrast thespectrum for MPSK can be presented as follows
SMPSK = int
infin
minusinfin
infin
sum
minusinfin
119860119892 (119905 minus 119899119879) 119890119895(2120587119891119888119905+2120587((119894minus1)120573))
119890minus1198952120587119891119905
119889119905
= sum
119894
119860 sin 119888 (119891) lowast 120575 (119891 minus 119891119888) 1198901198952120587((119894minus1)120573)
= sum
119894
119860 sin 119888 (119891 minus 119891119888) 1198901198952120587((119894minus1)120573)
(7)
where lowast stands for the convolution operation The spectrumof MPSK comes out to be a monotone decreasing sine func-tion with no impulse The calculation process of spectrumof MQAM is similar to MPSK and their consequences arealso similar Figure 1 shows the results of differentmodulationtypes by 120574 = 1 From (a) and (b) we can see that there isapparent impulse forMFSK just aswe analyze in theory quitedifferent from that in (c) and (d) which represent the resultsof MPSK and MQAM respectively That is to say we candistinguish MFSK from others by the spectrum of the signaland the number of pulses indicates the order of MFSK
For MPSK when 120574 lt 120573
S120574MPSK
= int
infin
minusinfin
infin
sum
minusinfin
119860120574
119892120574
(119905 minus 119899119879) 119890119895120574(2120587119891
119888119905+2120587((119894minus1)120573))
119890minus1198952120587119891119905
119889119905
= sum
119894
119860120574 sin 119888 (119891) lowast 120575 (119891 minus 120574119891
From the expression we can see that there is no impulse inthis condition However when 120574 = 120573
S120573MPSK
= int
infin
minusinfin
infin
sum
minusinfin
119860120573
119892120573
(119905 minus 119899119879) 119890119895120573(2120587119891
119888119905+2120587((119894minus1)120573))
119890minus1198952120587119891119905
119889119905
= int
infin
minusinfin
infin
sum
minusinfin
119860120573
119892120573
(119905 minus 119899119879) 119890minus1198952120587119891119905
119889119905
= 119860120573
int
infin
minusinfin
119890minus1198952120587119891119905
119889119905 = 119860120573
120575 (119891)
(9)
For 119892(119905 minus 119899119879) we choose rectangle filter Since119890119895120573(2120587119891
119888119905+2120587((119894minus1)120573))
= 1 suminfin
minusinfin119860120573
119892120573
(119905 minus 119899119879) becomes aconstant and the Fourier transform of it is an impulse Basedon it we compare the results of 120574 = 1 120574 = 2 120574 = 4 and120574 = 8 referring to Figure 2 It can be seen that the impulsefirstly appears when 120574 = 120573 with 120574 varying from small to bigwhich can be used to determine the order of MPSK
As forMQAM owing to similarity of the signal constella-tion the property of MQAM is similar with QPSK shown asFigure 3 It can be easily seen that the impulse firstly appearswhen 120574 = 4 and it has no relationship with the specific orderof it
The sparsity of this feature is in inverse proportion with119873 which represents length of the signal as well as the IFFTsize on condition that there appears the impulse
To sum up the only remaining problem is to distinguishQPSK and MQAM Therefore we construct another featurefor it
32 Feature 2 AComposition ofMultipleHigh-OrderMomentsof the Signal For a digital modulated communication signal119909(119898) = 119909(119905) | (119905 = 119898119879119904 119898 = 1 2 ) the mixed momentsof order 119901 + 119902 are defined as (6) at a zero delay vector [10]
119872119901+119902119902
(119909) = 119864 (119909 (119898)119901
(119909lowast
(119898))119902
) (10)
where the superscript lowast denotes conjugation and 119864(sdot) meanscalculating the mean value
In our system model we intend to acquire 11987221
(119910) and11987240
(119910) as recognition parameters which we call the high-order moment feature
With carrier known symbols in the digital signals can beregarded as points in the signal constellation [13 14] Sincepoints in digital signals of linear modulations are of equalprobabilities when the data size is large enough we can usepoints in the signal constellation to calculate the theoreticalvalues of 119872
21and 119872
40 as Table 1 shows
Referring to Table 111987240
of different modulation formatsare of different theoretical times compared to 119864
We take QPSK and 16QAM as examples According to(11) the theoretical values 120572 of QPSK and 16QAM respec-tively come out to be 1 and 068 If we get the identifica-tion characteristic 120572 of a signal we can then identify themodulation format by comparing 120572 with a suitable decisionthreshold
Since high-order moment is a kind of statistics we needsample several times Then to obtain 119872
Figure 2 Spectrum of 120574th power of signal modulated by MPSK
ℎ = 119896 the value comes out to be zero for the uncorrelationbetween symbols of the signal For R
11991040 119910ℎ119910119896corresponds
to (119873(ℎ minus 1) + 119896)th element of vecyy119879 When ℎ = 119896 therelationship is that 119910
ℎ
2 corresponds to the (119873(ℎ minus 1) + ℎ)thelement of vecyy119879 According to (10) 119864(119910
ℎ
4
) is the desiredvalue 119872
40(119910) so the (119873(ℎ minus 1) + ℎ)th diagonal elements
(ℎ = 1 2 119873) ofR11991040
are equal to11987240
(119910) Other elementsare zero for the uncorrelation between symbols of the signalThe theoretical figures of R
11991021and R
11991040are shown as
Figure 4It is obvious thatR
11991021andR
11991040in Figure 4 are sparse For
R11991021
all diagonal elements are nonzero meaning the sparsity
degree of it is 1119873 For R11991040
the ((ℎ minus 1) times119873+ ℎ)th elementsof vecR
119909119879 are nonzero meaning the sparsity degree of it is
11198733
4 Recovery of the Identification Features withCompressing Samples
In this section we introduce the approaches of recoveringthe two identification features based on CS We firstly buildthe linear relationships between compressive samples and thedefined features and then give a brief introduction of the
Figure 3 Spectrum of 120574th power of signal modulated by MQAM
020
4060
80
020
4060
800
05
1
15
t
Tau
Auto
corr
elat
ion
mat
rix o
f the
sign
al
(a) R11991021
050
100150
050
100150
0
02
04
06
08
The c
onstr
ucte
d m
atrix
Ry40
(b) R11991040
Figure 4 R11991021
and R11991040
of the signal modulated by 16QAM
reconstruction algorithm and the practical selection strategyfor the measurement matrix
41 Linear Relationships between Compressive Samples andthe Identification Features
411 Linear Relationships between Compressive Samples andthe Spectrum Feature It is obviously that the 120574th power of thesignal is a nonlinear transformation To get linear relationshipbetween compressive samples and the spectrumof the signalrsquos120574th power nonlinear transformation we choose the specialmeasurement matrix proposed in Section 2 According tothe nature of this certain-form matrix we can easily get thefollowing relationship based on (3)
z120574 = A1y120574 (15)
and A1is the measurement matrix for feature 1 Then refer-
ring to (5) we obtain
z120574 = A1FS120574= ΘS120574 (16)
whereΘ = A1F is the sensing matrix we needed
412 Linear Relationships between Compressive Samples andthe High-Order Moment Feature For this identification fea-ture the sampling matrix can be chosen as any one as long asit satisfies the restricted isometry property (RIP)
(i) R11991021
according to (3) and the nature of transpose weget the following relationship and A
2stands for the
measurement matrix for R11991021
zz119867 = A2(yy119867)A
2
119867
(17)
Mobile Information Systems 7
Take the average of both sides
119864 (zz119867) = A2sdot 119864 (yy119867) sdot A
2
119867
(18)
We useR11991121
to represent119864(zz119867) simultaneously referto (12) and then get
R11991121
= A2sdot R11991021
sdot A2
119867
(19)
Next we apply the property vecUXV = (V119879 otimes
U)vecX to transform (19) to (20) It is worth notic-ing that A
2
119867
= A2
119879 for A is a real-value matrix
vec R11991121
= A2otimes A2vec R
11991021 = Ψvec R
11991021 (20)
where Ψ = A2otimes A2can be regarded as the sensing
matrix with the scale of 1198722 times 1198732
(ii) R11991040
since the sparsity degree of R11991040
is far fewerthan that of R
11991021 the dimension of signal needed
and scale of measurement can also be very low Werepresent the measurement for R
11991040as A3 while the
only difference of it from A2is the dimension
Similar to (17) there is
zz119879 = A3(yy119879)A
3
119879
(21)
Then according to vecUXV = (V119879 otimes U)vecX wecan transform the two-dimensional relationship intoone-dimensional relationship
vec zz119879 = A3otimes A3vec yy119879 (22)
We can obtain
vec zz119879 vec119879 zz119879
= (A3otimes A3) vec yy119879 vec119879 yy119879 (A
3otimes A3)
(23)
Take the average of both sides
119864 (vec zz119879 vec119879 zz119879)
= (A3otimes A3) 119864 (vec yy119879 vec119879 yy119879) (A
3otimes A3)
(24)
Based on (13) we get the relationship
R11991140
= (A3otimes A3)R11991040
(A3otimes A3) (25)
where R11991140
denotes 119864(veczz119879vec119879zz119879) And thenwe have
vec R11991140
= (A3otimes A3) otimes (A
3otimes A3) vec R
11991040
= Φvec R11991040
(26)
whereΦ = (A3otimesA3)otimes(A3otimesA3) is the sensingmatrix
42 Reconstruction of Identification Features z120574 R11991121
andR11991140
can be calculated by the sampling value z With sensingmatrixes and measurement vectors known the reconstruc-tion of the sparse vectors can be regarded as the signalrecovery problem by solving the NP-hard puzzle as followstaking R
11991021as an example
vec Ry21 = argmin 10038171003817100381710038171003817vec R
11991021100381710038171003817100381710038170
st vec R11991121
= Φ vec R11991121
(27)
This can be transformed into a linear programming problem
minvecRy21
10038171003817100381710038171003817vec R
11991121 minusΦ vec R
1199102110038171003817100381710038171003817
12
2
+ 11989410038171003817100381710038171003817vec R
11991021100381710038171003817100381710038171
(28)
which is called 1198971-norm least square programming problemand is proved to be convex that there exists a unique optimumsolution 119894 gt 0 is a weighting scalar that balances the sparsityof the solution induced by the 1198971-norm term and the datareconstruction error reflected by the 1198972-norm LS term
In Section 41 we havementioned recovering three recog-nition features by using measurement matrixes A
1 A2 and
A3 respectively However practically only using A
1as the
compressive measurement may meet the requirement ofrecovering all of the features The reason is that A
2and
A3differ in the dimension but are both designed with the
constraint of RIP property only From the other aspect theprimary requirement of constructing matrix A
1is also the
RIP condition
5 Modulation Recognition withthe Identification Features
Given a received communication signalmodulated byMFSKMPSK or MQAM we firstly get compressive samples usingmeasurement matrixes present in Section 2 In this processdue to difference of sparsity we have analyzed in Section 3various features may apply various length of the signal andthis can be decided based on actual situations According tothe approaches proposed above the identification featurescan be easily obtained Then we can recognize the modu-lation format effectively referring to the flowchart shown inFigure 5 and specific steps are listed in the following
Step 1 Reconstruct the spectrum feature when 120574 = 1 withcompressive samples If there is impulse in the recoveredspectrum the modulation mode can be identified as MFSKand the number of impulses indicates the order of it How-ever if there is no impulse in the feature the communicationsignal is modulated by MPSK or MQAM and then Step 2should be conducted
Step 2 Reconstruct the spectrum feature when 120574 = 2 4 8
with compressive samples and observe value of 120574 when theimpulse firstly appears If 120574 = 4 when the impulse appearsthe modulation mode can be regarded as QPSK or MQAM
8 Mobile Information Systems
MFSK MPSK and MQAM
Order of MPSK
Order of MFSK
MPSK and MQAM
QPSK and MQAM
QPSK and order ofMQAM
The spectrumNumber of
impulses
No impulse
The spectrum
Impulse appears when
Impulse appears
feature (120574 = 1)
when 120574 ne 4
feature (120574 ne 1)
120574 = 4
(120573 ne 4)
The high-ordermoment feature
Value range of 120572
Figure 5 The process of digital modulation recognition
and then we go to Step 3 However if 120574 = 4when the impulseappears the signal is modulated by MPSK and this value of 120574is the order of it
Step 3 Reconstruct R11991021
and R11991040
of the signal with com-pressive samples get average values of the diagonal as119872
21(119910)
and 11987240
(119910) respectively and then calculate 120572 based on (11)Compare 120572 with the calculated boundary values shown inTable 1 and determine the modulation type
6 Numerical Results
This section presents the simulation results of our feature-based recognition method We firstly generate a stream ofsignals modulated by MPSK MFSK or MQAM All the sig-nals share the same bit rate 1 kbits and the carrier frequency2 kHz and the carrier spacing for MFSK is 025 kHz Forthe two proposed features the observation time is variousbecause data volume needed by the two features are all
120574 = 1
120574 = 2
120574 = 4
120574 = 8
0
01
02
03
04
05
06
07
08
09
1
Cor
rect
det
ectio
n ra
te
84 102 60minus2minus4minus6minus8minus10
SNR
Figure 6 Correct detection rate of impulse in reconstructed feature1
differentThe performance of reconstruction is closely relatedto the signal-to-noise ratio (SNR) which is set as a variable inour simulation and simulations at every SNR are carried outfor 500 times
As mentioned above information we need to capturein feature 1 is whether there are impulses and the numberof them rather than accurate numerical values Thereforewe apply correct detection rate of pulse to evaluate theperformance of reconstruction of spectrum feature whichis shown in Figure 6 We set a decision threshold whichequals two-thirds of the biggest reconstructed value and ifthere is no other value larger than the threshold the biggestvalue would be regarded as the impulse In this scenario thecompressive ratio is set as 03 which means 119872119873 = 03We calculate the detection rate for MFSK signal on 120574 = 1BPSK on 120574 = 2 QPSK and MQAM on 120574 = 4 and 8PSKon 120574 = 8 respectively It is obvious that the detection ratevaries a lot with 120574 The reason is that 120574th power of signal is anonlinear transform meaning that the uniformly distributednoise ismagnified and the degree ofmagnification extends asthe increasing of 120574Therefore detection rate of impulse when120574 = 8 is the worst one
Figure 7 shows the mean square error (MSE) of recon-structed feature 2 with respect to the theoretical ones Thatis
MSE = 119864
1003817100381710038171003817100381710038171003817
vec S120574 minus vec S
120574
1003817100381710038171003817100381710038171003817
2
2
10038171003817100381710038171003817vec S
12057410038171003817100381710038171003817
2
2
(29)
We give the MSE of reconstructed R11991021
and R11991040
respec-tively with the compressive ratio chosen as 03 and 045 FromFigure 7 we can see that the performance of reconstructionof R11991040
is closely related to the compressive ratio while theperformance of reconstruction of R
11991021is relatively perfect
Mobile Information Systems 9
5 6 7 8 9 10 11 12 13 140
005
015
025
03
02
01
035
04
045
SNR (dB)
MSE
Ry21
Ry40
Ry21
Ry40
MN = 03
MN = 03
MN = 045MN = 045
Figure 7 MSE of reconstructed R11991021
and R11991040
with differentcompressive ratio
even at a low compressive ratio Moreover we can easily getthe conclusion that when the compressive ratio is suitable theprecision of feature 2 is high enough as long as the SNR ishigher than 10 dB
Figure 8 shows the correct classification rate of differentmodulation modes at relatively low SNR Difference of thecorrect classification comes from various performance ofreconstruction of features which has been shown in Figures6 and 7 MFSK has high recognition rate larger than 093
even when SNR = minus6 dB For MPSK the correct recognitionrate declines as 120573 increases However for QPSK andMQAMthe performance is quite different and we give the followinganalysis
According to [14] we have the fact that 11987240
of just thesignal and mixture of noise and signal are of the same valueso the main cause of the error comes from 119872
21
As for 11987221 we have the following proof stating the
variation of the value in noisy condition and noiselesscondition To describe this clearly 119872
21(1199100) 11987221
(V) and11987221
(119910) are respectively used to replace 11987221
while beingin the following condition of signal only noise only and themixture of noise and signal
11987221
(1199100) = 119864 (119910
0ℎ1199100ℎ
lowast
)
11987221
(V) = 119864 (VℎVℎ
lowast
)
11987221
(119910) = 119864 ((1199100ℎ
+ Vℎ) (1199100ℎ
+ Vℎ)lowast
)
= 119864 ((1199100ℎ
+ Vℎ) (1199100ℎ
lowast
+ Vℎ
lowast
))
= 119864 (1199100ℎ1199100ℎ
lowast
+ 1199100ℎ
lowastVℎ+ 1199100ℎVℎ
lowast
+ VℎVℎ
lowast
)
= 119864 (1199100ℎ1199100ℎ
lowast
) + 119864 (1199100ℎ
lowastVℎ) + 119864 (119910
0ℎVℎ
lowast
)
+ 119864 (VℎVℎ
lowast
)
(30)
0
01
02
03
04
05
06
07
08
09
1
SNR (dB)C
orre
ct re
cogn
ition
rate
minus10 minus5 0 5 10 15
MFSKBPSK8PSK
QPSK16QAM
Figure 8 Correct classification rate of different modulation modes
V is zero-mean random measure noises with Gaussiandistribution which is independent from 119910 According to thenature of expectation we know that
119864 (1199100ℎ
lowastVℎ) = 119864 (119910
0ℎ1199100ℎ
lowast
) = 0 (31)
Therefore we can obtain the following relationship
11987221
(119910) = 119864 (1199100ℎ1199100ℎ
lowast
) + 119864 (VℎVℎ
lowast
)
= 11987221
(1199100) + 119872
21(V)
(32)
meaning11987221
(119910) is the sum of signal power and noise powerFrom (11) and (27) we can obtain the relationship of the
where 119875V denotes noise power and 1198751199100
denotes signal powerTo sum up 119872
21(119910) is added by the power of noise and
as a consequence the identification parameter 120572 becomes
10 Mobile Information Systems
smaller thus QPSKmay be recognized as 16QAMThereforethe correct recognition rate of 16QAM is much higher thanQPSK when SNR is lower than 10 dB as shown in Figure 8
7 Conclusion
To solve the problem of high sampling rate for digital modu-lation recognition in spectrum sensing we have proposed afeature-based method to identify the modulation formats ofdigital modulated communication signals using compressivesamples and have greatly lowered the sampling rate basedon CS Two features are constructed in our method oneof which is the spectrum of signalrsquos 120574th power nonlineartransformation and the other is a composition of multiplehigh-order moments of the signal both with desired sparsityBy these two features we have applied suitable measurementmatrixes and built linear relationships referring to themThemethod successfully avoids reconstructing original signalsand uses recognition features to classify signals directlydeclining the algorithm complexity effectively Simulationsshow that correct recognition rates are different for differentmodulation types but are all relatively ideal even in noisy sce-narios In actual situations the method can be decomposedaiming at variable demands and for further work we tend toimprove the performance of the whole method continuouslyespecially the noise elimination in the classification of QPSKand MQAM
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by The China National NaturalScience Fund under Grants 61271181 and 61171109 and theJoint Project withChina Southwest Institute of Electronic andTelecommunication Technology
References
[1] H Bogucka P Kryszkiewicz and A Kliks ldquoDynamic spectrumaggregation for future 5G communicationsrdquo IEEE Communica-tions Magazine vol 53 no 5 pp 35ndash43 2015
[2] T Irnich J Kronander and Y Selen ldquoSpectrum sharing sce-narios and resulting technical requirements for 5G systemsrdquoin Proceedings of the IEEE 24th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCWorkshops rsquo13) pp 127ndash132 IEEE London UK September2013
[3] S Fengpan Research on Modulation Classification for Compres-sive Sensing in Cognitive Radio Ningbo University 2013
[4] O A Dobre A Abdi Y Bar-Ness and W Su ldquoSurveyof automatic modulation classification techniques classicalapproaches and new trendsrdquo IET Communications vol 1 no2 pp 137ndash156 2007
[5] F Wang and X Wang ldquoFast and robust modulation classi-fication via Kolmogorov-Smirnov testrdquo IEEE Transactions onCommunications vol 58 no 8 pp 2324ndash2332 2010
[6] E Cands ldquoCompressive samplingrdquo inProceedings of the Interna-tional Congress ofMathematicians vol 3 pp 1433ndash1452MadridSpain 2006
[7] E J Candes and M B Wakin ldquoAn introduction to compressivesamplingrdquo IEEE Signal Processing Magazine vol 25 no 2 pp21ndash30 2008
[8] Z Tian Y Tafesse and B M Sadler ldquoCyclic feature detectionwith sub-nyquist sampling for wideband spectrum sensingrdquoIEEE Journal on Selected Topics in Signal Processing vol 6 no 1pp 58ndash69 2012
[9] L Zhou and H Man ldquoDistributed automatic modulationclassification based on cyclic feature via compressive sensingrdquoin Proceedings of the IEEEMilitary Communications Conference(MILCOM rsquo13) pp 40ndash45 IEEE San Diego Calif USANovember 2013
[10] J Reichert ldquoAutomatic classification of communication signalsusing higher order statisticsrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo92) vol 5 pp 221ndash224 San Francisco Calif USAMarch 1992
[11] V Orlic and M L Dukic ldquoAlgorithm for automatic modula-tion classification in multipath channel based on sixth-ordercumulantsrdquo inProceedings of the 9th International Conference onTelecommunication inModern Satellite Cable and BroadcastingServices (TELSIKS rsquo09) pp 423ndash426 IEEE Nis Serbia October2009
[12] D C Chang and P K Shih ldquoCumulants-based modulationclassification technique in multipath fading channelsrdquo IETCommunications vol 9 no 6 pp 828ndash835 2015
[13] B Wang and L Ge ldquoA novel algorithm for identification ofOFDM signalrdquo in Proceedings of the International Conference onWireless Communications Networking and Mobile Computing(WCNM rsquo05) pp 261ndash264 September 2005
[14] D Grimaldi S Rapuano and G Truglia ldquoAn automatic digitalmodulation classifier for measurement on telecommunicationnetworksrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology ConferencemdashConference Record pp1711ndash1720 Sorrento Italy 2002
Then we calculate the spectrum of [119910(119905)]120574 which is rep-
resented as y120574 with 120574 ranging from 0 to a larger number Wehave the following relationship
y120574 = FS120574 (5)
where F = [119890minus1198952120587119886119887119873
](119886119887)
represents the 119873-point IFFTmatrix
For different kinds of modulation modes the results arequite different which can be used to do the recognition andwe call this feature the spectrum feature below
ForMFSK according to (1) and (2) the spectrum of it canbe calculated as follows
120575 [119891 minus (119891119888+ 119894Δ119891)]
(6)
where 120575(sdot) stands for the impulse function Obviously thereis impulse in the spectrum of MFSK and the number ofthese impulses just corresponds to order 120573 As a contrast thespectrum for MPSK can be presented as follows
SMPSK = int
infin
minusinfin
infin
sum
minusinfin
119860119892 (119905 minus 119899119879) 119890119895(2120587119891119888119905+2120587((119894minus1)120573))
119890minus1198952120587119891119905
119889119905
= sum
119894
119860 sin 119888 (119891) lowast 120575 (119891 minus 119891119888) 1198901198952120587((119894minus1)120573)
= sum
119894
119860 sin 119888 (119891 minus 119891119888) 1198901198952120587((119894minus1)120573)
(7)
where lowast stands for the convolution operation The spectrumof MPSK comes out to be a monotone decreasing sine func-tion with no impulse The calculation process of spectrumof MQAM is similar to MPSK and their consequences arealso similar Figure 1 shows the results of differentmodulationtypes by 120574 = 1 From (a) and (b) we can see that there isapparent impulse forMFSK just aswe analyze in theory quitedifferent from that in (c) and (d) which represent the resultsof MPSK and MQAM respectively That is to say we candistinguish MFSK from others by the spectrum of the signaland the number of pulses indicates the order of MFSK
For MPSK when 120574 lt 120573
S120574MPSK
= int
infin
minusinfin
infin
sum
minusinfin
119860120574
119892120574
(119905 minus 119899119879) 119890119895120574(2120587119891
119888119905+2120587((119894minus1)120573))
119890minus1198952120587119891119905
119889119905
= sum
119894
119860120574 sin 119888 (119891) lowast 120575 (119891 minus 120574119891
From the expression we can see that there is no impulse inthis condition However when 120574 = 120573
S120573MPSK
= int
infin
minusinfin
infin
sum
minusinfin
119860120573
119892120573
(119905 minus 119899119879) 119890119895120573(2120587119891
119888119905+2120587((119894minus1)120573))
119890minus1198952120587119891119905
119889119905
= int
infin
minusinfin
infin
sum
minusinfin
119860120573
119892120573
(119905 minus 119899119879) 119890minus1198952120587119891119905
119889119905
= 119860120573
int
infin
minusinfin
119890minus1198952120587119891119905
119889119905 = 119860120573
120575 (119891)
(9)
For 119892(119905 minus 119899119879) we choose rectangle filter Since119890119895120573(2120587119891
119888119905+2120587((119894minus1)120573))
= 1 suminfin
minusinfin119860120573
119892120573
(119905 minus 119899119879) becomes aconstant and the Fourier transform of it is an impulse Basedon it we compare the results of 120574 = 1 120574 = 2 120574 = 4 and120574 = 8 referring to Figure 2 It can be seen that the impulsefirstly appears when 120574 = 120573 with 120574 varying from small to bigwhich can be used to determine the order of MPSK
As forMQAM owing to similarity of the signal constella-tion the property of MQAM is similar with QPSK shown asFigure 3 It can be easily seen that the impulse firstly appearswhen 120574 = 4 and it has no relationship with the specific orderof it
The sparsity of this feature is in inverse proportion with119873 which represents length of the signal as well as the IFFTsize on condition that there appears the impulse
To sum up the only remaining problem is to distinguishQPSK and MQAM Therefore we construct another featurefor it
32 Feature 2 AComposition ofMultipleHigh-OrderMomentsof the Signal For a digital modulated communication signal119909(119898) = 119909(119905) | (119905 = 119898119879119904 119898 = 1 2 ) the mixed momentsof order 119901 + 119902 are defined as (6) at a zero delay vector [10]
119872119901+119902119902
(119909) = 119864 (119909 (119898)119901
(119909lowast
(119898))119902
) (10)
where the superscript lowast denotes conjugation and 119864(sdot) meanscalculating the mean value
In our system model we intend to acquire 11987221
(119910) and11987240
(119910) as recognition parameters which we call the high-order moment feature
With carrier known symbols in the digital signals can beregarded as points in the signal constellation [13 14] Sincepoints in digital signals of linear modulations are of equalprobabilities when the data size is large enough we can usepoints in the signal constellation to calculate the theoreticalvalues of 119872
21and 119872
40 as Table 1 shows
Referring to Table 111987240
of different modulation formatsare of different theoretical times compared to 119864
We take QPSK and 16QAM as examples According to(11) the theoretical values 120572 of QPSK and 16QAM respec-tively come out to be 1 and 068 If we get the identifica-tion characteristic 120572 of a signal we can then identify themodulation format by comparing 120572 with a suitable decisionthreshold
Since high-order moment is a kind of statistics we needsample several times Then to obtain 119872
Figure 2 Spectrum of 120574th power of signal modulated by MPSK
ℎ = 119896 the value comes out to be zero for the uncorrelationbetween symbols of the signal For R
11991040 119910ℎ119910119896corresponds
to (119873(ℎ minus 1) + 119896)th element of vecyy119879 When ℎ = 119896 therelationship is that 119910
ℎ
2 corresponds to the (119873(ℎ minus 1) + ℎ)thelement of vecyy119879 According to (10) 119864(119910
ℎ
4
) is the desiredvalue 119872
40(119910) so the (119873(ℎ minus 1) + ℎ)th diagonal elements
(ℎ = 1 2 119873) ofR11991040
are equal to11987240
(119910) Other elementsare zero for the uncorrelation between symbols of the signalThe theoretical figures of R
11991021and R
11991040are shown as
Figure 4It is obvious thatR
11991021andR
11991040in Figure 4 are sparse For
R11991021
all diagonal elements are nonzero meaning the sparsity
degree of it is 1119873 For R11991040
the ((ℎ minus 1) times119873+ ℎ)th elementsof vecR
119909119879 are nonzero meaning the sparsity degree of it is
11198733
4 Recovery of the Identification Features withCompressing Samples
In this section we introduce the approaches of recoveringthe two identification features based on CS We firstly buildthe linear relationships between compressive samples and thedefined features and then give a brief introduction of the
Figure 3 Spectrum of 120574th power of signal modulated by MQAM
020
4060
80
020
4060
800
05
1
15
t
Tau
Auto
corr
elat
ion
mat
rix o
f the
sign
al
(a) R11991021
050
100150
050
100150
0
02
04
06
08
The c
onstr
ucte
d m
atrix
Ry40
(b) R11991040
Figure 4 R11991021
and R11991040
of the signal modulated by 16QAM
reconstruction algorithm and the practical selection strategyfor the measurement matrix
41 Linear Relationships between Compressive Samples andthe Identification Features
411 Linear Relationships between Compressive Samples andthe Spectrum Feature It is obviously that the 120574th power of thesignal is a nonlinear transformation To get linear relationshipbetween compressive samples and the spectrumof the signalrsquos120574th power nonlinear transformation we choose the specialmeasurement matrix proposed in Section 2 According tothe nature of this certain-form matrix we can easily get thefollowing relationship based on (3)
z120574 = A1y120574 (15)
and A1is the measurement matrix for feature 1 Then refer-
ring to (5) we obtain
z120574 = A1FS120574= ΘS120574 (16)
whereΘ = A1F is the sensing matrix we needed
412 Linear Relationships between Compressive Samples andthe High-Order Moment Feature For this identification fea-ture the sampling matrix can be chosen as any one as long asit satisfies the restricted isometry property (RIP)
(i) R11991021
according to (3) and the nature of transpose weget the following relationship and A
2stands for the
measurement matrix for R11991021
zz119867 = A2(yy119867)A
2
119867
(17)
Mobile Information Systems 7
Take the average of both sides
119864 (zz119867) = A2sdot 119864 (yy119867) sdot A
2
119867
(18)
We useR11991121
to represent119864(zz119867) simultaneously referto (12) and then get
R11991121
= A2sdot R11991021
sdot A2
119867
(19)
Next we apply the property vecUXV = (V119879 otimes
U)vecX to transform (19) to (20) It is worth notic-ing that A
2
119867
= A2
119879 for A is a real-value matrix
vec R11991121
= A2otimes A2vec R
11991021 = Ψvec R
11991021 (20)
where Ψ = A2otimes A2can be regarded as the sensing
matrix with the scale of 1198722 times 1198732
(ii) R11991040
since the sparsity degree of R11991040
is far fewerthan that of R
11991021 the dimension of signal needed
and scale of measurement can also be very low Werepresent the measurement for R
11991040as A3 while the
only difference of it from A2is the dimension
Similar to (17) there is
zz119879 = A3(yy119879)A
3
119879
(21)
Then according to vecUXV = (V119879 otimes U)vecX wecan transform the two-dimensional relationship intoone-dimensional relationship
vec zz119879 = A3otimes A3vec yy119879 (22)
We can obtain
vec zz119879 vec119879 zz119879
= (A3otimes A3) vec yy119879 vec119879 yy119879 (A
3otimes A3)
(23)
Take the average of both sides
119864 (vec zz119879 vec119879 zz119879)
= (A3otimes A3) 119864 (vec yy119879 vec119879 yy119879) (A
3otimes A3)
(24)
Based on (13) we get the relationship
R11991140
= (A3otimes A3)R11991040
(A3otimes A3) (25)
where R11991140
denotes 119864(veczz119879vec119879zz119879) And thenwe have
vec R11991140
= (A3otimes A3) otimes (A
3otimes A3) vec R
11991040
= Φvec R11991040
(26)
whereΦ = (A3otimesA3)otimes(A3otimesA3) is the sensingmatrix
42 Reconstruction of Identification Features z120574 R11991121
andR11991140
can be calculated by the sampling value z With sensingmatrixes and measurement vectors known the reconstruc-tion of the sparse vectors can be regarded as the signalrecovery problem by solving the NP-hard puzzle as followstaking R
11991021as an example
vec Ry21 = argmin 10038171003817100381710038171003817vec R
11991021100381710038171003817100381710038170
st vec R11991121
= Φ vec R11991121
(27)
This can be transformed into a linear programming problem
minvecRy21
10038171003817100381710038171003817vec R
11991121 minusΦ vec R
1199102110038171003817100381710038171003817
12
2
+ 11989410038171003817100381710038171003817vec R
11991021100381710038171003817100381710038171
(28)
which is called 1198971-norm least square programming problemand is proved to be convex that there exists a unique optimumsolution 119894 gt 0 is a weighting scalar that balances the sparsityof the solution induced by the 1198971-norm term and the datareconstruction error reflected by the 1198972-norm LS term
In Section 41 we havementioned recovering three recog-nition features by using measurement matrixes A
1 A2 and
A3 respectively However practically only using A
1as the
compressive measurement may meet the requirement ofrecovering all of the features The reason is that A
2and
A3differ in the dimension but are both designed with the
constraint of RIP property only From the other aspect theprimary requirement of constructing matrix A
1is also the
RIP condition
5 Modulation Recognition withthe Identification Features
Given a received communication signalmodulated byMFSKMPSK or MQAM we firstly get compressive samples usingmeasurement matrixes present in Section 2 In this processdue to difference of sparsity we have analyzed in Section 3various features may apply various length of the signal andthis can be decided based on actual situations According tothe approaches proposed above the identification featurescan be easily obtained Then we can recognize the modu-lation format effectively referring to the flowchart shown inFigure 5 and specific steps are listed in the following
Step 1 Reconstruct the spectrum feature when 120574 = 1 withcompressive samples If there is impulse in the recoveredspectrum the modulation mode can be identified as MFSKand the number of impulses indicates the order of it How-ever if there is no impulse in the feature the communicationsignal is modulated by MPSK or MQAM and then Step 2should be conducted
Step 2 Reconstruct the spectrum feature when 120574 = 2 4 8
with compressive samples and observe value of 120574 when theimpulse firstly appears If 120574 = 4 when the impulse appearsthe modulation mode can be regarded as QPSK or MQAM
8 Mobile Information Systems
MFSK MPSK and MQAM
Order of MPSK
Order of MFSK
MPSK and MQAM
QPSK and MQAM
QPSK and order ofMQAM
The spectrumNumber of
impulses
No impulse
The spectrum
Impulse appears when
Impulse appears
feature (120574 = 1)
when 120574 ne 4
feature (120574 ne 1)
120574 = 4
(120573 ne 4)
The high-ordermoment feature
Value range of 120572
Figure 5 The process of digital modulation recognition
and then we go to Step 3 However if 120574 = 4when the impulseappears the signal is modulated by MPSK and this value of 120574is the order of it
Step 3 Reconstruct R11991021
and R11991040
of the signal with com-pressive samples get average values of the diagonal as119872
21(119910)
and 11987240
(119910) respectively and then calculate 120572 based on (11)Compare 120572 with the calculated boundary values shown inTable 1 and determine the modulation type
6 Numerical Results
This section presents the simulation results of our feature-based recognition method We firstly generate a stream ofsignals modulated by MPSK MFSK or MQAM All the sig-nals share the same bit rate 1 kbits and the carrier frequency2 kHz and the carrier spacing for MFSK is 025 kHz Forthe two proposed features the observation time is variousbecause data volume needed by the two features are all
120574 = 1
120574 = 2
120574 = 4
120574 = 8
0
01
02
03
04
05
06
07
08
09
1
Cor
rect
det
ectio
n ra
te
84 102 60minus2minus4minus6minus8minus10
SNR
Figure 6 Correct detection rate of impulse in reconstructed feature1
differentThe performance of reconstruction is closely relatedto the signal-to-noise ratio (SNR) which is set as a variable inour simulation and simulations at every SNR are carried outfor 500 times
As mentioned above information we need to capturein feature 1 is whether there are impulses and the numberof them rather than accurate numerical values Thereforewe apply correct detection rate of pulse to evaluate theperformance of reconstruction of spectrum feature whichis shown in Figure 6 We set a decision threshold whichequals two-thirds of the biggest reconstructed value and ifthere is no other value larger than the threshold the biggestvalue would be regarded as the impulse In this scenario thecompressive ratio is set as 03 which means 119872119873 = 03We calculate the detection rate for MFSK signal on 120574 = 1BPSK on 120574 = 2 QPSK and MQAM on 120574 = 4 and 8PSKon 120574 = 8 respectively It is obvious that the detection ratevaries a lot with 120574 The reason is that 120574th power of signal is anonlinear transform meaning that the uniformly distributednoise ismagnified and the degree ofmagnification extends asthe increasing of 120574Therefore detection rate of impulse when120574 = 8 is the worst one
Figure 7 shows the mean square error (MSE) of recon-structed feature 2 with respect to the theoretical ones Thatis
MSE = 119864
1003817100381710038171003817100381710038171003817
vec S120574 minus vec S
120574
1003817100381710038171003817100381710038171003817
2
2
10038171003817100381710038171003817vec S
12057410038171003817100381710038171003817
2
2
(29)
We give the MSE of reconstructed R11991021
and R11991040
respec-tively with the compressive ratio chosen as 03 and 045 FromFigure 7 we can see that the performance of reconstructionof R11991040
is closely related to the compressive ratio while theperformance of reconstruction of R
11991021is relatively perfect
Mobile Information Systems 9
5 6 7 8 9 10 11 12 13 140
005
015
025
03
02
01
035
04
045
SNR (dB)
MSE
Ry21
Ry40
Ry21
Ry40
MN = 03
MN = 03
MN = 045MN = 045
Figure 7 MSE of reconstructed R11991021
and R11991040
with differentcompressive ratio
even at a low compressive ratio Moreover we can easily getthe conclusion that when the compressive ratio is suitable theprecision of feature 2 is high enough as long as the SNR ishigher than 10 dB
Figure 8 shows the correct classification rate of differentmodulation modes at relatively low SNR Difference of thecorrect classification comes from various performance ofreconstruction of features which has been shown in Figures6 and 7 MFSK has high recognition rate larger than 093
even when SNR = minus6 dB For MPSK the correct recognitionrate declines as 120573 increases However for QPSK andMQAMthe performance is quite different and we give the followinganalysis
According to [14] we have the fact that 11987240
of just thesignal and mixture of noise and signal are of the same valueso the main cause of the error comes from 119872
21
As for 11987221 we have the following proof stating the
variation of the value in noisy condition and noiselesscondition To describe this clearly 119872
21(1199100) 11987221
(V) and11987221
(119910) are respectively used to replace 11987221
while beingin the following condition of signal only noise only and themixture of noise and signal
11987221
(1199100) = 119864 (119910
0ℎ1199100ℎ
lowast
)
11987221
(V) = 119864 (VℎVℎ
lowast
)
11987221
(119910) = 119864 ((1199100ℎ
+ Vℎ) (1199100ℎ
+ Vℎ)lowast
)
= 119864 ((1199100ℎ
+ Vℎ) (1199100ℎ
lowast
+ Vℎ
lowast
))
= 119864 (1199100ℎ1199100ℎ
lowast
+ 1199100ℎ
lowastVℎ+ 1199100ℎVℎ
lowast
+ VℎVℎ
lowast
)
= 119864 (1199100ℎ1199100ℎ
lowast
) + 119864 (1199100ℎ
lowastVℎ) + 119864 (119910
0ℎVℎ
lowast
)
+ 119864 (VℎVℎ
lowast
)
(30)
0
01
02
03
04
05
06
07
08
09
1
SNR (dB)C
orre
ct re
cogn
ition
rate
minus10 minus5 0 5 10 15
MFSKBPSK8PSK
QPSK16QAM
Figure 8 Correct classification rate of different modulation modes
V is zero-mean random measure noises with Gaussiandistribution which is independent from 119910 According to thenature of expectation we know that
119864 (1199100ℎ
lowastVℎ) = 119864 (119910
0ℎ1199100ℎ
lowast
) = 0 (31)
Therefore we can obtain the following relationship
11987221
(119910) = 119864 (1199100ℎ1199100ℎ
lowast
) + 119864 (VℎVℎ
lowast
)
= 11987221
(1199100) + 119872
21(V)
(32)
meaning11987221
(119910) is the sum of signal power and noise powerFrom (11) and (27) we can obtain the relationship of the
where 119875V denotes noise power and 1198751199100
denotes signal powerTo sum up 119872
21(119910) is added by the power of noise and
as a consequence the identification parameter 120572 becomes
10 Mobile Information Systems
smaller thus QPSKmay be recognized as 16QAMThereforethe correct recognition rate of 16QAM is much higher thanQPSK when SNR is lower than 10 dB as shown in Figure 8
7 Conclusion
To solve the problem of high sampling rate for digital modu-lation recognition in spectrum sensing we have proposed afeature-based method to identify the modulation formats ofdigital modulated communication signals using compressivesamples and have greatly lowered the sampling rate basedon CS Two features are constructed in our method oneof which is the spectrum of signalrsquos 120574th power nonlineartransformation and the other is a composition of multiplehigh-order moments of the signal both with desired sparsityBy these two features we have applied suitable measurementmatrixes and built linear relationships referring to themThemethod successfully avoids reconstructing original signalsand uses recognition features to classify signals directlydeclining the algorithm complexity effectively Simulationsshow that correct recognition rates are different for differentmodulation types but are all relatively ideal even in noisy sce-narios In actual situations the method can be decomposedaiming at variable demands and for further work we tend toimprove the performance of the whole method continuouslyespecially the noise elimination in the classification of QPSKand MQAM
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by The China National NaturalScience Fund under Grants 61271181 and 61171109 and theJoint Project withChina Southwest Institute of Electronic andTelecommunication Technology
References
[1] H Bogucka P Kryszkiewicz and A Kliks ldquoDynamic spectrumaggregation for future 5G communicationsrdquo IEEE Communica-tions Magazine vol 53 no 5 pp 35ndash43 2015
[2] T Irnich J Kronander and Y Selen ldquoSpectrum sharing sce-narios and resulting technical requirements for 5G systemsrdquoin Proceedings of the IEEE 24th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCWorkshops rsquo13) pp 127ndash132 IEEE London UK September2013
[3] S Fengpan Research on Modulation Classification for Compres-sive Sensing in Cognitive Radio Ningbo University 2013
[4] O A Dobre A Abdi Y Bar-Ness and W Su ldquoSurveyof automatic modulation classification techniques classicalapproaches and new trendsrdquo IET Communications vol 1 no2 pp 137ndash156 2007
[5] F Wang and X Wang ldquoFast and robust modulation classi-fication via Kolmogorov-Smirnov testrdquo IEEE Transactions onCommunications vol 58 no 8 pp 2324ndash2332 2010
[6] E Cands ldquoCompressive samplingrdquo inProceedings of the Interna-tional Congress ofMathematicians vol 3 pp 1433ndash1452MadridSpain 2006
[7] E J Candes and M B Wakin ldquoAn introduction to compressivesamplingrdquo IEEE Signal Processing Magazine vol 25 no 2 pp21ndash30 2008
[8] Z Tian Y Tafesse and B M Sadler ldquoCyclic feature detectionwith sub-nyquist sampling for wideband spectrum sensingrdquoIEEE Journal on Selected Topics in Signal Processing vol 6 no 1pp 58ndash69 2012
[9] L Zhou and H Man ldquoDistributed automatic modulationclassification based on cyclic feature via compressive sensingrdquoin Proceedings of the IEEEMilitary Communications Conference(MILCOM rsquo13) pp 40ndash45 IEEE San Diego Calif USANovember 2013
[10] J Reichert ldquoAutomatic classification of communication signalsusing higher order statisticsrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo92) vol 5 pp 221ndash224 San Francisco Calif USAMarch 1992
[11] V Orlic and M L Dukic ldquoAlgorithm for automatic modula-tion classification in multipath channel based on sixth-ordercumulantsrdquo inProceedings of the 9th International Conference onTelecommunication inModern Satellite Cable and BroadcastingServices (TELSIKS rsquo09) pp 423ndash426 IEEE Nis Serbia October2009
[12] D C Chang and P K Shih ldquoCumulants-based modulationclassification technique in multipath fading channelsrdquo IETCommunications vol 9 no 6 pp 828ndash835 2015
[13] B Wang and L Ge ldquoA novel algorithm for identification ofOFDM signalrdquo in Proceedings of the International Conference onWireless Communications Networking and Mobile Computing(WCNM rsquo05) pp 261ndash264 September 2005
[14] D Grimaldi S Rapuano and G Truglia ldquoAn automatic digitalmodulation classifier for measurement on telecommunicationnetworksrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology ConferencemdashConference Record pp1711ndash1720 Sorrento Italy 2002
We take QPSK and 16QAM as examples According to(11) the theoretical values 120572 of QPSK and 16QAM respec-tively come out to be 1 and 068 If we get the identifica-tion characteristic 120572 of a signal we can then identify themodulation format by comparing 120572 with a suitable decisionthreshold
Since high-order moment is a kind of statistics we needsample several times Then to obtain 119872
Figure 2 Spectrum of 120574th power of signal modulated by MPSK
ℎ = 119896 the value comes out to be zero for the uncorrelationbetween symbols of the signal For R
11991040 119910ℎ119910119896corresponds
to (119873(ℎ minus 1) + 119896)th element of vecyy119879 When ℎ = 119896 therelationship is that 119910
ℎ
2 corresponds to the (119873(ℎ minus 1) + ℎ)thelement of vecyy119879 According to (10) 119864(119910
ℎ
4
) is the desiredvalue 119872
40(119910) so the (119873(ℎ minus 1) + ℎ)th diagonal elements
(ℎ = 1 2 119873) ofR11991040
are equal to11987240
(119910) Other elementsare zero for the uncorrelation between symbols of the signalThe theoretical figures of R
11991021and R
11991040are shown as
Figure 4It is obvious thatR
11991021andR
11991040in Figure 4 are sparse For
R11991021
all diagonal elements are nonzero meaning the sparsity
degree of it is 1119873 For R11991040
the ((ℎ minus 1) times119873+ ℎ)th elementsof vecR
119909119879 are nonzero meaning the sparsity degree of it is
11198733
4 Recovery of the Identification Features withCompressing Samples
In this section we introduce the approaches of recoveringthe two identification features based on CS We firstly buildthe linear relationships between compressive samples and thedefined features and then give a brief introduction of the
Figure 3 Spectrum of 120574th power of signal modulated by MQAM
020
4060
80
020
4060
800
05
1
15
t
Tau
Auto
corr
elat
ion
mat
rix o
f the
sign
al
(a) R11991021
050
100150
050
100150
0
02
04
06
08
The c
onstr
ucte
d m
atrix
Ry40
(b) R11991040
Figure 4 R11991021
and R11991040
of the signal modulated by 16QAM
reconstruction algorithm and the practical selection strategyfor the measurement matrix
41 Linear Relationships between Compressive Samples andthe Identification Features
411 Linear Relationships between Compressive Samples andthe Spectrum Feature It is obviously that the 120574th power of thesignal is a nonlinear transformation To get linear relationshipbetween compressive samples and the spectrumof the signalrsquos120574th power nonlinear transformation we choose the specialmeasurement matrix proposed in Section 2 According tothe nature of this certain-form matrix we can easily get thefollowing relationship based on (3)
z120574 = A1y120574 (15)
and A1is the measurement matrix for feature 1 Then refer-
ring to (5) we obtain
z120574 = A1FS120574= ΘS120574 (16)
whereΘ = A1F is the sensing matrix we needed
412 Linear Relationships between Compressive Samples andthe High-Order Moment Feature For this identification fea-ture the sampling matrix can be chosen as any one as long asit satisfies the restricted isometry property (RIP)
(i) R11991021
according to (3) and the nature of transpose weget the following relationship and A
2stands for the
measurement matrix for R11991021
zz119867 = A2(yy119867)A
2
119867
(17)
Mobile Information Systems 7
Take the average of both sides
119864 (zz119867) = A2sdot 119864 (yy119867) sdot A
2
119867
(18)
We useR11991121
to represent119864(zz119867) simultaneously referto (12) and then get
R11991121
= A2sdot R11991021
sdot A2
119867
(19)
Next we apply the property vecUXV = (V119879 otimes
U)vecX to transform (19) to (20) It is worth notic-ing that A
2
119867
= A2
119879 for A is a real-value matrix
vec R11991121
= A2otimes A2vec R
11991021 = Ψvec R
11991021 (20)
where Ψ = A2otimes A2can be regarded as the sensing
matrix with the scale of 1198722 times 1198732
(ii) R11991040
since the sparsity degree of R11991040
is far fewerthan that of R
11991021 the dimension of signal needed
and scale of measurement can also be very low Werepresent the measurement for R
11991040as A3 while the
only difference of it from A2is the dimension
Similar to (17) there is
zz119879 = A3(yy119879)A
3
119879
(21)
Then according to vecUXV = (V119879 otimes U)vecX wecan transform the two-dimensional relationship intoone-dimensional relationship
vec zz119879 = A3otimes A3vec yy119879 (22)
We can obtain
vec zz119879 vec119879 zz119879
= (A3otimes A3) vec yy119879 vec119879 yy119879 (A
3otimes A3)
(23)
Take the average of both sides
119864 (vec zz119879 vec119879 zz119879)
= (A3otimes A3) 119864 (vec yy119879 vec119879 yy119879) (A
3otimes A3)
(24)
Based on (13) we get the relationship
R11991140
= (A3otimes A3)R11991040
(A3otimes A3) (25)
where R11991140
denotes 119864(veczz119879vec119879zz119879) And thenwe have
vec R11991140
= (A3otimes A3) otimes (A
3otimes A3) vec R
11991040
= Φvec R11991040
(26)
whereΦ = (A3otimesA3)otimes(A3otimesA3) is the sensingmatrix
42 Reconstruction of Identification Features z120574 R11991121
andR11991140
can be calculated by the sampling value z With sensingmatrixes and measurement vectors known the reconstruc-tion of the sparse vectors can be regarded as the signalrecovery problem by solving the NP-hard puzzle as followstaking R
11991021as an example
vec Ry21 = argmin 10038171003817100381710038171003817vec R
11991021100381710038171003817100381710038170
st vec R11991121
= Φ vec R11991121
(27)
This can be transformed into a linear programming problem
minvecRy21
10038171003817100381710038171003817vec R
11991121 minusΦ vec R
1199102110038171003817100381710038171003817
12
2
+ 11989410038171003817100381710038171003817vec R
11991021100381710038171003817100381710038171
(28)
which is called 1198971-norm least square programming problemand is proved to be convex that there exists a unique optimumsolution 119894 gt 0 is a weighting scalar that balances the sparsityof the solution induced by the 1198971-norm term and the datareconstruction error reflected by the 1198972-norm LS term
In Section 41 we havementioned recovering three recog-nition features by using measurement matrixes A
1 A2 and
A3 respectively However practically only using A
1as the
compressive measurement may meet the requirement ofrecovering all of the features The reason is that A
2and
A3differ in the dimension but are both designed with the
constraint of RIP property only From the other aspect theprimary requirement of constructing matrix A
1is also the
RIP condition
5 Modulation Recognition withthe Identification Features
Given a received communication signalmodulated byMFSKMPSK or MQAM we firstly get compressive samples usingmeasurement matrixes present in Section 2 In this processdue to difference of sparsity we have analyzed in Section 3various features may apply various length of the signal andthis can be decided based on actual situations According tothe approaches proposed above the identification featurescan be easily obtained Then we can recognize the modu-lation format effectively referring to the flowchart shown inFigure 5 and specific steps are listed in the following
Step 1 Reconstruct the spectrum feature when 120574 = 1 withcompressive samples If there is impulse in the recoveredspectrum the modulation mode can be identified as MFSKand the number of impulses indicates the order of it How-ever if there is no impulse in the feature the communicationsignal is modulated by MPSK or MQAM and then Step 2should be conducted
Step 2 Reconstruct the spectrum feature when 120574 = 2 4 8
with compressive samples and observe value of 120574 when theimpulse firstly appears If 120574 = 4 when the impulse appearsthe modulation mode can be regarded as QPSK or MQAM
8 Mobile Information Systems
MFSK MPSK and MQAM
Order of MPSK
Order of MFSK
MPSK and MQAM
QPSK and MQAM
QPSK and order ofMQAM
The spectrumNumber of
impulses
No impulse
The spectrum
Impulse appears when
Impulse appears
feature (120574 = 1)
when 120574 ne 4
feature (120574 ne 1)
120574 = 4
(120573 ne 4)
The high-ordermoment feature
Value range of 120572
Figure 5 The process of digital modulation recognition
and then we go to Step 3 However if 120574 = 4when the impulseappears the signal is modulated by MPSK and this value of 120574is the order of it
Step 3 Reconstruct R11991021
and R11991040
of the signal with com-pressive samples get average values of the diagonal as119872
21(119910)
and 11987240
(119910) respectively and then calculate 120572 based on (11)Compare 120572 with the calculated boundary values shown inTable 1 and determine the modulation type
6 Numerical Results
This section presents the simulation results of our feature-based recognition method We firstly generate a stream ofsignals modulated by MPSK MFSK or MQAM All the sig-nals share the same bit rate 1 kbits and the carrier frequency2 kHz and the carrier spacing for MFSK is 025 kHz Forthe two proposed features the observation time is variousbecause data volume needed by the two features are all
120574 = 1
120574 = 2
120574 = 4
120574 = 8
0
01
02
03
04
05
06
07
08
09
1
Cor
rect
det
ectio
n ra
te
84 102 60minus2minus4minus6minus8minus10
SNR
Figure 6 Correct detection rate of impulse in reconstructed feature1
differentThe performance of reconstruction is closely relatedto the signal-to-noise ratio (SNR) which is set as a variable inour simulation and simulations at every SNR are carried outfor 500 times
As mentioned above information we need to capturein feature 1 is whether there are impulses and the numberof them rather than accurate numerical values Thereforewe apply correct detection rate of pulse to evaluate theperformance of reconstruction of spectrum feature whichis shown in Figure 6 We set a decision threshold whichequals two-thirds of the biggest reconstructed value and ifthere is no other value larger than the threshold the biggestvalue would be regarded as the impulse In this scenario thecompressive ratio is set as 03 which means 119872119873 = 03We calculate the detection rate for MFSK signal on 120574 = 1BPSK on 120574 = 2 QPSK and MQAM on 120574 = 4 and 8PSKon 120574 = 8 respectively It is obvious that the detection ratevaries a lot with 120574 The reason is that 120574th power of signal is anonlinear transform meaning that the uniformly distributednoise ismagnified and the degree ofmagnification extends asthe increasing of 120574Therefore detection rate of impulse when120574 = 8 is the worst one
Figure 7 shows the mean square error (MSE) of recon-structed feature 2 with respect to the theoretical ones Thatis
MSE = 119864
1003817100381710038171003817100381710038171003817
vec S120574 minus vec S
120574
1003817100381710038171003817100381710038171003817
2
2
10038171003817100381710038171003817vec S
12057410038171003817100381710038171003817
2
2
(29)
We give the MSE of reconstructed R11991021
and R11991040
respec-tively with the compressive ratio chosen as 03 and 045 FromFigure 7 we can see that the performance of reconstructionof R11991040
is closely related to the compressive ratio while theperformance of reconstruction of R
11991021is relatively perfect
Mobile Information Systems 9
5 6 7 8 9 10 11 12 13 140
005
015
025
03
02
01
035
04
045
SNR (dB)
MSE
Ry21
Ry40
Ry21
Ry40
MN = 03
MN = 03
MN = 045MN = 045
Figure 7 MSE of reconstructed R11991021
and R11991040
with differentcompressive ratio
even at a low compressive ratio Moreover we can easily getthe conclusion that when the compressive ratio is suitable theprecision of feature 2 is high enough as long as the SNR ishigher than 10 dB
Figure 8 shows the correct classification rate of differentmodulation modes at relatively low SNR Difference of thecorrect classification comes from various performance ofreconstruction of features which has been shown in Figures6 and 7 MFSK has high recognition rate larger than 093
even when SNR = minus6 dB For MPSK the correct recognitionrate declines as 120573 increases However for QPSK andMQAMthe performance is quite different and we give the followinganalysis
According to [14] we have the fact that 11987240
of just thesignal and mixture of noise and signal are of the same valueso the main cause of the error comes from 119872
21
As for 11987221 we have the following proof stating the
variation of the value in noisy condition and noiselesscondition To describe this clearly 119872
21(1199100) 11987221
(V) and11987221
(119910) are respectively used to replace 11987221
while beingin the following condition of signal only noise only and themixture of noise and signal
11987221
(1199100) = 119864 (119910
0ℎ1199100ℎ
lowast
)
11987221
(V) = 119864 (VℎVℎ
lowast
)
11987221
(119910) = 119864 ((1199100ℎ
+ Vℎ) (1199100ℎ
+ Vℎ)lowast
)
= 119864 ((1199100ℎ
+ Vℎ) (1199100ℎ
lowast
+ Vℎ
lowast
))
= 119864 (1199100ℎ1199100ℎ
lowast
+ 1199100ℎ
lowastVℎ+ 1199100ℎVℎ
lowast
+ VℎVℎ
lowast
)
= 119864 (1199100ℎ1199100ℎ
lowast
) + 119864 (1199100ℎ
lowastVℎ) + 119864 (119910
0ℎVℎ
lowast
)
+ 119864 (VℎVℎ
lowast
)
(30)
0
01
02
03
04
05
06
07
08
09
1
SNR (dB)C
orre
ct re
cogn
ition
rate
minus10 minus5 0 5 10 15
MFSKBPSK8PSK
QPSK16QAM
Figure 8 Correct classification rate of different modulation modes
V is zero-mean random measure noises with Gaussiandistribution which is independent from 119910 According to thenature of expectation we know that
119864 (1199100ℎ
lowastVℎ) = 119864 (119910
0ℎ1199100ℎ
lowast
) = 0 (31)
Therefore we can obtain the following relationship
11987221
(119910) = 119864 (1199100ℎ1199100ℎ
lowast
) + 119864 (VℎVℎ
lowast
)
= 11987221
(1199100) + 119872
21(V)
(32)
meaning11987221
(119910) is the sum of signal power and noise powerFrom (11) and (27) we can obtain the relationship of the
where 119875V denotes noise power and 1198751199100
denotes signal powerTo sum up 119872
21(119910) is added by the power of noise and
as a consequence the identification parameter 120572 becomes
10 Mobile Information Systems
smaller thus QPSKmay be recognized as 16QAMThereforethe correct recognition rate of 16QAM is much higher thanQPSK when SNR is lower than 10 dB as shown in Figure 8
7 Conclusion
To solve the problem of high sampling rate for digital modu-lation recognition in spectrum sensing we have proposed afeature-based method to identify the modulation formats ofdigital modulated communication signals using compressivesamples and have greatly lowered the sampling rate basedon CS Two features are constructed in our method oneof which is the spectrum of signalrsquos 120574th power nonlineartransformation and the other is a composition of multiplehigh-order moments of the signal both with desired sparsityBy these two features we have applied suitable measurementmatrixes and built linear relationships referring to themThemethod successfully avoids reconstructing original signalsand uses recognition features to classify signals directlydeclining the algorithm complexity effectively Simulationsshow that correct recognition rates are different for differentmodulation types but are all relatively ideal even in noisy sce-narios In actual situations the method can be decomposedaiming at variable demands and for further work we tend toimprove the performance of the whole method continuouslyespecially the noise elimination in the classification of QPSKand MQAM
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by The China National NaturalScience Fund under Grants 61271181 and 61171109 and theJoint Project withChina Southwest Institute of Electronic andTelecommunication Technology
References
[1] H Bogucka P Kryszkiewicz and A Kliks ldquoDynamic spectrumaggregation for future 5G communicationsrdquo IEEE Communica-tions Magazine vol 53 no 5 pp 35ndash43 2015
[2] T Irnich J Kronander and Y Selen ldquoSpectrum sharing sce-narios and resulting technical requirements for 5G systemsrdquoin Proceedings of the IEEE 24th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCWorkshops rsquo13) pp 127ndash132 IEEE London UK September2013
[3] S Fengpan Research on Modulation Classification for Compres-sive Sensing in Cognitive Radio Ningbo University 2013
[4] O A Dobre A Abdi Y Bar-Ness and W Su ldquoSurveyof automatic modulation classification techniques classicalapproaches and new trendsrdquo IET Communications vol 1 no2 pp 137ndash156 2007
[5] F Wang and X Wang ldquoFast and robust modulation classi-fication via Kolmogorov-Smirnov testrdquo IEEE Transactions onCommunications vol 58 no 8 pp 2324ndash2332 2010
[6] E Cands ldquoCompressive samplingrdquo inProceedings of the Interna-tional Congress ofMathematicians vol 3 pp 1433ndash1452MadridSpain 2006
[7] E J Candes and M B Wakin ldquoAn introduction to compressivesamplingrdquo IEEE Signal Processing Magazine vol 25 no 2 pp21ndash30 2008
[8] Z Tian Y Tafesse and B M Sadler ldquoCyclic feature detectionwith sub-nyquist sampling for wideband spectrum sensingrdquoIEEE Journal on Selected Topics in Signal Processing vol 6 no 1pp 58ndash69 2012
[9] L Zhou and H Man ldquoDistributed automatic modulationclassification based on cyclic feature via compressive sensingrdquoin Proceedings of the IEEEMilitary Communications Conference(MILCOM rsquo13) pp 40ndash45 IEEE San Diego Calif USANovember 2013
[10] J Reichert ldquoAutomatic classification of communication signalsusing higher order statisticsrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo92) vol 5 pp 221ndash224 San Francisco Calif USAMarch 1992
[11] V Orlic and M L Dukic ldquoAlgorithm for automatic modula-tion classification in multipath channel based on sixth-ordercumulantsrdquo inProceedings of the 9th International Conference onTelecommunication inModern Satellite Cable and BroadcastingServices (TELSIKS rsquo09) pp 423ndash426 IEEE Nis Serbia October2009
[12] D C Chang and P K Shih ldquoCumulants-based modulationclassification technique in multipath fading channelsrdquo IETCommunications vol 9 no 6 pp 828ndash835 2015
[13] B Wang and L Ge ldquoA novel algorithm for identification ofOFDM signalrdquo in Proceedings of the International Conference onWireless Communications Networking and Mobile Computing(WCNM rsquo05) pp 261ndash264 September 2005
[14] D Grimaldi S Rapuano and G Truglia ldquoAn automatic digitalmodulation classifier for measurement on telecommunicationnetworksrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology ConferencemdashConference Record pp1711ndash1720 Sorrento Italy 2002
Figure 2 Spectrum of 120574th power of signal modulated by MPSK
ℎ = 119896 the value comes out to be zero for the uncorrelationbetween symbols of the signal For R
11991040 119910ℎ119910119896corresponds
to (119873(ℎ minus 1) + 119896)th element of vecyy119879 When ℎ = 119896 therelationship is that 119910
ℎ
2 corresponds to the (119873(ℎ minus 1) + ℎ)thelement of vecyy119879 According to (10) 119864(119910
ℎ
4
) is the desiredvalue 119872
40(119910) so the (119873(ℎ minus 1) + ℎ)th diagonal elements
(ℎ = 1 2 119873) ofR11991040
are equal to11987240
(119910) Other elementsare zero for the uncorrelation between symbols of the signalThe theoretical figures of R
11991021and R
11991040are shown as
Figure 4It is obvious thatR
11991021andR
11991040in Figure 4 are sparse For
R11991021
all diagonal elements are nonzero meaning the sparsity
degree of it is 1119873 For R11991040
the ((ℎ minus 1) times119873+ ℎ)th elementsof vecR
119909119879 are nonzero meaning the sparsity degree of it is
11198733
4 Recovery of the Identification Features withCompressing Samples
In this section we introduce the approaches of recoveringthe two identification features based on CS We firstly buildthe linear relationships between compressive samples and thedefined features and then give a brief introduction of the
Figure 3 Spectrum of 120574th power of signal modulated by MQAM
020
4060
80
020
4060
800
05
1
15
t
Tau
Auto
corr
elat
ion
mat
rix o
f the
sign
al
(a) R11991021
050
100150
050
100150
0
02
04
06
08
The c
onstr
ucte
d m
atrix
Ry40
(b) R11991040
Figure 4 R11991021
and R11991040
of the signal modulated by 16QAM
reconstruction algorithm and the practical selection strategyfor the measurement matrix
41 Linear Relationships between Compressive Samples andthe Identification Features
411 Linear Relationships between Compressive Samples andthe Spectrum Feature It is obviously that the 120574th power of thesignal is a nonlinear transformation To get linear relationshipbetween compressive samples and the spectrumof the signalrsquos120574th power nonlinear transformation we choose the specialmeasurement matrix proposed in Section 2 According tothe nature of this certain-form matrix we can easily get thefollowing relationship based on (3)
z120574 = A1y120574 (15)
and A1is the measurement matrix for feature 1 Then refer-
ring to (5) we obtain
z120574 = A1FS120574= ΘS120574 (16)
whereΘ = A1F is the sensing matrix we needed
412 Linear Relationships between Compressive Samples andthe High-Order Moment Feature For this identification fea-ture the sampling matrix can be chosen as any one as long asit satisfies the restricted isometry property (RIP)
(i) R11991021
according to (3) and the nature of transpose weget the following relationship and A
2stands for the
measurement matrix for R11991021
zz119867 = A2(yy119867)A
2
119867
(17)
Mobile Information Systems 7
Take the average of both sides
119864 (zz119867) = A2sdot 119864 (yy119867) sdot A
2
119867
(18)
We useR11991121
to represent119864(zz119867) simultaneously referto (12) and then get
R11991121
= A2sdot R11991021
sdot A2
119867
(19)
Next we apply the property vecUXV = (V119879 otimes
U)vecX to transform (19) to (20) It is worth notic-ing that A
2
119867
= A2
119879 for A is a real-value matrix
vec R11991121
= A2otimes A2vec R
11991021 = Ψvec R
11991021 (20)
where Ψ = A2otimes A2can be regarded as the sensing
matrix with the scale of 1198722 times 1198732
(ii) R11991040
since the sparsity degree of R11991040
is far fewerthan that of R
11991021 the dimension of signal needed
and scale of measurement can also be very low Werepresent the measurement for R
11991040as A3 while the
only difference of it from A2is the dimension
Similar to (17) there is
zz119879 = A3(yy119879)A
3
119879
(21)
Then according to vecUXV = (V119879 otimes U)vecX wecan transform the two-dimensional relationship intoone-dimensional relationship
vec zz119879 = A3otimes A3vec yy119879 (22)
We can obtain
vec zz119879 vec119879 zz119879
= (A3otimes A3) vec yy119879 vec119879 yy119879 (A
3otimes A3)
(23)
Take the average of both sides
119864 (vec zz119879 vec119879 zz119879)
= (A3otimes A3) 119864 (vec yy119879 vec119879 yy119879) (A
3otimes A3)
(24)
Based on (13) we get the relationship
R11991140
= (A3otimes A3)R11991040
(A3otimes A3) (25)
where R11991140
denotes 119864(veczz119879vec119879zz119879) And thenwe have
vec R11991140
= (A3otimes A3) otimes (A
3otimes A3) vec R
11991040
= Φvec R11991040
(26)
whereΦ = (A3otimesA3)otimes(A3otimesA3) is the sensingmatrix
42 Reconstruction of Identification Features z120574 R11991121
andR11991140
can be calculated by the sampling value z With sensingmatrixes and measurement vectors known the reconstruc-tion of the sparse vectors can be regarded as the signalrecovery problem by solving the NP-hard puzzle as followstaking R
11991021as an example
vec Ry21 = argmin 10038171003817100381710038171003817vec R
11991021100381710038171003817100381710038170
st vec R11991121
= Φ vec R11991121
(27)
This can be transformed into a linear programming problem
minvecRy21
10038171003817100381710038171003817vec R
11991121 minusΦ vec R
1199102110038171003817100381710038171003817
12
2
+ 11989410038171003817100381710038171003817vec R
11991021100381710038171003817100381710038171
(28)
which is called 1198971-norm least square programming problemand is proved to be convex that there exists a unique optimumsolution 119894 gt 0 is a weighting scalar that balances the sparsityof the solution induced by the 1198971-norm term and the datareconstruction error reflected by the 1198972-norm LS term
In Section 41 we havementioned recovering three recog-nition features by using measurement matrixes A
1 A2 and
A3 respectively However practically only using A
1as the
compressive measurement may meet the requirement ofrecovering all of the features The reason is that A
2and
A3differ in the dimension but are both designed with the
constraint of RIP property only From the other aspect theprimary requirement of constructing matrix A
1is also the
RIP condition
5 Modulation Recognition withthe Identification Features
Given a received communication signalmodulated byMFSKMPSK or MQAM we firstly get compressive samples usingmeasurement matrixes present in Section 2 In this processdue to difference of sparsity we have analyzed in Section 3various features may apply various length of the signal andthis can be decided based on actual situations According tothe approaches proposed above the identification featurescan be easily obtained Then we can recognize the modu-lation format effectively referring to the flowchart shown inFigure 5 and specific steps are listed in the following
Step 1 Reconstruct the spectrum feature when 120574 = 1 withcompressive samples If there is impulse in the recoveredspectrum the modulation mode can be identified as MFSKand the number of impulses indicates the order of it How-ever if there is no impulse in the feature the communicationsignal is modulated by MPSK or MQAM and then Step 2should be conducted
Step 2 Reconstruct the spectrum feature when 120574 = 2 4 8
with compressive samples and observe value of 120574 when theimpulse firstly appears If 120574 = 4 when the impulse appearsthe modulation mode can be regarded as QPSK or MQAM
8 Mobile Information Systems
MFSK MPSK and MQAM
Order of MPSK
Order of MFSK
MPSK and MQAM
QPSK and MQAM
QPSK and order ofMQAM
The spectrumNumber of
impulses
No impulse
The spectrum
Impulse appears when
Impulse appears
feature (120574 = 1)
when 120574 ne 4
feature (120574 ne 1)
120574 = 4
(120573 ne 4)
The high-ordermoment feature
Value range of 120572
Figure 5 The process of digital modulation recognition
and then we go to Step 3 However if 120574 = 4when the impulseappears the signal is modulated by MPSK and this value of 120574is the order of it
Step 3 Reconstruct R11991021
and R11991040
of the signal with com-pressive samples get average values of the diagonal as119872
21(119910)
and 11987240
(119910) respectively and then calculate 120572 based on (11)Compare 120572 with the calculated boundary values shown inTable 1 and determine the modulation type
6 Numerical Results
This section presents the simulation results of our feature-based recognition method We firstly generate a stream ofsignals modulated by MPSK MFSK or MQAM All the sig-nals share the same bit rate 1 kbits and the carrier frequency2 kHz and the carrier spacing for MFSK is 025 kHz Forthe two proposed features the observation time is variousbecause data volume needed by the two features are all
120574 = 1
120574 = 2
120574 = 4
120574 = 8
0
01
02
03
04
05
06
07
08
09
1
Cor
rect
det
ectio
n ra
te
84 102 60minus2minus4minus6minus8minus10
SNR
Figure 6 Correct detection rate of impulse in reconstructed feature1
differentThe performance of reconstruction is closely relatedto the signal-to-noise ratio (SNR) which is set as a variable inour simulation and simulations at every SNR are carried outfor 500 times
As mentioned above information we need to capturein feature 1 is whether there are impulses and the numberof them rather than accurate numerical values Thereforewe apply correct detection rate of pulse to evaluate theperformance of reconstruction of spectrum feature whichis shown in Figure 6 We set a decision threshold whichequals two-thirds of the biggest reconstructed value and ifthere is no other value larger than the threshold the biggestvalue would be regarded as the impulse In this scenario thecompressive ratio is set as 03 which means 119872119873 = 03We calculate the detection rate for MFSK signal on 120574 = 1BPSK on 120574 = 2 QPSK and MQAM on 120574 = 4 and 8PSKon 120574 = 8 respectively It is obvious that the detection ratevaries a lot with 120574 The reason is that 120574th power of signal is anonlinear transform meaning that the uniformly distributednoise ismagnified and the degree ofmagnification extends asthe increasing of 120574Therefore detection rate of impulse when120574 = 8 is the worst one
Figure 7 shows the mean square error (MSE) of recon-structed feature 2 with respect to the theoretical ones Thatis
MSE = 119864
1003817100381710038171003817100381710038171003817
vec S120574 minus vec S
120574
1003817100381710038171003817100381710038171003817
2
2
10038171003817100381710038171003817vec S
12057410038171003817100381710038171003817
2
2
(29)
We give the MSE of reconstructed R11991021
and R11991040
respec-tively with the compressive ratio chosen as 03 and 045 FromFigure 7 we can see that the performance of reconstructionof R11991040
is closely related to the compressive ratio while theperformance of reconstruction of R
11991021is relatively perfect
Mobile Information Systems 9
5 6 7 8 9 10 11 12 13 140
005
015
025
03
02
01
035
04
045
SNR (dB)
MSE
Ry21
Ry40
Ry21
Ry40
MN = 03
MN = 03
MN = 045MN = 045
Figure 7 MSE of reconstructed R11991021
and R11991040
with differentcompressive ratio
even at a low compressive ratio Moreover we can easily getthe conclusion that when the compressive ratio is suitable theprecision of feature 2 is high enough as long as the SNR ishigher than 10 dB
Figure 8 shows the correct classification rate of differentmodulation modes at relatively low SNR Difference of thecorrect classification comes from various performance ofreconstruction of features which has been shown in Figures6 and 7 MFSK has high recognition rate larger than 093
even when SNR = minus6 dB For MPSK the correct recognitionrate declines as 120573 increases However for QPSK andMQAMthe performance is quite different and we give the followinganalysis
According to [14] we have the fact that 11987240
of just thesignal and mixture of noise and signal are of the same valueso the main cause of the error comes from 119872
21
As for 11987221 we have the following proof stating the
variation of the value in noisy condition and noiselesscondition To describe this clearly 119872
21(1199100) 11987221
(V) and11987221
(119910) are respectively used to replace 11987221
while beingin the following condition of signal only noise only and themixture of noise and signal
11987221
(1199100) = 119864 (119910
0ℎ1199100ℎ
lowast
)
11987221
(V) = 119864 (VℎVℎ
lowast
)
11987221
(119910) = 119864 ((1199100ℎ
+ Vℎ) (1199100ℎ
+ Vℎ)lowast
)
= 119864 ((1199100ℎ
+ Vℎ) (1199100ℎ
lowast
+ Vℎ
lowast
))
= 119864 (1199100ℎ1199100ℎ
lowast
+ 1199100ℎ
lowastVℎ+ 1199100ℎVℎ
lowast
+ VℎVℎ
lowast
)
= 119864 (1199100ℎ1199100ℎ
lowast
) + 119864 (1199100ℎ
lowastVℎ) + 119864 (119910
0ℎVℎ
lowast
)
+ 119864 (VℎVℎ
lowast
)
(30)
0
01
02
03
04
05
06
07
08
09
1
SNR (dB)C
orre
ct re
cogn
ition
rate
minus10 minus5 0 5 10 15
MFSKBPSK8PSK
QPSK16QAM
Figure 8 Correct classification rate of different modulation modes
V is zero-mean random measure noises with Gaussiandistribution which is independent from 119910 According to thenature of expectation we know that
119864 (1199100ℎ
lowastVℎ) = 119864 (119910
0ℎ1199100ℎ
lowast
) = 0 (31)
Therefore we can obtain the following relationship
11987221
(119910) = 119864 (1199100ℎ1199100ℎ
lowast
) + 119864 (VℎVℎ
lowast
)
= 11987221
(1199100) + 119872
21(V)
(32)
meaning11987221
(119910) is the sum of signal power and noise powerFrom (11) and (27) we can obtain the relationship of the
where 119875V denotes noise power and 1198751199100
denotes signal powerTo sum up 119872
21(119910) is added by the power of noise and
as a consequence the identification parameter 120572 becomes
10 Mobile Information Systems
smaller thus QPSKmay be recognized as 16QAMThereforethe correct recognition rate of 16QAM is much higher thanQPSK when SNR is lower than 10 dB as shown in Figure 8
7 Conclusion
To solve the problem of high sampling rate for digital modu-lation recognition in spectrum sensing we have proposed afeature-based method to identify the modulation formats ofdigital modulated communication signals using compressivesamples and have greatly lowered the sampling rate basedon CS Two features are constructed in our method oneof which is the spectrum of signalrsquos 120574th power nonlineartransformation and the other is a composition of multiplehigh-order moments of the signal both with desired sparsityBy these two features we have applied suitable measurementmatrixes and built linear relationships referring to themThemethod successfully avoids reconstructing original signalsand uses recognition features to classify signals directlydeclining the algorithm complexity effectively Simulationsshow that correct recognition rates are different for differentmodulation types but are all relatively ideal even in noisy sce-narios In actual situations the method can be decomposedaiming at variable demands and for further work we tend toimprove the performance of the whole method continuouslyespecially the noise elimination in the classification of QPSKand MQAM
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by The China National NaturalScience Fund under Grants 61271181 and 61171109 and theJoint Project withChina Southwest Institute of Electronic andTelecommunication Technology
References
[1] H Bogucka P Kryszkiewicz and A Kliks ldquoDynamic spectrumaggregation for future 5G communicationsrdquo IEEE Communica-tions Magazine vol 53 no 5 pp 35ndash43 2015
[2] T Irnich J Kronander and Y Selen ldquoSpectrum sharing sce-narios and resulting technical requirements for 5G systemsrdquoin Proceedings of the IEEE 24th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCWorkshops rsquo13) pp 127ndash132 IEEE London UK September2013
[3] S Fengpan Research on Modulation Classification for Compres-sive Sensing in Cognitive Radio Ningbo University 2013
[4] O A Dobre A Abdi Y Bar-Ness and W Su ldquoSurveyof automatic modulation classification techniques classicalapproaches and new trendsrdquo IET Communications vol 1 no2 pp 137ndash156 2007
[5] F Wang and X Wang ldquoFast and robust modulation classi-fication via Kolmogorov-Smirnov testrdquo IEEE Transactions onCommunications vol 58 no 8 pp 2324ndash2332 2010
[6] E Cands ldquoCompressive samplingrdquo inProceedings of the Interna-tional Congress ofMathematicians vol 3 pp 1433ndash1452MadridSpain 2006
[7] E J Candes and M B Wakin ldquoAn introduction to compressivesamplingrdquo IEEE Signal Processing Magazine vol 25 no 2 pp21ndash30 2008
[8] Z Tian Y Tafesse and B M Sadler ldquoCyclic feature detectionwith sub-nyquist sampling for wideband spectrum sensingrdquoIEEE Journal on Selected Topics in Signal Processing vol 6 no 1pp 58ndash69 2012
[9] L Zhou and H Man ldquoDistributed automatic modulationclassification based on cyclic feature via compressive sensingrdquoin Proceedings of the IEEEMilitary Communications Conference(MILCOM rsquo13) pp 40ndash45 IEEE San Diego Calif USANovember 2013
[10] J Reichert ldquoAutomatic classification of communication signalsusing higher order statisticsrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo92) vol 5 pp 221ndash224 San Francisco Calif USAMarch 1992
[11] V Orlic and M L Dukic ldquoAlgorithm for automatic modula-tion classification in multipath channel based on sixth-ordercumulantsrdquo inProceedings of the 9th International Conference onTelecommunication inModern Satellite Cable and BroadcastingServices (TELSIKS rsquo09) pp 423ndash426 IEEE Nis Serbia October2009
[12] D C Chang and P K Shih ldquoCumulants-based modulationclassification technique in multipath fading channelsrdquo IETCommunications vol 9 no 6 pp 828ndash835 2015
[13] B Wang and L Ge ldquoA novel algorithm for identification ofOFDM signalrdquo in Proceedings of the International Conference onWireless Communications Networking and Mobile Computing(WCNM rsquo05) pp 261ndash264 September 2005
[14] D Grimaldi S Rapuano and G Truglia ldquoAn automatic digitalmodulation classifier for measurement on telecommunicationnetworksrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology ConferencemdashConference Record pp1711ndash1720 Sorrento Italy 2002
Figure 3 Spectrum of 120574th power of signal modulated by MQAM
020
4060
80
020
4060
800
05
1
15
t
Tau
Auto
corr
elat
ion
mat
rix o
f the
sign
al
(a) R11991021
050
100150
050
100150
0
02
04
06
08
The c
onstr
ucte
d m
atrix
Ry40
(b) R11991040
Figure 4 R11991021
and R11991040
of the signal modulated by 16QAM
reconstruction algorithm and the practical selection strategyfor the measurement matrix
41 Linear Relationships between Compressive Samples andthe Identification Features
411 Linear Relationships between Compressive Samples andthe Spectrum Feature It is obviously that the 120574th power of thesignal is a nonlinear transformation To get linear relationshipbetween compressive samples and the spectrumof the signalrsquos120574th power nonlinear transformation we choose the specialmeasurement matrix proposed in Section 2 According tothe nature of this certain-form matrix we can easily get thefollowing relationship based on (3)
z120574 = A1y120574 (15)
and A1is the measurement matrix for feature 1 Then refer-
ring to (5) we obtain
z120574 = A1FS120574= ΘS120574 (16)
whereΘ = A1F is the sensing matrix we needed
412 Linear Relationships between Compressive Samples andthe High-Order Moment Feature For this identification fea-ture the sampling matrix can be chosen as any one as long asit satisfies the restricted isometry property (RIP)
(i) R11991021
according to (3) and the nature of transpose weget the following relationship and A
2stands for the
measurement matrix for R11991021
zz119867 = A2(yy119867)A
2
119867
(17)
Mobile Information Systems 7
Take the average of both sides
119864 (zz119867) = A2sdot 119864 (yy119867) sdot A
2
119867
(18)
We useR11991121
to represent119864(zz119867) simultaneously referto (12) and then get
R11991121
= A2sdot R11991021
sdot A2
119867
(19)
Next we apply the property vecUXV = (V119879 otimes
U)vecX to transform (19) to (20) It is worth notic-ing that A
2
119867
= A2
119879 for A is a real-value matrix
vec R11991121
= A2otimes A2vec R
11991021 = Ψvec R
11991021 (20)
where Ψ = A2otimes A2can be regarded as the sensing
matrix with the scale of 1198722 times 1198732
(ii) R11991040
since the sparsity degree of R11991040
is far fewerthan that of R
11991021 the dimension of signal needed
and scale of measurement can also be very low Werepresent the measurement for R
11991040as A3 while the
only difference of it from A2is the dimension
Similar to (17) there is
zz119879 = A3(yy119879)A
3
119879
(21)
Then according to vecUXV = (V119879 otimes U)vecX wecan transform the two-dimensional relationship intoone-dimensional relationship
vec zz119879 = A3otimes A3vec yy119879 (22)
We can obtain
vec zz119879 vec119879 zz119879
= (A3otimes A3) vec yy119879 vec119879 yy119879 (A
3otimes A3)
(23)
Take the average of both sides
119864 (vec zz119879 vec119879 zz119879)
= (A3otimes A3) 119864 (vec yy119879 vec119879 yy119879) (A
3otimes A3)
(24)
Based on (13) we get the relationship
R11991140
= (A3otimes A3)R11991040
(A3otimes A3) (25)
where R11991140
denotes 119864(veczz119879vec119879zz119879) And thenwe have
vec R11991140
= (A3otimes A3) otimes (A
3otimes A3) vec R
11991040
= Φvec R11991040
(26)
whereΦ = (A3otimesA3)otimes(A3otimesA3) is the sensingmatrix
42 Reconstruction of Identification Features z120574 R11991121
andR11991140
can be calculated by the sampling value z With sensingmatrixes and measurement vectors known the reconstruc-tion of the sparse vectors can be regarded as the signalrecovery problem by solving the NP-hard puzzle as followstaking R
11991021as an example
vec Ry21 = argmin 10038171003817100381710038171003817vec R
11991021100381710038171003817100381710038170
st vec R11991121
= Φ vec R11991121
(27)
This can be transformed into a linear programming problem
minvecRy21
10038171003817100381710038171003817vec R
11991121 minusΦ vec R
1199102110038171003817100381710038171003817
12
2
+ 11989410038171003817100381710038171003817vec R
11991021100381710038171003817100381710038171
(28)
which is called 1198971-norm least square programming problemand is proved to be convex that there exists a unique optimumsolution 119894 gt 0 is a weighting scalar that balances the sparsityof the solution induced by the 1198971-norm term and the datareconstruction error reflected by the 1198972-norm LS term
In Section 41 we havementioned recovering three recog-nition features by using measurement matrixes A
1 A2 and
A3 respectively However practically only using A
1as the
compressive measurement may meet the requirement ofrecovering all of the features The reason is that A
2and
A3differ in the dimension but are both designed with the
constraint of RIP property only From the other aspect theprimary requirement of constructing matrix A
1is also the
RIP condition
5 Modulation Recognition withthe Identification Features
Given a received communication signalmodulated byMFSKMPSK or MQAM we firstly get compressive samples usingmeasurement matrixes present in Section 2 In this processdue to difference of sparsity we have analyzed in Section 3various features may apply various length of the signal andthis can be decided based on actual situations According tothe approaches proposed above the identification featurescan be easily obtained Then we can recognize the modu-lation format effectively referring to the flowchart shown inFigure 5 and specific steps are listed in the following
Step 1 Reconstruct the spectrum feature when 120574 = 1 withcompressive samples If there is impulse in the recoveredspectrum the modulation mode can be identified as MFSKand the number of impulses indicates the order of it How-ever if there is no impulse in the feature the communicationsignal is modulated by MPSK or MQAM and then Step 2should be conducted
Step 2 Reconstruct the spectrum feature when 120574 = 2 4 8
with compressive samples and observe value of 120574 when theimpulse firstly appears If 120574 = 4 when the impulse appearsthe modulation mode can be regarded as QPSK or MQAM
8 Mobile Information Systems
MFSK MPSK and MQAM
Order of MPSK
Order of MFSK
MPSK and MQAM
QPSK and MQAM
QPSK and order ofMQAM
The spectrumNumber of
impulses
No impulse
The spectrum
Impulse appears when
Impulse appears
feature (120574 = 1)
when 120574 ne 4
feature (120574 ne 1)
120574 = 4
(120573 ne 4)
The high-ordermoment feature
Value range of 120572
Figure 5 The process of digital modulation recognition
and then we go to Step 3 However if 120574 = 4when the impulseappears the signal is modulated by MPSK and this value of 120574is the order of it
Step 3 Reconstruct R11991021
and R11991040
of the signal with com-pressive samples get average values of the diagonal as119872
21(119910)
and 11987240
(119910) respectively and then calculate 120572 based on (11)Compare 120572 with the calculated boundary values shown inTable 1 and determine the modulation type
6 Numerical Results
This section presents the simulation results of our feature-based recognition method We firstly generate a stream ofsignals modulated by MPSK MFSK or MQAM All the sig-nals share the same bit rate 1 kbits and the carrier frequency2 kHz and the carrier spacing for MFSK is 025 kHz Forthe two proposed features the observation time is variousbecause data volume needed by the two features are all
120574 = 1
120574 = 2
120574 = 4
120574 = 8
0
01
02
03
04
05
06
07
08
09
1
Cor
rect
det
ectio
n ra
te
84 102 60minus2minus4minus6minus8minus10
SNR
Figure 6 Correct detection rate of impulse in reconstructed feature1
differentThe performance of reconstruction is closely relatedto the signal-to-noise ratio (SNR) which is set as a variable inour simulation and simulations at every SNR are carried outfor 500 times
As mentioned above information we need to capturein feature 1 is whether there are impulses and the numberof them rather than accurate numerical values Thereforewe apply correct detection rate of pulse to evaluate theperformance of reconstruction of spectrum feature whichis shown in Figure 6 We set a decision threshold whichequals two-thirds of the biggest reconstructed value and ifthere is no other value larger than the threshold the biggestvalue would be regarded as the impulse In this scenario thecompressive ratio is set as 03 which means 119872119873 = 03We calculate the detection rate for MFSK signal on 120574 = 1BPSK on 120574 = 2 QPSK and MQAM on 120574 = 4 and 8PSKon 120574 = 8 respectively It is obvious that the detection ratevaries a lot with 120574 The reason is that 120574th power of signal is anonlinear transform meaning that the uniformly distributednoise ismagnified and the degree ofmagnification extends asthe increasing of 120574Therefore detection rate of impulse when120574 = 8 is the worst one
Figure 7 shows the mean square error (MSE) of recon-structed feature 2 with respect to the theoretical ones Thatis
MSE = 119864
1003817100381710038171003817100381710038171003817
vec S120574 minus vec S
120574
1003817100381710038171003817100381710038171003817
2
2
10038171003817100381710038171003817vec S
12057410038171003817100381710038171003817
2
2
(29)
We give the MSE of reconstructed R11991021
and R11991040
respec-tively with the compressive ratio chosen as 03 and 045 FromFigure 7 we can see that the performance of reconstructionof R11991040
is closely related to the compressive ratio while theperformance of reconstruction of R
11991021is relatively perfect
Mobile Information Systems 9
5 6 7 8 9 10 11 12 13 140
005
015
025
03
02
01
035
04
045
SNR (dB)
MSE
Ry21
Ry40
Ry21
Ry40
MN = 03
MN = 03
MN = 045MN = 045
Figure 7 MSE of reconstructed R11991021
and R11991040
with differentcompressive ratio
even at a low compressive ratio Moreover we can easily getthe conclusion that when the compressive ratio is suitable theprecision of feature 2 is high enough as long as the SNR ishigher than 10 dB
Figure 8 shows the correct classification rate of differentmodulation modes at relatively low SNR Difference of thecorrect classification comes from various performance ofreconstruction of features which has been shown in Figures6 and 7 MFSK has high recognition rate larger than 093
even when SNR = minus6 dB For MPSK the correct recognitionrate declines as 120573 increases However for QPSK andMQAMthe performance is quite different and we give the followinganalysis
According to [14] we have the fact that 11987240
of just thesignal and mixture of noise and signal are of the same valueso the main cause of the error comes from 119872
21
As for 11987221 we have the following proof stating the
variation of the value in noisy condition and noiselesscondition To describe this clearly 119872
21(1199100) 11987221
(V) and11987221
(119910) are respectively used to replace 11987221
while beingin the following condition of signal only noise only and themixture of noise and signal
11987221
(1199100) = 119864 (119910
0ℎ1199100ℎ
lowast
)
11987221
(V) = 119864 (VℎVℎ
lowast
)
11987221
(119910) = 119864 ((1199100ℎ
+ Vℎ) (1199100ℎ
+ Vℎ)lowast
)
= 119864 ((1199100ℎ
+ Vℎ) (1199100ℎ
lowast
+ Vℎ
lowast
))
= 119864 (1199100ℎ1199100ℎ
lowast
+ 1199100ℎ
lowastVℎ+ 1199100ℎVℎ
lowast
+ VℎVℎ
lowast
)
= 119864 (1199100ℎ1199100ℎ
lowast
) + 119864 (1199100ℎ
lowastVℎ) + 119864 (119910
0ℎVℎ
lowast
)
+ 119864 (VℎVℎ
lowast
)
(30)
0
01
02
03
04
05
06
07
08
09
1
SNR (dB)C
orre
ct re
cogn
ition
rate
minus10 minus5 0 5 10 15
MFSKBPSK8PSK
QPSK16QAM
Figure 8 Correct classification rate of different modulation modes
V is zero-mean random measure noises with Gaussiandistribution which is independent from 119910 According to thenature of expectation we know that
119864 (1199100ℎ
lowastVℎ) = 119864 (119910
0ℎ1199100ℎ
lowast
) = 0 (31)
Therefore we can obtain the following relationship
11987221
(119910) = 119864 (1199100ℎ1199100ℎ
lowast
) + 119864 (VℎVℎ
lowast
)
= 11987221
(1199100) + 119872
21(V)
(32)
meaning11987221
(119910) is the sum of signal power and noise powerFrom (11) and (27) we can obtain the relationship of the
where 119875V denotes noise power and 1198751199100
denotes signal powerTo sum up 119872
21(119910) is added by the power of noise and
as a consequence the identification parameter 120572 becomes
10 Mobile Information Systems
smaller thus QPSKmay be recognized as 16QAMThereforethe correct recognition rate of 16QAM is much higher thanQPSK when SNR is lower than 10 dB as shown in Figure 8
7 Conclusion
To solve the problem of high sampling rate for digital modu-lation recognition in spectrum sensing we have proposed afeature-based method to identify the modulation formats ofdigital modulated communication signals using compressivesamples and have greatly lowered the sampling rate basedon CS Two features are constructed in our method oneof which is the spectrum of signalrsquos 120574th power nonlineartransformation and the other is a composition of multiplehigh-order moments of the signal both with desired sparsityBy these two features we have applied suitable measurementmatrixes and built linear relationships referring to themThemethod successfully avoids reconstructing original signalsand uses recognition features to classify signals directlydeclining the algorithm complexity effectively Simulationsshow that correct recognition rates are different for differentmodulation types but are all relatively ideal even in noisy sce-narios In actual situations the method can be decomposedaiming at variable demands and for further work we tend toimprove the performance of the whole method continuouslyespecially the noise elimination in the classification of QPSKand MQAM
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by The China National NaturalScience Fund under Grants 61271181 and 61171109 and theJoint Project withChina Southwest Institute of Electronic andTelecommunication Technology
References
[1] H Bogucka P Kryszkiewicz and A Kliks ldquoDynamic spectrumaggregation for future 5G communicationsrdquo IEEE Communica-tions Magazine vol 53 no 5 pp 35ndash43 2015
[2] T Irnich J Kronander and Y Selen ldquoSpectrum sharing sce-narios and resulting technical requirements for 5G systemsrdquoin Proceedings of the IEEE 24th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCWorkshops rsquo13) pp 127ndash132 IEEE London UK September2013
[3] S Fengpan Research on Modulation Classification for Compres-sive Sensing in Cognitive Radio Ningbo University 2013
[4] O A Dobre A Abdi Y Bar-Ness and W Su ldquoSurveyof automatic modulation classification techniques classicalapproaches and new trendsrdquo IET Communications vol 1 no2 pp 137ndash156 2007
[5] F Wang and X Wang ldquoFast and robust modulation classi-fication via Kolmogorov-Smirnov testrdquo IEEE Transactions onCommunications vol 58 no 8 pp 2324ndash2332 2010
[6] E Cands ldquoCompressive samplingrdquo inProceedings of the Interna-tional Congress ofMathematicians vol 3 pp 1433ndash1452MadridSpain 2006
[7] E J Candes and M B Wakin ldquoAn introduction to compressivesamplingrdquo IEEE Signal Processing Magazine vol 25 no 2 pp21ndash30 2008
[8] Z Tian Y Tafesse and B M Sadler ldquoCyclic feature detectionwith sub-nyquist sampling for wideband spectrum sensingrdquoIEEE Journal on Selected Topics in Signal Processing vol 6 no 1pp 58ndash69 2012
[9] L Zhou and H Man ldquoDistributed automatic modulationclassification based on cyclic feature via compressive sensingrdquoin Proceedings of the IEEEMilitary Communications Conference(MILCOM rsquo13) pp 40ndash45 IEEE San Diego Calif USANovember 2013
[10] J Reichert ldquoAutomatic classification of communication signalsusing higher order statisticsrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo92) vol 5 pp 221ndash224 San Francisco Calif USAMarch 1992
[11] V Orlic and M L Dukic ldquoAlgorithm for automatic modula-tion classification in multipath channel based on sixth-ordercumulantsrdquo inProceedings of the 9th International Conference onTelecommunication inModern Satellite Cable and BroadcastingServices (TELSIKS rsquo09) pp 423ndash426 IEEE Nis Serbia October2009
[12] D C Chang and P K Shih ldquoCumulants-based modulationclassification technique in multipath fading channelsrdquo IETCommunications vol 9 no 6 pp 828ndash835 2015
[13] B Wang and L Ge ldquoA novel algorithm for identification ofOFDM signalrdquo in Proceedings of the International Conference onWireless Communications Networking and Mobile Computing(WCNM rsquo05) pp 261ndash264 September 2005
[14] D Grimaldi S Rapuano and G Truglia ldquoAn automatic digitalmodulation classifier for measurement on telecommunicationnetworksrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology ConferencemdashConference Record pp1711ndash1720 Sorrento Italy 2002
119864 (zz119867) = A2sdot 119864 (yy119867) sdot A
2
119867
(18)
We useR11991121
to represent119864(zz119867) simultaneously referto (12) and then get
R11991121
= A2sdot R11991021
sdot A2
119867
(19)
Next we apply the property vecUXV = (V119879 otimes
U)vecX to transform (19) to (20) It is worth notic-ing that A
2
119867
= A2
119879 for A is a real-value matrix
vec R11991121
= A2otimes A2vec R
11991021 = Ψvec R
11991021 (20)
where Ψ = A2otimes A2can be regarded as the sensing
matrix with the scale of 1198722 times 1198732
(ii) R11991040
since the sparsity degree of R11991040
is far fewerthan that of R
11991021 the dimension of signal needed
and scale of measurement can also be very low Werepresent the measurement for R
11991040as A3 while the
only difference of it from A2is the dimension
Similar to (17) there is
zz119879 = A3(yy119879)A
3
119879
(21)
Then according to vecUXV = (V119879 otimes U)vecX wecan transform the two-dimensional relationship intoone-dimensional relationship
vec zz119879 = A3otimes A3vec yy119879 (22)
We can obtain
vec zz119879 vec119879 zz119879
= (A3otimes A3) vec yy119879 vec119879 yy119879 (A
3otimes A3)
(23)
Take the average of both sides
119864 (vec zz119879 vec119879 zz119879)
= (A3otimes A3) 119864 (vec yy119879 vec119879 yy119879) (A
3otimes A3)
(24)
Based on (13) we get the relationship
R11991140
= (A3otimes A3)R11991040
(A3otimes A3) (25)
where R11991140
denotes 119864(veczz119879vec119879zz119879) And thenwe have
vec R11991140
= (A3otimes A3) otimes (A
3otimes A3) vec R
11991040
= Φvec R11991040
(26)
whereΦ = (A3otimesA3)otimes(A3otimesA3) is the sensingmatrix
42 Reconstruction of Identification Features z120574 R11991121
andR11991140
can be calculated by the sampling value z With sensingmatrixes and measurement vectors known the reconstruc-tion of the sparse vectors can be regarded as the signalrecovery problem by solving the NP-hard puzzle as followstaking R
11991021as an example
vec Ry21 = argmin 10038171003817100381710038171003817vec R
11991021100381710038171003817100381710038170
st vec R11991121
= Φ vec R11991121
(27)
This can be transformed into a linear programming problem
minvecRy21
10038171003817100381710038171003817vec R
11991121 minusΦ vec R
1199102110038171003817100381710038171003817
12
2
+ 11989410038171003817100381710038171003817vec R
11991021100381710038171003817100381710038171
(28)
which is called 1198971-norm least square programming problemand is proved to be convex that there exists a unique optimumsolution 119894 gt 0 is a weighting scalar that balances the sparsityof the solution induced by the 1198971-norm term and the datareconstruction error reflected by the 1198972-norm LS term
In Section 41 we havementioned recovering three recog-nition features by using measurement matrixes A
1 A2 and
A3 respectively However practically only using A
1as the
compressive measurement may meet the requirement ofrecovering all of the features The reason is that A
2and
A3differ in the dimension but are both designed with the
constraint of RIP property only From the other aspect theprimary requirement of constructing matrix A
1is also the
RIP condition
5 Modulation Recognition withthe Identification Features
Given a received communication signalmodulated byMFSKMPSK or MQAM we firstly get compressive samples usingmeasurement matrixes present in Section 2 In this processdue to difference of sparsity we have analyzed in Section 3various features may apply various length of the signal andthis can be decided based on actual situations According tothe approaches proposed above the identification featurescan be easily obtained Then we can recognize the modu-lation format effectively referring to the flowchart shown inFigure 5 and specific steps are listed in the following
Step 1 Reconstruct the spectrum feature when 120574 = 1 withcompressive samples If there is impulse in the recoveredspectrum the modulation mode can be identified as MFSKand the number of impulses indicates the order of it How-ever if there is no impulse in the feature the communicationsignal is modulated by MPSK or MQAM and then Step 2should be conducted
Step 2 Reconstruct the spectrum feature when 120574 = 2 4 8
with compressive samples and observe value of 120574 when theimpulse firstly appears If 120574 = 4 when the impulse appearsthe modulation mode can be regarded as QPSK or MQAM
8 Mobile Information Systems
MFSK MPSK and MQAM
Order of MPSK
Order of MFSK
MPSK and MQAM
QPSK and MQAM
QPSK and order ofMQAM
The spectrumNumber of
impulses
No impulse
The spectrum
Impulse appears when
Impulse appears
feature (120574 = 1)
when 120574 ne 4
feature (120574 ne 1)
120574 = 4
(120573 ne 4)
The high-ordermoment feature
Value range of 120572
Figure 5 The process of digital modulation recognition
and then we go to Step 3 However if 120574 = 4when the impulseappears the signal is modulated by MPSK and this value of 120574is the order of it
Step 3 Reconstruct R11991021
and R11991040
of the signal with com-pressive samples get average values of the diagonal as119872
21(119910)
and 11987240
(119910) respectively and then calculate 120572 based on (11)Compare 120572 with the calculated boundary values shown inTable 1 and determine the modulation type
6 Numerical Results
This section presents the simulation results of our feature-based recognition method We firstly generate a stream ofsignals modulated by MPSK MFSK or MQAM All the sig-nals share the same bit rate 1 kbits and the carrier frequency2 kHz and the carrier spacing for MFSK is 025 kHz Forthe two proposed features the observation time is variousbecause data volume needed by the two features are all
120574 = 1
120574 = 2
120574 = 4
120574 = 8
0
01
02
03
04
05
06
07
08
09
1
Cor
rect
det
ectio
n ra
te
84 102 60minus2minus4minus6minus8minus10
SNR
Figure 6 Correct detection rate of impulse in reconstructed feature1
differentThe performance of reconstruction is closely relatedto the signal-to-noise ratio (SNR) which is set as a variable inour simulation and simulations at every SNR are carried outfor 500 times
As mentioned above information we need to capturein feature 1 is whether there are impulses and the numberof them rather than accurate numerical values Thereforewe apply correct detection rate of pulse to evaluate theperformance of reconstruction of spectrum feature whichis shown in Figure 6 We set a decision threshold whichequals two-thirds of the biggest reconstructed value and ifthere is no other value larger than the threshold the biggestvalue would be regarded as the impulse In this scenario thecompressive ratio is set as 03 which means 119872119873 = 03We calculate the detection rate for MFSK signal on 120574 = 1BPSK on 120574 = 2 QPSK and MQAM on 120574 = 4 and 8PSKon 120574 = 8 respectively It is obvious that the detection ratevaries a lot with 120574 The reason is that 120574th power of signal is anonlinear transform meaning that the uniformly distributednoise ismagnified and the degree ofmagnification extends asthe increasing of 120574Therefore detection rate of impulse when120574 = 8 is the worst one
Figure 7 shows the mean square error (MSE) of recon-structed feature 2 with respect to the theoretical ones Thatis
MSE = 119864
1003817100381710038171003817100381710038171003817
vec S120574 minus vec S
120574
1003817100381710038171003817100381710038171003817
2
2
10038171003817100381710038171003817vec S
12057410038171003817100381710038171003817
2
2
(29)
We give the MSE of reconstructed R11991021
and R11991040
respec-tively with the compressive ratio chosen as 03 and 045 FromFigure 7 we can see that the performance of reconstructionof R11991040
is closely related to the compressive ratio while theperformance of reconstruction of R
11991021is relatively perfect
Mobile Information Systems 9
5 6 7 8 9 10 11 12 13 140
005
015
025
03
02
01
035
04
045
SNR (dB)
MSE
Ry21
Ry40
Ry21
Ry40
MN = 03
MN = 03
MN = 045MN = 045
Figure 7 MSE of reconstructed R11991021
and R11991040
with differentcompressive ratio
even at a low compressive ratio Moreover we can easily getthe conclusion that when the compressive ratio is suitable theprecision of feature 2 is high enough as long as the SNR ishigher than 10 dB
Figure 8 shows the correct classification rate of differentmodulation modes at relatively low SNR Difference of thecorrect classification comes from various performance ofreconstruction of features which has been shown in Figures6 and 7 MFSK has high recognition rate larger than 093
even when SNR = minus6 dB For MPSK the correct recognitionrate declines as 120573 increases However for QPSK andMQAMthe performance is quite different and we give the followinganalysis
According to [14] we have the fact that 11987240
of just thesignal and mixture of noise and signal are of the same valueso the main cause of the error comes from 119872
21
As for 11987221 we have the following proof stating the
variation of the value in noisy condition and noiselesscondition To describe this clearly 119872
21(1199100) 11987221
(V) and11987221
(119910) are respectively used to replace 11987221
while beingin the following condition of signal only noise only and themixture of noise and signal
11987221
(1199100) = 119864 (119910
0ℎ1199100ℎ
lowast
)
11987221
(V) = 119864 (VℎVℎ
lowast
)
11987221
(119910) = 119864 ((1199100ℎ
+ Vℎ) (1199100ℎ
+ Vℎ)lowast
)
= 119864 ((1199100ℎ
+ Vℎ) (1199100ℎ
lowast
+ Vℎ
lowast
))
= 119864 (1199100ℎ1199100ℎ
lowast
+ 1199100ℎ
lowastVℎ+ 1199100ℎVℎ
lowast
+ VℎVℎ
lowast
)
= 119864 (1199100ℎ1199100ℎ
lowast
) + 119864 (1199100ℎ
lowastVℎ) + 119864 (119910
0ℎVℎ
lowast
)
+ 119864 (VℎVℎ
lowast
)
(30)
0
01
02
03
04
05
06
07
08
09
1
SNR (dB)C
orre
ct re
cogn
ition
rate
minus10 minus5 0 5 10 15
MFSKBPSK8PSK
QPSK16QAM
Figure 8 Correct classification rate of different modulation modes
V is zero-mean random measure noises with Gaussiandistribution which is independent from 119910 According to thenature of expectation we know that
119864 (1199100ℎ
lowastVℎ) = 119864 (119910
0ℎ1199100ℎ
lowast
) = 0 (31)
Therefore we can obtain the following relationship
11987221
(119910) = 119864 (1199100ℎ1199100ℎ
lowast
) + 119864 (VℎVℎ
lowast
)
= 11987221
(1199100) + 119872
21(V)
(32)
meaning11987221
(119910) is the sum of signal power and noise powerFrom (11) and (27) we can obtain the relationship of the
where 119875V denotes noise power and 1198751199100
denotes signal powerTo sum up 119872
21(119910) is added by the power of noise and
as a consequence the identification parameter 120572 becomes
10 Mobile Information Systems
smaller thus QPSKmay be recognized as 16QAMThereforethe correct recognition rate of 16QAM is much higher thanQPSK when SNR is lower than 10 dB as shown in Figure 8
7 Conclusion
To solve the problem of high sampling rate for digital modu-lation recognition in spectrum sensing we have proposed afeature-based method to identify the modulation formats ofdigital modulated communication signals using compressivesamples and have greatly lowered the sampling rate basedon CS Two features are constructed in our method oneof which is the spectrum of signalrsquos 120574th power nonlineartransformation and the other is a composition of multiplehigh-order moments of the signal both with desired sparsityBy these two features we have applied suitable measurementmatrixes and built linear relationships referring to themThemethod successfully avoids reconstructing original signalsand uses recognition features to classify signals directlydeclining the algorithm complexity effectively Simulationsshow that correct recognition rates are different for differentmodulation types but are all relatively ideal even in noisy sce-narios In actual situations the method can be decomposedaiming at variable demands and for further work we tend toimprove the performance of the whole method continuouslyespecially the noise elimination in the classification of QPSKand MQAM
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by The China National NaturalScience Fund under Grants 61271181 and 61171109 and theJoint Project withChina Southwest Institute of Electronic andTelecommunication Technology
References
[1] H Bogucka P Kryszkiewicz and A Kliks ldquoDynamic spectrumaggregation for future 5G communicationsrdquo IEEE Communica-tions Magazine vol 53 no 5 pp 35ndash43 2015
[2] T Irnich J Kronander and Y Selen ldquoSpectrum sharing sce-narios and resulting technical requirements for 5G systemsrdquoin Proceedings of the IEEE 24th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCWorkshops rsquo13) pp 127ndash132 IEEE London UK September2013
[3] S Fengpan Research on Modulation Classification for Compres-sive Sensing in Cognitive Radio Ningbo University 2013
[4] O A Dobre A Abdi Y Bar-Ness and W Su ldquoSurveyof automatic modulation classification techniques classicalapproaches and new trendsrdquo IET Communications vol 1 no2 pp 137ndash156 2007
[5] F Wang and X Wang ldquoFast and robust modulation classi-fication via Kolmogorov-Smirnov testrdquo IEEE Transactions onCommunications vol 58 no 8 pp 2324ndash2332 2010
[6] E Cands ldquoCompressive samplingrdquo inProceedings of the Interna-tional Congress ofMathematicians vol 3 pp 1433ndash1452MadridSpain 2006
[7] E J Candes and M B Wakin ldquoAn introduction to compressivesamplingrdquo IEEE Signal Processing Magazine vol 25 no 2 pp21ndash30 2008
[8] Z Tian Y Tafesse and B M Sadler ldquoCyclic feature detectionwith sub-nyquist sampling for wideband spectrum sensingrdquoIEEE Journal on Selected Topics in Signal Processing vol 6 no 1pp 58ndash69 2012
[9] L Zhou and H Man ldquoDistributed automatic modulationclassification based on cyclic feature via compressive sensingrdquoin Proceedings of the IEEEMilitary Communications Conference(MILCOM rsquo13) pp 40ndash45 IEEE San Diego Calif USANovember 2013
[10] J Reichert ldquoAutomatic classification of communication signalsusing higher order statisticsrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo92) vol 5 pp 221ndash224 San Francisco Calif USAMarch 1992
[11] V Orlic and M L Dukic ldquoAlgorithm for automatic modula-tion classification in multipath channel based on sixth-ordercumulantsrdquo inProceedings of the 9th International Conference onTelecommunication inModern Satellite Cable and BroadcastingServices (TELSIKS rsquo09) pp 423ndash426 IEEE Nis Serbia October2009
[12] D C Chang and P K Shih ldquoCumulants-based modulationclassification technique in multipath fading channelsrdquo IETCommunications vol 9 no 6 pp 828ndash835 2015
[13] B Wang and L Ge ldquoA novel algorithm for identification ofOFDM signalrdquo in Proceedings of the International Conference onWireless Communications Networking and Mobile Computing(WCNM rsquo05) pp 261ndash264 September 2005
[14] D Grimaldi S Rapuano and G Truglia ldquoAn automatic digitalmodulation classifier for measurement on telecommunicationnetworksrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology ConferencemdashConference Record pp1711ndash1720 Sorrento Italy 2002
Figure 5 The process of digital modulation recognition
and then we go to Step 3 However if 120574 = 4when the impulseappears the signal is modulated by MPSK and this value of 120574is the order of it
Step 3 Reconstruct R11991021
and R11991040
of the signal with com-pressive samples get average values of the diagonal as119872
21(119910)
and 11987240
(119910) respectively and then calculate 120572 based on (11)Compare 120572 with the calculated boundary values shown inTable 1 and determine the modulation type
6 Numerical Results
This section presents the simulation results of our feature-based recognition method We firstly generate a stream ofsignals modulated by MPSK MFSK or MQAM All the sig-nals share the same bit rate 1 kbits and the carrier frequency2 kHz and the carrier spacing for MFSK is 025 kHz Forthe two proposed features the observation time is variousbecause data volume needed by the two features are all
120574 = 1
120574 = 2
120574 = 4
120574 = 8
0
01
02
03
04
05
06
07
08
09
1
Cor
rect
det
ectio
n ra
te
84 102 60minus2minus4minus6minus8minus10
SNR
Figure 6 Correct detection rate of impulse in reconstructed feature1
differentThe performance of reconstruction is closely relatedto the signal-to-noise ratio (SNR) which is set as a variable inour simulation and simulations at every SNR are carried outfor 500 times
As mentioned above information we need to capturein feature 1 is whether there are impulses and the numberof them rather than accurate numerical values Thereforewe apply correct detection rate of pulse to evaluate theperformance of reconstruction of spectrum feature whichis shown in Figure 6 We set a decision threshold whichequals two-thirds of the biggest reconstructed value and ifthere is no other value larger than the threshold the biggestvalue would be regarded as the impulse In this scenario thecompressive ratio is set as 03 which means 119872119873 = 03We calculate the detection rate for MFSK signal on 120574 = 1BPSK on 120574 = 2 QPSK and MQAM on 120574 = 4 and 8PSKon 120574 = 8 respectively It is obvious that the detection ratevaries a lot with 120574 The reason is that 120574th power of signal is anonlinear transform meaning that the uniformly distributednoise ismagnified and the degree ofmagnification extends asthe increasing of 120574Therefore detection rate of impulse when120574 = 8 is the worst one
Figure 7 shows the mean square error (MSE) of recon-structed feature 2 with respect to the theoretical ones Thatis
MSE = 119864
1003817100381710038171003817100381710038171003817
vec S120574 minus vec S
120574
1003817100381710038171003817100381710038171003817
2
2
10038171003817100381710038171003817vec S
12057410038171003817100381710038171003817
2
2
(29)
We give the MSE of reconstructed R11991021
and R11991040
respec-tively with the compressive ratio chosen as 03 and 045 FromFigure 7 we can see that the performance of reconstructionof R11991040
is closely related to the compressive ratio while theperformance of reconstruction of R
11991021is relatively perfect
Mobile Information Systems 9
5 6 7 8 9 10 11 12 13 140
005
015
025
03
02
01
035
04
045
SNR (dB)
MSE
Ry21
Ry40
Ry21
Ry40
MN = 03
MN = 03
MN = 045MN = 045
Figure 7 MSE of reconstructed R11991021
and R11991040
with differentcompressive ratio
even at a low compressive ratio Moreover we can easily getthe conclusion that when the compressive ratio is suitable theprecision of feature 2 is high enough as long as the SNR ishigher than 10 dB
Figure 8 shows the correct classification rate of differentmodulation modes at relatively low SNR Difference of thecorrect classification comes from various performance ofreconstruction of features which has been shown in Figures6 and 7 MFSK has high recognition rate larger than 093
even when SNR = minus6 dB For MPSK the correct recognitionrate declines as 120573 increases However for QPSK andMQAMthe performance is quite different and we give the followinganalysis
According to [14] we have the fact that 11987240
of just thesignal and mixture of noise and signal are of the same valueso the main cause of the error comes from 119872
21
As for 11987221 we have the following proof stating the
variation of the value in noisy condition and noiselesscondition To describe this clearly 119872
21(1199100) 11987221
(V) and11987221
(119910) are respectively used to replace 11987221
while beingin the following condition of signal only noise only and themixture of noise and signal
11987221
(1199100) = 119864 (119910
0ℎ1199100ℎ
lowast
)
11987221
(V) = 119864 (VℎVℎ
lowast
)
11987221
(119910) = 119864 ((1199100ℎ
+ Vℎ) (1199100ℎ
+ Vℎ)lowast
)
= 119864 ((1199100ℎ
+ Vℎ) (1199100ℎ
lowast
+ Vℎ
lowast
))
= 119864 (1199100ℎ1199100ℎ
lowast
+ 1199100ℎ
lowastVℎ+ 1199100ℎVℎ
lowast
+ VℎVℎ
lowast
)
= 119864 (1199100ℎ1199100ℎ
lowast
) + 119864 (1199100ℎ
lowastVℎ) + 119864 (119910
0ℎVℎ
lowast
)
+ 119864 (VℎVℎ
lowast
)
(30)
0
01
02
03
04
05
06
07
08
09
1
SNR (dB)C
orre
ct re
cogn
ition
rate
minus10 minus5 0 5 10 15
MFSKBPSK8PSK
QPSK16QAM
Figure 8 Correct classification rate of different modulation modes
V is zero-mean random measure noises with Gaussiandistribution which is independent from 119910 According to thenature of expectation we know that
119864 (1199100ℎ
lowastVℎ) = 119864 (119910
0ℎ1199100ℎ
lowast
) = 0 (31)
Therefore we can obtain the following relationship
11987221
(119910) = 119864 (1199100ℎ1199100ℎ
lowast
) + 119864 (VℎVℎ
lowast
)
= 11987221
(1199100) + 119872
21(V)
(32)
meaning11987221
(119910) is the sum of signal power and noise powerFrom (11) and (27) we can obtain the relationship of the
where 119875V denotes noise power and 1198751199100
denotes signal powerTo sum up 119872
21(119910) is added by the power of noise and
as a consequence the identification parameter 120572 becomes
10 Mobile Information Systems
smaller thus QPSKmay be recognized as 16QAMThereforethe correct recognition rate of 16QAM is much higher thanQPSK when SNR is lower than 10 dB as shown in Figure 8
7 Conclusion
To solve the problem of high sampling rate for digital modu-lation recognition in spectrum sensing we have proposed afeature-based method to identify the modulation formats ofdigital modulated communication signals using compressivesamples and have greatly lowered the sampling rate basedon CS Two features are constructed in our method oneof which is the spectrum of signalrsquos 120574th power nonlineartransformation and the other is a composition of multiplehigh-order moments of the signal both with desired sparsityBy these two features we have applied suitable measurementmatrixes and built linear relationships referring to themThemethod successfully avoids reconstructing original signalsand uses recognition features to classify signals directlydeclining the algorithm complexity effectively Simulationsshow that correct recognition rates are different for differentmodulation types but are all relatively ideal even in noisy sce-narios In actual situations the method can be decomposedaiming at variable demands and for further work we tend toimprove the performance of the whole method continuouslyespecially the noise elimination in the classification of QPSKand MQAM
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by The China National NaturalScience Fund under Grants 61271181 and 61171109 and theJoint Project withChina Southwest Institute of Electronic andTelecommunication Technology
References
[1] H Bogucka P Kryszkiewicz and A Kliks ldquoDynamic spectrumaggregation for future 5G communicationsrdquo IEEE Communica-tions Magazine vol 53 no 5 pp 35ndash43 2015
[2] T Irnich J Kronander and Y Selen ldquoSpectrum sharing sce-narios and resulting technical requirements for 5G systemsrdquoin Proceedings of the IEEE 24th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCWorkshops rsquo13) pp 127ndash132 IEEE London UK September2013
[3] S Fengpan Research on Modulation Classification for Compres-sive Sensing in Cognitive Radio Ningbo University 2013
[4] O A Dobre A Abdi Y Bar-Ness and W Su ldquoSurveyof automatic modulation classification techniques classicalapproaches and new trendsrdquo IET Communications vol 1 no2 pp 137ndash156 2007
[5] F Wang and X Wang ldquoFast and robust modulation classi-fication via Kolmogorov-Smirnov testrdquo IEEE Transactions onCommunications vol 58 no 8 pp 2324ndash2332 2010
[6] E Cands ldquoCompressive samplingrdquo inProceedings of the Interna-tional Congress ofMathematicians vol 3 pp 1433ndash1452MadridSpain 2006
[7] E J Candes and M B Wakin ldquoAn introduction to compressivesamplingrdquo IEEE Signal Processing Magazine vol 25 no 2 pp21ndash30 2008
[8] Z Tian Y Tafesse and B M Sadler ldquoCyclic feature detectionwith sub-nyquist sampling for wideband spectrum sensingrdquoIEEE Journal on Selected Topics in Signal Processing vol 6 no 1pp 58ndash69 2012
[9] L Zhou and H Man ldquoDistributed automatic modulationclassification based on cyclic feature via compressive sensingrdquoin Proceedings of the IEEEMilitary Communications Conference(MILCOM rsquo13) pp 40ndash45 IEEE San Diego Calif USANovember 2013
[10] J Reichert ldquoAutomatic classification of communication signalsusing higher order statisticsrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo92) vol 5 pp 221ndash224 San Francisco Calif USAMarch 1992
[11] V Orlic and M L Dukic ldquoAlgorithm for automatic modula-tion classification in multipath channel based on sixth-ordercumulantsrdquo inProceedings of the 9th International Conference onTelecommunication inModern Satellite Cable and BroadcastingServices (TELSIKS rsquo09) pp 423ndash426 IEEE Nis Serbia October2009
[12] D C Chang and P K Shih ldquoCumulants-based modulationclassification technique in multipath fading channelsrdquo IETCommunications vol 9 no 6 pp 828ndash835 2015
[13] B Wang and L Ge ldquoA novel algorithm for identification ofOFDM signalrdquo in Proceedings of the International Conference onWireless Communications Networking and Mobile Computing(WCNM rsquo05) pp 261ndash264 September 2005
[14] D Grimaldi S Rapuano and G Truglia ldquoAn automatic digitalmodulation classifier for measurement on telecommunicationnetworksrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology ConferencemdashConference Record pp1711ndash1720 Sorrento Italy 2002
even at a low compressive ratio Moreover we can easily getthe conclusion that when the compressive ratio is suitable theprecision of feature 2 is high enough as long as the SNR ishigher than 10 dB
Figure 8 shows the correct classification rate of differentmodulation modes at relatively low SNR Difference of thecorrect classification comes from various performance ofreconstruction of features which has been shown in Figures6 and 7 MFSK has high recognition rate larger than 093
even when SNR = minus6 dB For MPSK the correct recognitionrate declines as 120573 increases However for QPSK andMQAMthe performance is quite different and we give the followinganalysis
According to [14] we have the fact that 11987240
of just thesignal and mixture of noise and signal are of the same valueso the main cause of the error comes from 119872
21
As for 11987221 we have the following proof stating the
variation of the value in noisy condition and noiselesscondition To describe this clearly 119872
21(1199100) 11987221
(V) and11987221
(119910) are respectively used to replace 11987221
while beingin the following condition of signal only noise only and themixture of noise and signal
11987221
(1199100) = 119864 (119910
0ℎ1199100ℎ
lowast
)
11987221
(V) = 119864 (VℎVℎ
lowast
)
11987221
(119910) = 119864 ((1199100ℎ
+ Vℎ) (1199100ℎ
+ Vℎ)lowast
)
= 119864 ((1199100ℎ
+ Vℎ) (1199100ℎ
lowast
+ Vℎ
lowast
))
= 119864 (1199100ℎ1199100ℎ
lowast
+ 1199100ℎ
lowastVℎ+ 1199100ℎVℎ
lowast
+ VℎVℎ
lowast
)
= 119864 (1199100ℎ1199100ℎ
lowast
) + 119864 (1199100ℎ
lowastVℎ) + 119864 (119910
0ℎVℎ
lowast
)
+ 119864 (VℎVℎ
lowast
)
(30)
0
01
02
03
04
05
06
07
08
09
1
SNR (dB)C
orre
ct re
cogn
ition
rate
minus10 minus5 0 5 10 15
MFSKBPSK8PSK
QPSK16QAM
Figure 8 Correct classification rate of different modulation modes
V is zero-mean random measure noises with Gaussiandistribution which is independent from 119910 According to thenature of expectation we know that
119864 (1199100ℎ
lowastVℎ) = 119864 (119910
0ℎ1199100ℎ
lowast
) = 0 (31)
Therefore we can obtain the following relationship
11987221
(119910) = 119864 (1199100ℎ1199100ℎ
lowast
) + 119864 (VℎVℎ
lowast
)
= 11987221
(1199100) + 119872
21(V)
(32)
meaning11987221
(119910) is the sum of signal power and noise powerFrom (11) and (27) we can obtain the relationship of the
where 119875V denotes noise power and 1198751199100
denotes signal powerTo sum up 119872
21(119910) is added by the power of noise and
as a consequence the identification parameter 120572 becomes
10 Mobile Information Systems
smaller thus QPSKmay be recognized as 16QAMThereforethe correct recognition rate of 16QAM is much higher thanQPSK when SNR is lower than 10 dB as shown in Figure 8
7 Conclusion
To solve the problem of high sampling rate for digital modu-lation recognition in spectrum sensing we have proposed afeature-based method to identify the modulation formats ofdigital modulated communication signals using compressivesamples and have greatly lowered the sampling rate basedon CS Two features are constructed in our method oneof which is the spectrum of signalrsquos 120574th power nonlineartransformation and the other is a composition of multiplehigh-order moments of the signal both with desired sparsityBy these two features we have applied suitable measurementmatrixes and built linear relationships referring to themThemethod successfully avoids reconstructing original signalsand uses recognition features to classify signals directlydeclining the algorithm complexity effectively Simulationsshow that correct recognition rates are different for differentmodulation types but are all relatively ideal even in noisy sce-narios In actual situations the method can be decomposedaiming at variable demands and for further work we tend toimprove the performance of the whole method continuouslyespecially the noise elimination in the classification of QPSKand MQAM
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by The China National NaturalScience Fund under Grants 61271181 and 61171109 and theJoint Project withChina Southwest Institute of Electronic andTelecommunication Technology
References
[1] H Bogucka P Kryszkiewicz and A Kliks ldquoDynamic spectrumaggregation for future 5G communicationsrdquo IEEE Communica-tions Magazine vol 53 no 5 pp 35ndash43 2015
[2] T Irnich J Kronander and Y Selen ldquoSpectrum sharing sce-narios and resulting technical requirements for 5G systemsrdquoin Proceedings of the IEEE 24th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCWorkshops rsquo13) pp 127ndash132 IEEE London UK September2013
[3] S Fengpan Research on Modulation Classification for Compres-sive Sensing in Cognitive Radio Ningbo University 2013
[4] O A Dobre A Abdi Y Bar-Ness and W Su ldquoSurveyof automatic modulation classification techniques classicalapproaches and new trendsrdquo IET Communications vol 1 no2 pp 137ndash156 2007
[5] F Wang and X Wang ldquoFast and robust modulation classi-fication via Kolmogorov-Smirnov testrdquo IEEE Transactions onCommunications vol 58 no 8 pp 2324ndash2332 2010
[6] E Cands ldquoCompressive samplingrdquo inProceedings of the Interna-tional Congress ofMathematicians vol 3 pp 1433ndash1452MadridSpain 2006
[7] E J Candes and M B Wakin ldquoAn introduction to compressivesamplingrdquo IEEE Signal Processing Magazine vol 25 no 2 pp21ndash30 2008
[8] Z Tian Y Tafesse and B M Sadler ldquoCyclic feature detectionwith sub-nyquist sampling for wideband spectrum sensingrdquoIEEE Journal on Selected Topics in Signal Processing vol 6 no 1pp 58ndash69 2012
[9] L Zhou and H Man ldquoDistributed automatic modulationclassification based on cyclic feature via compressive sensingrdquoin Proceedings of the IEEEMilitary Communications Conference(MILCOM rsquo13) pp 40ndash45 IEEE San Diego Calif USANovember 2013
[10] J Reichert ldquoAutomatic classification of communication signalsusing higher order statisticsrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo92) vol 5 pp 221ndash224 San Francisco Calif USAMarch 1992
[11] V Orlic and M L Dukic ldquoAlgorithm for automatic modula-tion classification in multipath channel based on sixth-ordercumulantsrdquo inProceedings of the 9th International Conference onTelecommunication inModern Satellite Cable and BroadcastingServices (TELSIKS rsquo09) pp 423ndash426 IEEE Nis Serbia October2009
[12] D C Chang and P K Shih ldquoCumulants-based modulationclassification technique in multipath fading channelsrdquo IETCommunications vol 9 no 6 pp 828ndash835 2015
[13] B Wang and L Ge ldquoA novel algorithm for identification ofOFDM signalrdquo in Proceedings of the International Conference onWireless Communications Networking and Mobile Computing(WCNM rsquo05) pp 261ndash264 September 2005
[14] D Grimaldi S Rapuano and G Truglia ldquoAn automatic digitalmodulation classifier for measurement on telecommunicationnetworksrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology ConferencemdashConference Record pp1711ndash1720 Sorrento Italy 2002
smaller thus QPSKmay be recognized as 16QAMThereforethe correct recognition rate of 16QAM is much higher thanQPSK when SNR is lower than 10 dB as shown in Figure 8
7 Conclusion
To solve the problem of high sampling rate for digital modu-lation recognition in spectrum sensing we have proposed afeature-based method to identify the modulation formats ofdigital modulated communication signals using compressivesamples and have greatly lowered the sampling rate basedon CS Two features are constructed in our method oneof which is the spectrum of signalrsquos 120574th power nonlineartransformation and the other is a composition of multiplehigh-order moments of the signal both with desired sparsityBy these two features we have applied suitable measurementmatrixes and built linear relationships referring to themThemethod successfully avoids reconstructing original signalsand uses recognition features to classify signals directlydeclining the algorithm complexity effectively Simulationsshow that correct recognition rates are different for differentmodulation types but are all relatively ideal even in noisy sce-narios In actual situations the method can be decomposedaiming at variable demands and for further work we tend toimprove the performance of the whole method continuouslyespecially the noise elimination in the classification of QPSKand MQAM
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by The China National NaturalScience Fund under Grants 61271181 and 61171109 and theJoint Project withChina Southwest Institute of Electronic andTelecommunication Technology
References
[1] H Bogucka P Kryszkiewicz and A Kliks ldquoDynamic spectrumaggregation for future 5G communicationsrdquo IEEE Communica-tions Magazine vol 53 no 5 pp 35ndash43 2015
[2] T Irnich J Kronander and Y Selen ldquoSpectrum sharing sce-narios and resulting technical requirements for 5G systemsrdquoin Proceedings of the IEEE 24th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCWorkshops rsquo13) pp 127ndash132 IEEE London UK September2013
[3] S Fengpan Research on Modulation Classification for Compres-sive Sensing in Cognitive Radio Ningbo University 2013
[4] O A Dobre A Abdi Y Bar-Ness and W Su ldquoSurveyof automatic modulation classification techniques classicalapproaches and new trendsrdquo IET Communications vol 1 no2 pp 137ndash156 2007
[5] F Wang and X Wang ldquoFast and robust modulation classi-fication via Kolmogorov-Smirnov testrdquo IEEE Transactions onCommunications vol 58 no 8 pp 2324ndash2332 2010
[6] E Cands ldquoCompressive samplingrdquo inProceedings of the Interna-tional Congress ofMathematicians vol 3 pp 1433ndash1452MadridSpain 2006
[7] E J Candes and M B Wakin ldquoAn introduction to compressivesamplingrdquo IEEE Signal Processing Magazine vol 25 no 2 pp21ndash30 2008
[8] Z Tian Y Tafesse and B M Sadler ldquoCyclic feature detectionwith sub-nyquist sampling for wideband spectrum sensingrdquoIEEE Journal on Selected Topics in Signal Processing vol 6 no 1pp 58ndash69 2012
[9] L Zhou and H Man ldquoDistributed automatic modulationclassification based on cyclic feature via compressive sensingrdquoin Proceedings of the IEEEMilitary Communications Conference(MILCOM rsquo13) pp 40ndash45 IEEE San Diego Calif USANovember 2013
[10] J Reichert ldquoAutomatic classification of communication signalsusing higher order statisticsrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo92) vol 5 pp 221ndash224 San Francisco Calif USAMarch 1992
[11] V Orlic and M L Dukic ldquoAlgorithm for automatic modula-tion classification in multipath channel based on sixth-ordercumulantsrdquo inProceedings of the 9th International Conference onTelecommunication inModern Satellite Cable and BroadcastingServices (TELSIKS rsquo09) pp 423ndash426 IEEE Nis Serbia October2009
[12] D C Chang and P K Shih ldquoCumulants-based modulationclassification technique in multipath fading channelsrdquo IETCommunications vol 9 no 6 pp 828ndash835 2015
[13] B Wang and L Ge ldquoA novel algorithm for identification ofOFDM signalrdquo in Proceedings of the International Conference onWireless Communications Networking and Mobile Computing(WCNM rsquo05) pp 261ndash264 September 2005
[14] D Grimaldi S Rapuano and G Truglia ldquoAn automatic digitalmodulation classifier for measurement on telecommunicationnetworksrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology ConferencemdashConference Record pp1711ndash1720 Sorrento Italy 2002
