1.School of Aeronautics and Astronautics,University of Electronic Science and Technology of China,Chengdu 611731,China;2.School of Information Science and Engineering,Hangzhou Normal University,Hangzhou 311121,China;3.School of Electrical and Electronic Engineering,Nanyang Technological University,Singapore 639798,Singapore

Short-time Lv transform and its application for non-linear FM signal detection

Shan Luo1,Xiumei Li2,*,and Guoan Bi3

1.School of Aeronautics and Astronautics,University of Electronic Science and Technology of China,Chengdu 611731,China;
2.School of Information Science and Engineering,Hangzhou Normal University,Hangzhou 311121,China;
3.School of Electrical and Electronic Engineering,Nanyang Technological University,Singapore 639798,Singapore

A new time-frequency transform,known as short-time Lv transform(STLVT),is proposed by applying the inverse Lv distribution to process consecutive segments of long data sequence. Compared with other time-frequency representations,the STLVT is able to achieve better energy concentration in the time-frequency domain for signals containing multiple linear and/or non-linear frequency modulated components.The merits of the STLVT are demonstrated in terms of the effects of window length and overlap length between adjacent segments on signal energy concentration in the time-frequency domain,and the required computational complexity.An application on the spectrum sensing for cognitive ratio(CR)by using a joint use of the STLVT and Hough transform (HT)is proposed and simulated.

Lv distribution,time-frequency transform,frequency modulated signal,spectrum sensing.

1.Introduction

Many signals in geology,wireless communication and radar are non-stationary and belong to frequency modulated(FM)signals[1,2].It is widely recognizedthat to better characterize these signals,time-frequency transforms (TFTs),such as short-time Fourier transform(STFT)and Wigner-Ville distribution(WVD),are much more useful than the simple Fourier transform(FT)[3–9].Various approaches on time-frequencyanalysis have been proposed with advantages and shortcomings[10–15],e.g., the WVD has the best energy concentration but suffers from the cross-terms;the Hilbert-Huang transform[16] is only suitable for low noise corrupted signals;the independent component analysis based[17]one can detect in heavy noise but it does not accomplish the revealing of non-linear FM signals and costs much on computational complexity.

Recently,the Lv distribution(LVD)is proposed to effectively deal with the linear FM(LFM)signals[18].It canprovideaccurateinformationonthecentroidfrequency and chirp rate directly without using any searching process which is generally needed by other TFTs,such as, the fractional Fourier transform[19]and the local polynomial time-frequencytransform(LPTFT)[20,21].Based on the LVD,inverse LVD(ILVD)is also reported to generate the time-frequency representation(TFR)of LFM signals[18,22].Compared with other TFTs,the ILVD has a distinct capability of dealing with multiple LFM components with very small cross-terms among signal components in the time-frequency domain.Another report[23] shows the ILVD based TFR has a very good performance on the energyconcentration,which is better than the WVD and LPTFT when dealing with multi-componentLFM signals.However,the ILVD has difficultie in dealing with longdata sequence.Furthermore,it cannotbe directlyused to processnon-linearFM signals becausethe ILVD is valid only for signals having constant chirp rates.

Based onthe conceptthathas beenused inthe STFT,we proposethe short-time Lv transform(STLVT)in this paper by cascading the ILVD of consecutive segments of a long data sequence.In this way,the STLVT is able to reveal the time-frequency characteristics of both linear and nonlinear FM signals with reduced computational complexity.After providing the detailed procedures of segmenting the input data sequence,the performances in terms of signal concentration in the time-frequency domain achieved by the proposed STLVT,the smoothed pseudo WVD(SP-WVD)[24]and the LPTFT[21,22]are comparedbased on MonteCarlosimulationresults.Otherissues suchastheeffects on the signal concentration due to the types and the length of the segmentation window and the overlap length between adjacent segments are also discussed.Then,computational complexities required by the above mentioned three methods are compared.

Finally,an application on the spectrum sensing for the cognitive ratio(CR)communications by using a joint use of the STLVT and HT is proposed and simulated.Dealing with the wireless microphone(WM)signal,which is non-linear FM,an effective sensing scheme based on the STLVT-HT(SLHT)firstl produces the TFRs with sinusoid curves,then accumulates energy along the curves by using the generalized HT,and finall detects peaks on the Hough plane.Discussion and comparison are also presented.

This paper is organized as follows.Section 2 provides a brief review on the LVD and ILVD.In Section 3,the STLVT is proposed and defined Section 4 provides the performances of the STLVT,and compares it with other methods in terms of distribution concentration rate and required computational complexity.The application on the spectrumsensingfornon-linearFM signaldetectionis presented in Section 5.Finally conclusions are drawn in Section 6.

2.Background review

A signal,x(t),containing multiple LFM components,is expressed as

where K is the number of LFM components,Ak,fkand γkdenote the constant amplitude,the centroid frequency and the chirp rate of the kth component,respectively,and v(t)is the additive white Gaussian noise.The parametric symmetric instantaneousautocorrelationfunction(PSIAF) of x(t)[18]is define as

where a denotes a constant time-delay,is the crossterm between the ith and jth signal components,oris the cross-term between the kth signal component and the noise,andrepresents the noise term.

In(2),the time variable t and lag variable τ are coupled with each other in the exponential phase of the signal auto-term,i.e.,j2πγk(τ+a)t.Thisleads totheblurredrepresentation of the LFM components on the time-frequency plane since f and γ are also coupled together.It is necessary to decouple t and τ in the phase so that f and γ have no effect on each other when the FTs in terms of t and τ, respectively,are performed.This can be done by a scaling operation on a phase function G define as

where h is a scaling factor.Let us perform the scaling operation on the PSIAF in(2)to obtain

wheret andτ aredecoupledinthe phase.Afterperforming the FTs in terms of τ and tnin(4),respectively,we have

where F{·}means the FT,and the signal auto-term Lskis expressed as

Equation(5)is called the LVD and Lx(f,γ)is obtained in the frequency and chirp rate(FCR)domain.From(6), it is seen that f and γ are used in different delta functions. Therefore,the parameters of each LFM component,i.e., the centroid frequency and chirp rate,are obtained by thelocation,i.e.,(fk,γk),of the corresponding peak in the FCR domain.The parameters a=1 and h=1 are preferred to obtain a desirable FCR representation[18].

For a signal containing multiple LFM components,the LVD has a property of asymptotic linearity because the cross-terms of these components are trivial compared with their auto-terms.We therefore have

Assuming that x(t)is represented by a sequence of N points,an N/2×N/2 data matrix is produced by(2)in terms of both t and τ.It is noted that the output of(2)is not valid if x(t)is shifted by more than N/2 samples.The scaling operationon t in(2)is performedby using a scaled FT define as

where η=(2mTs+a)h/N and Tsis the sampling interval of x(t).The scaling operation in(4)is equivalently multiplying an exponential term to x(t).Other scaling techniques for time-frequencydistribution can be found in literature such as[25].

As a TFR,the ILVD of x(t)is obtained[19]by

where F?1{·},Γ?1[·]and M[·]represent the inverse FT, inverse scaling transform and masking operation,respectively.The masking operator is define as

where Xkdenotes the support of the auto-term of the kth component in the FCR domain.The ILVD has very good energy concentration in the time-frequency domain and suffers slightly from noise because the masking operation define in(10)covers only the concentrated area of signal components.

3.STLVT

Similar to most transforms that require a block of input data,the ILVD is not suitable to deal with long data sequence due to the limited memory size of the computing devices.Therefore,the ILVD cannot be directly used to deal with continuousdata streams.Moreover,the LVD and ILVD are not appropriateto process non-linear FM signals since(2)is valid only for LFM signals.To avoid these limitations for practical applications,the similar concept to that used in the STFT is applied to cascade the processed segments to formulate the time-frequency distribution of the signal.In particular,this practice allows us to process non-linear FM signals by assuming the chirp rates of the FM components in each short segment are constant.With this arrangement,we can obtain an approximated ILVD of the input signal by setting an appropriatelength of the data segments and the length of overlapping between the adjacent segments.Let us name this processing method as STLVT.

To fin the length of the overlapping between the adjacent segments,let us assume each segment contains N samples.The PSIAF function define in(2)requires the delay operation of the input segments.As the delay,τ,increases,x(t+(τ+a)/2)and x?(t?(τ+a)/2)are shifted into opposite directions,respectively,and there are only N/2 valid output samples producedfor each data segment. Therefore,the overlap length between two adjacent segments should be N?L,where L≤N/2 is the number of valid output samples used to form the fina TFR.Therefore,this segmentation process is described by

where the data stream x has an increasing index starting from 0,xqis the qth segment,and h[n](0≤n＜N)is the window function.Fig.1 shows the input segmentation process and the output cascading process of the STLVT. The selection of L and N values depends on the time resolutionandfrequencyresolutionneededbythe applications.

Fig.1 Segmentation example for the STLVT(L＜N/2)

The computational complexityrequired by the LVD has been reported to be in the order of[19] by assuming the FFT requires computation complexity of O(N log2N)[26].Let us consider the number of complex multiplications required by the LVD and ILVD basedon the FFT requiringcomplex multiplications. According to the definition in(2)–(5),the LVD requires

(i)N(N+2)/4 complex multiplications in(2);

(ii)N/2×N/2 complex multiplications in(4);

The main computation steps of STLVT are as follows:

Step 1Select appropriate values ofNandLfor the data stream according to required resolution and computational complexity;

Step 2Implement the segmentation process according to(11);

Step 3Do the ILVD for eachN-length segment,and only keep the middleL-length output;

Step4CascadeeachL-datatobethefina outputwhich is a TFR by the STLVT.

4.Experimental results

In this section,simulated experimental results are presented and discussed.The LFM signal follows the form in(1)and is denoted assLFM(t)withK=3,Ak=1 for allk,centroid frequenciesf1=?2 Hz,f2=10 Hz andf3=0 Hz,chirp ratesγ1=6.5 Hz/s,γ2=5 Hz/s andγ3=?6.25 Hz/s and sampling frequencyfs=128 Hz.

Fig.2(a),(b)and(c)present the TFRs ofsLFM(t)obtained by the proposed STLVT withN=128,512 and 1 024,respectively.It is seen that the STLVT provides a TFR with excellent signal concentration without obvious cross-terms among different components.Due to the window effect,a shorter window length(N=128)leads to poorer signal concentration compared with that withN=1 024.

The proposed STLVT is also used to deal with signals that have non-linear FM components.As an example,let us consider the sinusoidal FM signal define by

where the centroid frequencyfc=4 Hz,frequency deviation constantkf=28.647 9 Hz,modulating frequencyfm=0.1 Hz,andfs=128 Hz.Fig.2(d)presents a well concentrated TFR of the signal containing an LFM component and a sinusoidal FM component.The LVD-based transformresult of the same signal in Fig.2(d)is presented in Fig.3,showing only one peak which is related to the LFM component.It is indicated the sinusoidal FM component is not concentrated in the LVD plane since the LVD is not able to deal with non-linear FM signals.In general, for the window selecting in STLVT,the faster the IF of the signal changes,the shorter window length is used to meet the assumption that the chirp rates of the signal segments are constant.

Fig.2 TFRs of the multi-component signals based on STLVT with different segment lengths

Fig.3 LVD-based transform result of the example containing LFM and sinusoidal FM components

Let us compare the performances of signal concentration in the time-frequency domain obtained by the STLVT,SPWVD,LPTFT and local polynomial periodogram(LPP)[21].(The LPP basically is an energy form of the LPTFT,therefore it also has cross-terms as the LPTFT.)The last three methods have been recently reported in the literature to obtain higher signal concentration of multi-component signals in the time-frequency domain.Because other well known methods,such as theSTFT,short-time fractional Fourier transform,and pseudo WVD[21,27]have obvious problems for dealing with multi-component signals in low SNR environments,performance comparisons with these methods are not presented.

Followingthe conceptused in[28],the distributionconcentrationrate(DCR)is define tomeasurethesignal-term energy concentration in the time-frequency domain:

whereave[?]2means the average operation,Sdenotes the instantaneous frequency region of the signal in the timefrequency domain.

Fig.4 presents the DCR values of the multi-component LFM signalsLFM(t)achieved by the SPWVD,LPTFT, LPP and the proposed STLVT.In this simulation,the LPTFT,LPP and STLVT methodsuse the normalizedrectangular window for segmenting the input data stream and the lengths of the time and frequency windows used by the SPWVD method are 41 and 165,respectively,to obtain the highest DCR value.Fig.4 shows that the DCRs obtained by the proposed STLVT withN=256 and 512 are larger than those achieved by the LPTFT,LPP and SPWVD.In particular,the STLVT has achieved high DCR values when the SNR is as low as–5 dB,while the DCRs achieved by other methods decline when SNR≤2 dB. Fig.4 also shows that the use of longer window achieves higher DCRs for our synthesized signal.

Fig.4 DCRs of the noise-corruptedsLFM(t)obtained by different methods with different window lengths

Another issue is the effects of window types on the DCR values.Fig.5 shows the DCR values obtained by the STLVT using different types of windows,such as Gaussian,Hamming and rectangular windows of the same length,i.e.,N= 512 andL=N/2.It is observed that the DCR values are related to the main lobe widths of the windows.For example,the Gaussian window withα=2.5 has the widest main lob width in the frequency domain among the four windows used in our simulation and achieves the lowest DCR,while the rectangular window has the most narrow main lobe width and obtains the highest DCR.Therefore,the rectangular window is generally preferred to obtain the best possible DCRs without requiring any computation for the segmentation process.

Fig.5 DCRs of the noise-corruptedsLFM(t)obtained by STLVT with different window types and overlap lengths

The last important issue to be considered is the length of the overlapping between the input segments.It is generally true that the signal concentration in the timefrequency domain can be improved substantially by using those output samples containing the information most relevant to the input signal,which can be done by increasing the overlapping lengthN?L.Fig.5 also compares the DCRs obtained by using the rectangular windows with different lengths of overlapping.It is seen that substantial increase in DCRs is obtained whenLis deduced fromN/2 toN/4(the overlap length is equivalently increased fromN/2 to 3N/4).However,marginal DCR increase is achievedifLis furtherdecreased,forexample,from3N/8 toN/4.It means that using too much overlapbetween segments is not necessary to achieve higher DCRs.Similar situations are shown for the Gaussian and Hamming windows.

It is shown in Section 3 that the STLVT requirescomplex multiplications.The num-ber of complex multiplications required by the LPTFT isIn addition,the LPTFT also needs parameter estimation that substantially increases the computational complexity.In general,the computational complexities required by the LPTFT and the proposed STLVT are in the same order,i.e.,According to [24],the number of complex multiplications required by the SPWVD isdenote the timeandfrequencysmoothingwindowlengths,respectively.In the simulations for Fig.4,the STLVT requires less computational time compared with those needed by the LPTFT and SPWVD.

According to the above experiments,we can summarize the performance of STLVT.It is able to present a well concentrated TFR of a signal containing linear and nonlinear FM components without cross-terms.Based on the concept of DCR,the concentration of STLVT are much better than both the LPTFT and SPWVD.Moreover,the STLVT does not require more computational complexity than the LPTFT and SPWVD.Also,the use of longer window achieves better performance of concentration.Different window types lead to different performances as the reason mainly depends on the main lobe width,and the rectangular window obtains better concentration.Finally, the DCR would increase when the overlaps increase,until some marginal value such as L=N/4 in simulation.

5.Application on spectrum sensing for CR

As the STLVT has good ability to reveal a linear or nonlinear frequency modulated signal with multiple components in time-frequency domain with high signal energy concentration,thissectionwillemployitonspectrumsensingofWM signalsforthe CR system.In[29],we proposed and tested an effective method to sense the WM signals by using a combination scheme of the LPP and Hough transform(HT)[30,31],or called the LPP-HT(LHT).It performs very well in sensing single-component WM signals in negative SNR environments,better than some famous methods such as the cyclostationary based sensing. However,as we discussed in[22],the LPP would produce cross-terms that degrade the detection when multicomponent signal inputs.The method based on the LHT may fail to sense the WM signals when several microphones work at the same time.This case is probably to happen because many scenarios have several addressors or performers.As the concentration comparison in Fig.4 implies that the STLVT concentrates are better than the LPP mainly because of no cross-term,we employ it instead of the LPP to sense multi-componentWM signals.Since signaldetectingdirectlyintime-frequencydomainis complex and ineffective,a more proper method is detecting it in an integrated domain,such as the domain of HT.Many good examples have been reported in[27,29,32,33]which show that frequencyvaryingsignals can be detected easily in the HT domain.Therefore we adopt this concept to detect the WM signals by a combination of the STLVT-HT(SLHT), i.e.,we use the STLVT to produce the TFR of received WM signals,then perform the HT on this TFR and detect in the HT domain.The following contents will firstl introduce the backgrounds of CR,spectrum sensing,WM signals and generalized HT,then detection based on the SLHT is presented and simulated.

5.1 Backgrounds

Spectrum sensing is a crucial step in CR system to fin out available spectrum holes,which mean the absence of primary signals,for CR communications by using proper signaldetectionmethods[34–37].Since the primaryusers are licensed users while CR users are not,the CR system should be capable of deciding whether the primary signals exist or not in very low SNRs,such as＜?10 dB.Obviously,signal detection in such heavy noising environments plays an important role in spectrum sensing.

The WM signal is oneof the licensedsignals in the local TV bands,which have been permitted for CR communications on a basis of IEEE 802.22[38].They are generated by frequency modulation and has complex time-frequency features,resulting that many existing sensing approaches such as power spectral density and cyclostationary feature based methods may not be available to detect them when SNR is smallerthan?10dB[39–41].As theirfrequencies changewithtime,a suitabletime-frequencytechniqueusuallycanrevealanddetectthemeffectively[20,42].Herewe will propose a new spectrum sensing method for the WM signals based on the joint use of STLVT and HT,which is known as SLHT.

As a well-known pattern detector in image processing,the HT is a one-to-many mapping method from a data plane to a parameter plane,and converts the difficul problem of global detection in the data space to the easy problem of local peak detection in the parameter space[30,43,44].The concept of line detection by the HT has been applied to fin the LFM signals from the time-frequencyrepresentationproducedby the STFT[32], WVD[33]and LPP[27].It is shown that the HT is able to furtheraccumulatethe signal energyby the line integration operationalong the direction of the LFM signal in the time frequency domain.For the non-linear FM signals,such as the WM signals,a generalized HT is employed.It is able to detect any parameterizablecurves such as circles and ellipse[31].For example,for circle detection,a circle can be parameterized aswhere the parameters arex0,y0andR.Therefore we can construct a 3D accumulatorA(x0,y0,R).For each point (x0,y0),the correspondingRis computed by using the above circle equation,and matrixAis updated.A searching operation inAis then made to fin the peaks.Therefore,any curve can be detected based on the HT as long as the curve equation is obtained.As a 3D matrix is used in the algorithm,the complexity and computation time of the simulation is inevitably increased.

5.2Detection procedure

In this paper,the WM signals are detected based on the joint use of STLVT and HT,i.e.,SLHT,by the following steps:

Step 1Compute the representation of the WM signals in the time-frequency domain by using the STLVT;

Step 2Convert the representation obtained by the previous step into the parameterdomainobtainedby usingthe generalized HT;

Step 3Detect the peaks in the parameter domain(or Houghplane),obtain the coordinatesof the selected peaks, compare with the parameters of WM signals and make the sensing decision.

Reference[29]detects the WM signals based on the constantfalse alarmrate(CFAR)principlethat it firs gives a false alarm probability,then computes a corresponding threshold,compares with the peak in Hough plane and makes fina decision.The CFAR approach is a common scheme in radar detection.However,it may be not appropriate for the CR application.The CR system is different from radars,where it must guarantee the communication of primary users.That means,a CR user should make the detection probability as high as possible,no matter how large the false alarm probability is.There are two reasons, oneis that the primaryusers arelicensedusers so theyhave the right to communicate in their bands while the CR users are actually not;the other is that the high false alarm rate would not cost much for CR users,because they only need to fin another spectrum hole.Therefore in this section, we abandon the CFAR detection and use a more effective method,which firs searches the peaks in the entire Hough plane,obtains their coordinates information,then compares with the parameters of WM signals(usually the CR usersareassumedtohaveaprioriknowledgeoftheprimary signals when they are attempting to access the channels),and finall makes the decision of“primary user exists”if the information of peaks fit that of WM signals.

5.3Simulations

Experimental results will be presented in this section.A

WM signal is described as

where the term of cos(2πfmτ),as a modulating signal, represents the voice signal.The frequency deviationkfis the frequencysensitivity ofthe modulatorandfcis the carrier frequency.In this paper,fc=40 MHz which may be allocated for TV services.Basically there are three types of WM signals reported in[38]as follows:

(i)Silent:The modulatingfrequencyfm=32 KHz andkfis±5 KHz.The carrier signal is tuned so that it falls within any available TV channel;

(ii)Soft-speaker:fm=3.9 KHz,kf=±15 KHz;

(iii)Loud-speaker:fm=13.4 KHz,kf=±32.6 KHz.

From the signal function in(14),the IF of the WM signal is a sinusoid function described by

where the values ofkfandfmdepend on the three situations mentioned above.As we have seen in Fig.2(d),the STLVT can be applied to reveal the TFR of WM signals by assuming it in each short segment.

Fig.6 shows the STLVT and LPP of a two-component WM signal containing silent and loud situations by using segmentlengthN=512,andtheirHTresults.It is clearto see cross-terms appear in the LPP plane,while none in the STLVT plane,resulting that the two peaks in SLHT domain are more concentrated and easy to be detected than those in the LHT domain.

Fig.6 Transform resultsfor the two-component WMsignal containing silent and loud situations without noise

Noise-corruptedresults are further presented in Fig.7 at SNR=?14 dB,indicating that the SLHT based method can sense these two components while the LHT based onefails in such low SNR situation.A DCR comparison of the STLVT and LPP is shown in Table 1,and the STLVT concentrates better than the LPP,which is implied in the TFRs in Figs.6 and 7.Except the two-componentone,the DCRs for signal with three components also have the same performance,indicating that the SLHT based sensing method would perform better than the LHT for more microphones.

Fig.7 Transform results for thetwo-component WMsignalcontaining silent and loud situations at SNR=?14 dB

Table 1 Comparison of DCRs based on STLVT and LPP for WM signals

Fig.8 Detection probability of the two WM components signal (silent+loud)based on the energy detection,LHT and SLHT with 512 samples

Next a detection simulation based on the Monte Carlo trials will be implemented.Fig.8 presents the detection probability of the two-component signal having silent and loud cases by using the ED,LHT and SLHT.The ED approach[45]is introducedas a basic sensing method,which is used as a comparison reference.It is seen that the SLHT can sense the WM signal in heavier noise than the LHT and ED,thanks to the high performance on signal concentration of the STLVT when dealing with multi-component signals.The SLHT could be consideredas a robust sensing method against strong noise.

6.Conclusions

This paper presents the STLVT as a new time-frequency analysis method.Compared with other methods,the STLVT is particularly useful to deal with signals that contain multiple FM components in low SNR environments. Our simulation results show that the STLVT has achieved better signal concentration in the time-frequency domain with less computational complexity.An application of the joint use of STLVT and HT on spectrum sensing for CR is presented.It shows that our scheme can work at very low SNR,and can achieve better performance than the LHT-based sensing.

[1]M.Dorfle.Time-frequency analysis for music signals:a mathematical approach.Journal of New Music Research, 2001,30(1):3–12.

[2]R.J.McAulay,T.F.Quatieri.Pitch estimation and voicing detection based on a sinusoidal speech model.Proc.of the IEEE International Conference on Acoustics,Speech,and Signal Processing,1990:249–252.

[3]L.Cohen.Time-frequency analysis.Upper Saddle River,NJ: Prentice Hall,1995.

[4]S.Stankovic,I.Djurovic,I.Pitas.Watermarking in the space/spatial-frequency domain using two-dimensional Radon-Wigner distribution.IEEE Trans.on Image Processing,2001,10(4):650–658.

[5]K.Muneeswaran,L.Ganesan,S.Arumugam,et al.Texture image segmentation using combined features from spatial and spectral distribution.Pattern Recognition Letters,2006,27(7): 755–764.

[6]V.C.Chen,S.Qian.Joint time-frequency transform for radar range-Doppler imaging.IEEE Trans.on Aerospace and Electronic Systems,1998,34(2):486–499.

[7]K.Kim,I.Choi,H.Kim.Efficien radar target classificatio using adaptive joint time-frequency processing.IEEE Trans. on Antennas and Propagation,2000,48(12):1789–1801.

[8]X.Ouyang,M.G.Amin.Short-time Fourier transform receiver for nonstationary interference excision in direct sequence spread spectrum communications.IEEE Trans.on Signal Processing,2001,49(4):851–863.

[9]J.Alm,J.Walker.Time-frequency analysis of musical instruments.Society for Industrial and Applied Mathematics Review,2002,44(3):457–476.

[10]X.Xia.Discrete chirp-Fourier transform and its application to chirp rate estimation.IEEE Trans.on Signal Processing,2000, 48(11):3122–3133.

[11]Y.Wei,G.Bi.Efficien analysis of time-varying multicomponent signals with modifie LPTFT.EURASIP Journal on Applied Signal Processing,2005,2005(1):1261–1268.

[12]V.Namias.The fractional order fourier transform and itsapplication to quantum mechanics.IMA Journal of Applied Mathematics,1980,25(3):241–265.

[13]M.Z.Ikram,K.Abed-Meraim,Y.Hua.Fast quadratic phase transform for estimating the parameters of multicomponent chirp signals.DigitalSignalProcessing,1997,7(2):127–135.

[14]L.B.Almeida.The fractional fourier transform and timefrequency representations.IEEE Trans.on Signal Processing, 1994,42(11):3084–3091.

[15]H.M.Ozaktas,O.Arikan,M.A.Kutay,et al.Digitalcomputation of the fractional Fourier transform.IEEE Trans.on Signal Processing,1996,44(9):2141–2150.

[16]N.E.Huang,Z.Shen,S.Long,et al.The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis.Proceedings of the Royal Society,1998,454:903–995.

[17]Q.Guo,Y.Li,C.Wang.A new method of detecting multicomponent LFM signals based on blind signal processing.Journal of Computers,2011,6(9):1976–1982.

[18]X.L¨u,G.Bi,C.Wan,et al.Lv’s distribution:principle,implementation,properties and performance.IEEE Trans.on Signal Processing,2011,59(8):3576–3591.

[19]E.Sejdic,I.Djurovic,L.Stankovick.Fractional Fourier transform as a signal processing tool:an overview of recent developments.Signal Processing,2011,91(6):1351–1369.

[20]V.Katkovnik.A new form of the Fourier transform for timevarying frequency estimation.Signal Processing,1995,47(2): 187–200.

[21]X.Li,G.Bi,S.Stankovic,et al.Local polynomial Fourier transform:a review on recent developments and applications. Signal Processing,2011,91(6):1370–1393.

[22]S.Luo,X.L¨u,G.Bi.Lv’s distribution for time-frequency analysis.Proc.of the 2nd International Conference on Circuits, Systems,Control,Signals,2011:110–115.

[23]S.Luo,G.Bi,X.L¨u,et al.Performance analysis on Lv distribution and its application.Digital Signal Processing,2013, 23(3):797–807.

[24]N.Ma,J.Wang.An adaptive method of parameters selection for SPWVD.Proc.of the Third International Conference on Measuring Technology and Mechatronics Automation,2011: 363–366.

[25]L.Stankovic.Highly concentrated time-frequency distributions:pseudo quantum signal representation.IEEE Trans.on Signal Processing,1997,45(3):543–551.

[26]G.Bi,Y.Chen.Fast generalized DFT and DHT algorithms. Signal Processing,1998,65(3):383–390.

[27]G.Bi,X.Li,C.See.LFM signal detection using LPP-Hough transform.Signal Processing,2011,91(6):1432–1443.

[28]S.Zabin,H.Poor.Efficien estimation of class a noise parameters viatheEMalgorithm.IEEE Trans.onInformation Theory, 1991,37(1):60–72.

[29]S.Luo,G.Bi.Spectrum sensing of wireless microphone signals based on LHT.Proc.of the 8th International Conference on Information,Communications and Signal Processing, 2011:1–5.

[30]P.Hough.A method and means for recognizing complex patterns.U.S.Patent 3069654,1962.

[31]D.Ballard.Generalizing the Hough transform to detect arbitrary shapes.Pattern Recognition,1981,13(2):111–122.

[32]Y.Sun,P.Willett.Hough transform for long chirp detection.IEEE Trans.on Aerospace and Electronic Systems,2002, 38(2):553–569.

[33]S.Barbarossa.Analysis of multicomponent LFM signals by a combined Wigner-Hough transform.IEEE Trans.on Signal Processing,1995,43(6):1511–1515.

[34]J.Mitola,J.G.Q.Maguire.Cognitive radio:making software radios more personal.IEEE Personal Communications,1999, 6(4):13–18.

[35]J.Mitola.Cognitive radio:an integrated agent architecture for software define radio.Stockholm,Sweden:Royal Institute Technology,2000.

[36]A.Ghasemi,E.S.Sousa.Spectrum sensing in cognitive radio networks:requirements,challenges and design trade-offs. IEEE Communications Magazine,2008,46(4):32–39.

[37]I.F.Akyildiz,W.Lee,M.C.Vuran,et al.A survey on spectrum management in cognitive radio networks.IEEE Communications Magazine,2008,46(4):40–48.

[38]M.Kenkel,C.Clanton,Y.Tang.Wireless microphone signal simulation method.IEEE 802.22-07/0124r0,2007.

[39]H.Chen,W.Gao,D.G.Daut.Spectrum sensing for wireless microphone signals.Proc.of the IEEE Annual Communications Society Conference on Sensor,Mesh and Ad Hoc Communications and Networks Workshops,2008:1–5.

[40]A.Mossa,V.Jeoti.Cyclostationarity-based spectrum sensing for analog TV and wireless microphone signals.Proc.of the International Conference on Computational Intelligence, Communication Systems and Networks,2009:380–385.

[41]B.Adoum,V.Jeoti.Cyclostationary feature based multiresolution spectrum sensing approach for DVB-T and wireless microphone signals.Proc.of the International Conference on Computer and Communication Engineering,2010:1–6.

[42]F.Zhang,Y.Chen,G.Bi.Adaptiveharmonic fractional Fourier transform.IEEE Signal Processing Letters,1999,6(11):281–283.

[43]J.Illingworth,J.Kittler.A survey of the Hough transform. Computer Vision,Graphics,and Image Processing,1988, 44(1):87–116.

[44]R.Duda,P.Hart.Use of the Hough transformation to detect the lines and curves in pictures.Communications of the ACM, 1972,15(1):11–15.

[45]H.Urkowitz.Energy detection of unknown deterministic signals.Proceedings of the IEEE,1967,55(4):523–531.

Biographies

Shan Luowas born in 1985.She received her B.S.degree in electrical information engineering in 2007 and M.S.degree in signal and information processing in 2010 both from University of Electronic Science and Technology of China,and Ph.D degree in information engineering from Nanyang Technological University in 2014.Her research interests include time-frequency analysis,signal processing for wireless communications and image processing.

E-mail:luoshan@uestc.edu.cn

Xiumei Liwas born in 1978.She received her B.S. degree in electrical engineering from Lanzhou University in 1999,M.S.degree in information and signal processing from Institute of Acoustics,Chinese Academy of Sciences in 2002,and Ph.D degree in information engineering from Nanyang Technological University in 2010.Her research interests include time-frequency analysis and its applications, signal detection and estimation,and fast algorithms of signal processing methods.

E-mail:lixiumei@pmail.ntu.edu.sg

Guoan Biwas born in 1954.He received his B.S. degree in radio communications from Dalian University of Technology in 1982,M.S.degree in telecommunication systems and Ph.D.degree in electronics systems from Essex University,UK in 1985 and 1988,respectively.His research areas include computational fast algorithm,time-frequency analysis,signal detection and parameter estimation for applications in communications,radar and sonar systems.

E-mail:egbi@ntu.edu.sg

10.1109/JSEE.2015.00126

Manuscript received August 27,2014.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(61571174),the Zhejiang Provincial Natural Science Foundation of China(LY15F010010),the Open Project of Zhejiang Key Laboratory for Signal Processing(ZJKL 4 SP–OP2013–02),the Scientifi Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry[2013]693 and[2015]1098,the Fundamental Research Fundsfor the Central Universities(ZYGX2014J097)and the Technology Foundation for Selected Overseas Chinese Scholar.