Research Article The Wigner-Ville Distribution Based on ...Wigner-Ville distribution based on linear...

Research ArticleThe Wigner-Ville Distribution Based onthe Linear Canonical Transform and Its Applications forQFM Signal Parameters Estimation

Yu-E Song12 Xiao-Yan Zhang2 Chun-Heng Shang1 Hong-Xia Bu13 and Xiao-Yan Wang4

1 School of Information and Electronics Beijing Institute of Technology Beijing 100081 China2Department of Information Engineering Beijing Polytechnic College Beijing 100042 China3 College of Physics Science and Information Engineering Hebei Normal University Shijiazhuang 050016 China4Department of Fundamental Courses Academy of Armored Forces Engineering Beijing 100072 China

Correspondence should be addressed to Yu-E Song aeaeae623163com

Received 10 February 2014 Revised 8 April 2014 Accepted 8 April 2014 Published 13 May 2014

Academic Editor Baolin Wang

Copyright copy 2014 Yu-E Song et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The Wigner-Ville distribution (WVD) based on the linear canonical transform (LCT) (WDL) not only has the advantages of theLCT but also has the good properties of WVD In this paper some new and important properties of the WDL are derived and therelationships between WDL and some other time-frequency distributions are discussed such as the ambiguity function based onLCT (LCTAF) the short-time Fourier transform (STFT) and the wavelet transform (WT) The WDLs of some signals are alsodeduced A novel definition of the WVD based on the LCT and generalized instantaneous autocorrelation function (GWDL)is proposed and its applications in the estimation of parameters for QFM signals are also discussed The GWDL of the QFMsignal generates an impulse and the third-order phase coefficient of QFM signal can be estimated in accordance with the positioninformation of such impulse The proposed algorithm is fast because it only requires 1-dimensional maximization Also the newalgorithm only has fourth-order nonlinearity thus it has accurate estimation and low signal-to-noise ratio (SNR) threshold Thesimulation results are provided to support the theoretical results

1 Introduction

The main elements of the modern signal processing arenonstationary non-Gaussian and nonlinear signals Amongthese signals the development of the nonstationary signalprocessing theory is especially remarkable There are manytime-frequency analysis tools for nonstationary signals suchas short-time Fourier transform (STFT) fractional Fouriertransform (FRFT) Gabor transform (GT) Wigner-Ville dis-tribution function (WVD) ambiguity function (AF) linearcanonical transform (LCT) and so forth [1] The WVD isregarded as the mother of all the time-frequency distributionand has become an important distribution in signal analysisand processing especially in the nonstationary signal analysisand processing [2ndash4] The LCT as the generalization of theFourier transform (FT) and the FRFT was first introduced byMoshinsky and Quesne [5] and Collins and Stuart [6] Now

it has been applied for filter designing time-frequency signalseparating signal synthesis and signal encryption [7ndash9]

The quadratic frequency modulated (QFM) signal existswidely in nature and is an important nonlinear module inthe signal processing field It is applied widely in radarsonar speech and communication fields mostly in radarsystems [10] There are many algorithms for estimating theparameters of QFM signal such as the maximum likeli-hood (ML) method [11] the adaptive short-time Fouriertransform method [12] the polynomial Wigner-Ville dis-tributions (PWVDs) [13] the product high-order matched-phase transform (PHMT) [14] and the ambiguity functionbased on the LCT method (LCTAF) [15] Because the MLmethod and LCTAF method need 3-dimensional (3D) and2D maximizations respectively these methods suffer fromcomputational burdenThe adaptive STFTmethod has lowerresolution The PWVDs and PHMT algorithms need high

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 516457 8 pageshttpdxdoiorg1011552014516457

2 Journal of Applied Mathematics

order of nonlinearity (sixth-order to be exact) and this leadsto high signal-to-noise ratio (SNR) threshold Thereforemethods to estimate the QFM signal parameters quickly andaccurately are still an important issue to be solved

In [15 16] Tao et al and Bai et al have defined theWigner-Ville distribution based on linear canonical trans-form (WDL) separately For the WDL Bai has derived someproperties and used them to detect the linear frequencymodulated (LFM) signal The WDL is a new and importantsignal processing tool but they have not discussed theWDL in depth enough In this paper we deduce some newproperties of WDL and investigate the relationship betweenWDL and other transforms We also derive WDLs of somecommon signals In order to estimate QFM signal param-eters we define a new kind of Wigner-Ville distributionmdashthe generalized Wigner-Ville distribution based on the linearcanonical transform (GWDL) The GWDL algorithm justneeds 1D maximization so the amount of calculations issmaller compared to ML method and LCTAF method Andthe new algorithm only needs fourth-order nonlinearity soit has lower SNR threshold than the PHMT algorithm andPWVDs algorithm The simulation results are provided tosupport the theoretical results

The remainder of this paper is organized as followsSection 2 reviews the preliminaries about the WVD and theLCT In Section 3 some new properties of WDL are deducedand the relationship between WDL and other transforms isinvestigated The GWDL is defined and its application toQFM signal parameter estimation is illustrated in Section 4Finally Section 5 gives the conclusion

2 Preliminary

21 The Winger-Ville Distribution The instantaneous auto-correlation function 119896119891(119905 120591) of signal 119891(119905) is

119896119891 (119905 120591) = 119891(119905 +120591

2)119891

lowast(119905 minus

120591

2) (1)

The WVD of 119891(119905) is defined as the Fourier transform of119896119891(119905 120591) for 120591

119882119891 (119905 120596) = int

+infin

minusinfin119896119891 (119905 120591) 119890

minus119895120596120591119889120591

= int

+infin

minusinfin119891(119905 +

120591

2)119891

lowast(119905 minus

120591

2) 119890

minus119895120596120591119889120591

(2)

Another definition of WVD is

119882119865 (119905 120596) =1

2120587int

+infin


V2)119865

lowast(120596 minus

V2) 119890

119895V119905119889V (3)

where 119865(120596) is the Fourier transform of 119891(119905)The WVD has many important properties such as con-

jugation symmetry property time marginal property andenergy distribution property For more results about theWVD one can refer to [17ndash19]

22 Linear Canonical Transform (LCT) The LCT of a signal119891(119905) with parameter matrix 119860 = (119886 119887 119888 119889) is defined as

119865119860 (119906) = 119871119860 [119891] (119906)

=

int

+infin

minusinfin119891 (119905)119870119860 (119906 119905) 119889119905 119887 = 0

radic119889119890(1198951198881198892)1199062

119891 (119889119906) 119887 = 0

(4)

where the kernel function119870119860(119906 119905) is

119870119860 (119906 119905) =1

radic1198952120587119887exp(119895 119889

21198871199062minus 119895

119906119905

119887+ 119895

119886

21198871199052) (5)

and parameters 119886 119887 119888 119889 isin 119877 and satisfy 119886119889minus119887119888 = 1The LCThas additive property

119871 (1198862119887211988821198892) [119871 (119886

1119887111988811198891) [119891 (119905)]] = 119871 (119890119891119892ℎ) [119891 (119905)] (6)

where ( 119890 119891119892 ℎ ) = (11988621198872

11988821198892

) (11988611198871

11988811198891

) and reversible property

119871119860 [119871119860minus1 [119891 (119905)]] = 119891 (119905) (7)

Other properties of LCT such as sampling and dis-cretization uncertainty principles product and convolutiontheorems and Hilbert Transform are discussed in detail in[20ndash23]

When the parameter matrix119860 is with some special casesthe LCT reduces to FT FRFT Fresnel transform and scalingoperation [24] So the LCT is the generalization of thesetransforms From (4) we can see that when the parameter119887 = 0 the LCT is a scaling transform operation multiplying alinear frequency modulation signal so we suppose that 119887 = 0

in the following discussion

3 The Wigner-Ville Distribution Based onLinear Canonical Transform

In [16] the authors have defined the WDL according tothe actual needs but they have missed some importantproperties In this section we deduce some new propertiesof WDL and investigate the relationship between WDL andother transforms

31 The Definition of WDL

Definition 1 Keeping the instantaneous autocorrelationfunction 119896119891(119905 120591) unchanged and replacing the kernel func-tion 119890minus119895119908120591 of FT by the kernel function119870119860(119906 120591) of LCT in theWVD definition the WDL is defined as [15]

WD119860119891 (119905 119906) = int+infin

minusinfin119896119891 (119905 120591)119870119860 (119906 120591) 119889120591 (8)

where the 119896119891(119905 120591) is given by (1) and119870119860(119906 120591) is shown by (5)

Obviously when 119860 = (0 1 minus1 0) the LCT reduces to FTAccordingly the WDL reduces to classical WVD

WD119860119891 (119905 119906) = radic1

1198952120587119882119891 (119905 119906) (9)

Journal of Applied Mathematics 3

When 119860 = (cos120572 sin120572 minus sin120572 cos120572)

WD119860119891 (119905 119906) = radic119890minus119895120572119882120572119891 (119905 119906) (10)

From (9) and (10) one can see that the WDL is a gener-alization of WVD based on the Fourier transform and thefractional Fourier transform

We know that some nonbandlimited signals in the clas-sical Fourier domain especially some nonstationary signalsand non-Gaussian signals in the Fourier domain can bebandlimited in the LCT domain [20] The WVD can be seenas a FT of the instantaneous autocorrelation function and theWDL can be seen as a LCT of the instantaneous autocorre-lation function So signals which are nonbandlimited afterWVDmay be bandlimited in the WDL domain and then wecan use bandlimited theory to process this kind of signalsThe traditional nonbandlimited signals processing problemsin the Fourier domain can be solved in the LCT domainThisis one of the reasons why we discuss the WVD based on theLCT

The cross-WVD is the FT of crosscorrelation functionfor 120591 It has more information and making full use of thisinformation can improve signal processing ability So weprovide the definition of the cross-WVD based on the LCThere

Definition 2 The cross-WVD based on LCT of signal 119891(119905)and signal 119892(119905) is defined as

WD119860119891119892 (119905 119906) = int+infin


120591

2) 119892

lowast(119905 minus

120591

2)119870119860 (119906 120591) 119889120591

(11)

32 Basic Properties of WDL In this subsection some newproperties of WDL which are different from those in [16] areinvestigated and the proofs of some complex properties willbe given in detail All the properties are based on the fact thatthe WDL of 119891(119905) is WD119860119891(119905 119906)

Property 1 (symmetry and conjugation property) The WDLof signal 119891lowast(minus119905) is

WD119860119891lowast(minus119905) (119905 119906) = [WD119860minus1

119891 (minus119905 119906)]lowast

(12)

Property 2 (shifting properties) TheWDL of signal 119891(119905)1198901198951205961199052

is

WD119860119891(119905)119890119895120596119905

2 (119905 119906) = 1198901198952119889120596119905119906minus119895211988711988912059621205912WD119860119891 (119905 119906 minus 2119887120596119905) (13)

TheWDL of signal 119891(119905 minus 1199050)119890119895120596119905 is

WD119860119891(119905minus1199050)119890

119895120596119905 (119905 119906) = 119890119895(119906119889120596minus11988711988912059622)

timesWD119860119891 (119905 minus 1199050 119906 minus 1205961119887) (14)

TheWDL of signal 119892(119905) = 119891(119905)1198901198951205961119905+11989512059621199052

is

WD119860119892(119905) (119905 119906) = 119890119895(119906119889(120596

1+21205962119905)minus119887119889(120596

1+21205962119905)22)

timesWD119860119891 (119905 119906 minus 1205961119887 minus 21198871205962119905) (15)

Especially when 119891(119905) = 1 119892(119905) = 1198901198951205961119905+11989512059621199052

is the LFM signaland its WDL is

WD119860119892(119905) (119905 119906)

=

radic2120587119887

119895119890119895(1198892119887)1199062

120575 (21205962119887119905 + 1205961119887 minus 119906) 119886 = 0

radic1

119886119890119895(1198892119887)1199062minus119895((119906minus120596

1119887minus2119887120596

2119905)22119886119887)

119886 = 0

(16)

Proof Firstly a useful formula is given below that will be usedin this paper

int

+infin

minusinfin119890(minus1198741199052plusmn2119875119905+119876)

119889119905 = radic120587

119874119890(1198752119874+119876)

(17)

where 119874 119875 119876 isin C 119874 = 0 and Re(119874) ge 0Equation (15) is easy to get so we only prove (16)

WD119860119892(119905) (119905 119906) = int

+infin


120591

2) 119892

lowast(119905 +

120591

2)119870119860 (119906 120591) 119889120591

= radic1

1198952120587119887119890119895(1198892119887)1199062

times int

+infin

minusinfin119890119895(1199081+21205962119905minus119906119887)120591minus(minus119895(1198862119887))1205912

119889120591

(18)

So 119874 = minus119895(1198862119887) 119875 = (1199081119887 minus 119906)2119887 + 1205962119905When 119886 = 0 (18) is

WD119860119892(119905) (119905 119906) = radic1

1198952120587119887119890119895(1198892119887)1199062

int

+infin

minusinfin119890119895(1205961+21205962119905minus119906119887)120591

119889120591

= radic2120587119887

119895119890119895(1198892119887)1199062

120575 (21205962119905119887 + 1205961119887 minus 119906)

(19)

When 119886 = 0 119874 = 0 and the condition Re(119874) ge 0 is metAccording to (24) we can get

WD119860119892(119905) (119905 119906) = radic1

119886119890119895(1198892119887)1199062minus119895((2119887120596

2119905+1205961119887minus119906)22119886119887)

(20)

Then we obtain (16)

Property 3 (scaling property) If 119891(119905) = radic|120582|119891(120582119905) and 120582 = 0then the WDL of 119891(119905) is

WD119860ℎ (119905 119906) = radic1

|120582|WD119861119891 (120582119905 119906) (21)

where 119861 = (119886120582 119887120582 119888120582 119889120582)


33 The Relationships between WDL and OtherTime-Frequency Analysis Tools

331 The Relationship between WDL and the AmbiguityFunction Based on LCT (LCTAF) The LCTAF is defined as[15]

AF119860119891 (120591 119906) = int+infin

minusinfin119896119891 (119905 120591)119870119860 (119906 119905) 119889119905 (22)

Theorem3 The relationship betweenWDL and LCTAF is [15]

119882119863119860119891 (119905 119906) = ∬

+infin

minusinfin119860119865

119860119891 (120591 V) 119870119860119860minus1 (120591 V 119906 119905) 119889120591 119889V (23)

where 119870119860119860minus1(120591 V 119906 119905) = 119870119860(120591 119906)119870119860minus1(V 119905) is the kernelfunction of 2-D LCT [25]

Equation (23) indicates that the WDL is the 2D LCTof LCTAF and two parameter matrixes corresponding tothe kernel function of 2D LCT are reciprocal matrixes Thisindicates that not only classical AF and WVD but also theLCTAF and the WDL have close relationships

332 The Relationship between WDL and the STFT TheSTFT of signal 119891(119905) is defined as

119878119908119891 (119905 120596) = int

+infin

minusinfin119891 (119906) 119892

lowast(119906 minus 119905) 119890

minus119895120596119906119889119906 (24)

Theorem 4 TheWDL can be expressed by STFT

119882119863119860119891 (119905 119906)

= radic1

1198952120587119887∬

+infin

minusinfin119890119895((1198892119887)V2+V1205912119887minus311988612059128119887)

sdot 119878119882119891 (120591

2V minus 119886120591119887

)

times 119870119860119860minus1 (120591 V 119906 119905) 119889120591 119889V

(25)

where 119878119882119891 (120591 (2V minus 119886120591)2119887) = int

+infin

minusinfin119891(119905

1015840)[119891(119905

1015840minus 120591)

119890minus119895(1198862119887)(1199051015840minus120591)

2

]lowast119890minus1198951199051015840(V119887minus(1198862119887)120591)

1198891199051015840

Proof From [15] we know that the LCTAF can be expressedby STFT as

AF119860119891 (120591 119906) = radic1

1198952120587119887119890119895((1198892119887)1199062+1199061205912119887minus311988612059128119887)

sdot 119878119882119891 (120591

2119906 minus 119886120591

119887)

(26)

Taking (26) into (23) we can get (25)

333 The Relationship between WDL and Wavelet Transform(WT) TheWT of signal 119891(119905) is defined as

WT119891 (119886 119887) =1

radic119886int

+infin

minusinfin119891 (119905) 120595

lowast(119905 minus 119887

119886)119889119905 (27)

where 120595119886119887(119905) = (1radic119886)120595((119905 minus 119887)119886) is the basic function ofWT

Theorem 5 TheWDL can be expressed by WT as

119882119863119860119891 (119905 119906) = radic

1

1198952120587119887

times∬

+infin

minusinfin119890119895((1198892119887)V2minusV1205912119887+11988612059128119887)

sdotWT119891 (1 120591)

times 119870119860119860minus1 (120591 V 119906 119905) 119889120591 119889V(28)

where119882119879119891(1 120591) = 119891(119905 minus 120591)119890minus119895(1198862119887)(119905minus120591)2+119895(119905minus120591)((2119906minus119886120591)2119887)


AF119860119891 (120591 119906) = radic1

1198952120587119887119890119895((1198892119887)1199062minus1199061205912119887+11988612059128119887)

sdotWT119891 (1 120591)

(29)


34 WDLs of Some Common Signals Table 1 gives WDLs ofsome common signals They are easy to get so we do notprove them here

4 The QFM Signal Parameter EstimationAlgorithm Based on the Generalized WDL

41 The Algorithm for the Parameter Estimation of QFMSignal In [16] the authors have used theWDL for estimatingLFM signals but is it appropriate for dealing with QFMsignal Now let us discuss it The QFM signal considered inthis paper confirms themodel119891(119905) = 1198600119890

119895(1198861119905+11988621199052+11988631199053) where

1198600 is the amplitude and 1198861 1198862 and 1198863 (1198863 = 0) are the phasecoefficients to be determined For the QFM signal its WDLis

WD119860119891 (119905 119906) = radic1

119895212058711988711986020119890119895(1198892119887)1199062

times int

+infin

minusinfin119890119895[(1198861+21198862119905+311988631199052minus119906119887)120591+(1198862119887)1205912+119886

312059134]

119889120591

(30)

Formula (30) shows that we cannot get any informationabout phase coefficients so it is difficult to estimate param-eters of QFM signals But we can change the instantaneousautocorrelation function 119896119891(119905 120591) of WDL to achieve this


Table 1 WDLs of some common signals

Signals WDLs1 radic119886minus1119890

(1198951198882119886) 1199062

120575(119905 minus 1199050) 120575(2(119905 minus 1199050))119870119860(119906 2119905 minus 21199050)

119890119895120582119905

radic2120587119887

119895119890119895(1198892119887) 1199062

120575(119906 minus 120582119887) 119886 = 0

radic1

119886119890119895(1198892119887)1199062minus119895((119906minus1205961119887)

22119886119887) 119886 = 0

11989011989512058311990522

radic2120587119887

119895119890119895(1198892119887)1199062

120575(119906 minus 2120583119887119905) 119886 = 0

radic1

119886119890119895(1198892119887)1199062minus119895((119906minus2119887120583119905)22119886119887)

119886 = 0

11989011989512058311990522+119895120582119905+119895120574

radic2120587119887

119895119890119895(1198892119887)1199062

120575(119906 minus 120582119887 minus 2120583119887119905) 119886 = 0

radic1

119886119890119895(1198892119887)1199062 minus119895((119906minus120582119887minus2119887120583119905)22119886119887)

119886 = 0

119890minus11990522

radic2120587119887

119895119890119895(1198892119887)1199062

120575(119895119887119905 minus 119906) 119886 = 0

radic1

119886119890(1198951198882119886)1199062minus(119905119906119886)+119895(11990521198872119886)

119886 = 0

goal We define this new WDL as the generalized Wigner-Ville distribution based on the linear canonical transform(GWDL)

Definition 6 Given 1198961015840119891(119905 120591) = 119891(119905 + 1205912)119891(119905 minus 1205912)119891lowast(minus119905 +

1205912)119891lowast(minus119905 minus 1205912) one defines the GWDL of signal 119891(119905) as

follows

GWD119860119891 (119905 119906) = int+infin

minusinfin1198961015840119891 (119905 120591)119870119860 (119906 120591) 119889120591 (31)

where the superscript lowast denotes the conjugate and 119870119860(119906 120591)is given by (5) The definition of GWDL in (31) indicates thatthe GWDL has fourth-order nonlinearity Substituting theQFM signal 119891(119905) = 1198600119890

119895(1198861119905+11988621199052+11988631199053) into 1198961015840119891(119905 120591) we obtain

1198961015840119891(119905 120591) = 119860

40119890119895(4(1198861119905+11988631199053)+3119886

31199051205912) So the phase of 1198961015840119891(119905 120591) is

quadratic for 120591 while the phase of 119896119891(119905 120591) is cubic for 120591 Bythis change we can estimate theQFM signal parameters usingGWDL

Theorem 7 The GWDL of QFM signal is

GWD119860119891 (119905 119906)

=

11986040radic

2120587119887

119895119890119895((1198892119887)1199062+4(119886

1119905+11988631199053))120575 (119906) 119886 + 61198871198863119905 = 0

11986040

radic119886 + 61198871198863119905

times119890119895((1198892119887)1199062+4(119886

1119905+11988631199053)minus1199062(2119886119887+241198872119905119886

3)) 119886 + 61198871198863119905 = 0

(32)

Proof Substituting 1198961015840119891(119905 120591) = 11986040119890119895(4(1198861119905+11988631199053)+3119886

31199051205912) into the

definition of GWDL we obtain

GWD119860119891 (119905 119906) =11986040

radic1198952120587119887119890119895((1198892119887)1199062+4(119886

1119905+11988631199053))

times int

+infin

minusinfin119890minus119895(119906120591119887)+119895(1198862119887+3119886

3119905)1205912119889120591

(33)

According to (17) it is easy to obtain (32) by (33)

Formula (32) indicates that the GWDL of QFM signalwill generate an impulse at the point (minus11988661198871198863 0) in the (119905 119906)plane and the energy will gather along the line 119906 = 0 Sowe can get the location information (1199051015840 0) of the impulse bysearching the peak According to the formula 1199051015840 = minus11988661198871198863the parameter 1198863 can be estimated as

1198863 = minus119886

6119887 argmax119905 |GWDL (119905 0)| (34)

Equation (34) shows that the condition 119886 = 0 must besatisfied After 1198863 has been estimated the signal 119891(119905)119890minus1198951198863119905

3

can be approximated to a LFM signal So other parameterscan be estimated by algorithms for estimating the LFM signalsuch as WDL algorithm [16] FRFT algorithm [26] and thecubic phase function (CPF) algorithm [27] (in Section 42 wechoose the CPF algorithm to estimate the LFM signal)

For the discrete QFM signal 119891(119899) the specification of theproposed algorithm is as follows

Step 1 Compute the GWDL of 119891(119899) and search for the peakin the time-119906 frequency plane to get the location information(1198991015840 0) then estimate 1198863 according to (34)

Step 2 Multiply 119891(119899)with 119890minus11989511988631198993

and do CPF for 119891(119899)119890minus11989511988631198993

then estimate 1198862 according to

1198862 =argmaxΩ |CPF (119899 Ω)|

2 (35)

where CPF(119899 Ω) is the CPF of 119891(119899)119890minus11989511988631198993

Step 3 Estimate 1198861 by dechirping and finding the Fouriertransform peak

1198861 = argmax120596

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119873minus1)2

sum

119899=minus(119873minus1)2

119891 (119905) 119890minus119895(11988621198992+11988631198993)minus119895120596119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

(36)

Step 4 Estimate 1198600 by evaluating

1198600 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873

(119873minus1)2

sum

119899=minus(119873minus1)2

119891 (119905) 119890minus119895(1198861119899+11988621198992+11988631198993)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(37)

For the discrete signal to avoid ambiguities due to theperiodicity of digital spectra it is also assumed that [28]

10038161003816100381610038161003816119886119894Δ

11989410038161003816100381610038161003816le

120587

119894((119873 minus 1) 2)119894minus1 119894 = 1 2 3 (38)

where 119873 is the length of the discrete signal and Δ is thesampling interval


300

250

200

150

100

50

01000

500

0

t

0200

400600

8001000

u

Am

plitu

de

The GWDL of f(t) = 2exp(j(t + 01t2+ 001t

3))

500

t200

400600

800

u

Figure 1 The GWDL of QFM signal

42 Simulations The parameter estimation algorithm pro-posed above is applied to a QFM signal here The QFMsignal 119891(119905) is considered and its parameter values are setto be 1198600 = 2 1198861 = 1 1198862 = 01 and 1198863 = 001 thesampling interval is Δ = 002 s and the observing timeis 119905 = 20 s Obviously parameter values that we choseabove meet condition (38) Figure 1 indicates the GWDLof 119891(119905) when (119886 119887 119888 119889) = (002 124 0 50) It shows thatthe energy gathers along the line 119906 = 0 Figure 2 is the 119905-amplitude distribution of GWDL Figure 3 is the 119906-amplitudedistribution of GWDL and Figure 4 is the 119905-119906 distributionof QFM signal in the GWDL domain Using the GWDLalgorithm described above we get that the estimate valuesof 1198600 1198861 1198862 and 1198863 are 1198600 = 19059 1198861 = 09415 1198862 =

01020 and 1198863 = 00101 respectively The simulation resultsindicate that the algorithm is accurate and the GWDLapplying to the parameter estimation of the QFM signal isuseful and effective

43 Comparison with Other Methods We know that thedimension of maximization for an algorithm leads to itscomputational complexity and efficiency whereas the nonlin-earity order of an algorithmdetermines its SNR threshold andaccuracy So we compare the proposed method with othermethods in the aspects of the dimension ofmaximization andthe nonlinearity order

The algorithm we proposed above shows that it onlyneeds three times 1Dmaximization for estimating all the fourparameters while the MLmethod requires 3Dmaximizationand the LCTAF method needs 2D maximization So thisalgorithm does not have heavy computational burden and isefficient

The PWVDs method [13] and the PHMT method [14]both need the same dimension of maximization as theproposed method (1D to be exact) but they have sixth-ordernonlinearity which decreases the estimation accuracy andincreases the SNR threshold Our method only has fourth-order nonlinearity so the estimation values aremore accurateand have lower SNR threshold

300

250

200

150

100

100 200 300 400 500 600 700 800 900 1000

50

0

0

t

Am

plitu

de


3))

Figure 2 119905-amplitude distribution of QFM signal in GWDLdomain

300

250

200

150

100

100 200 300 400 500 600 700 800 900 1000

50

0

0

u

Am

plitu

deThe GWDL of f(t) = 2exp(j(t + 01t

2+ 001t

3))


5 Conclusions

Some theories of the WDL are investigated in this paperWe first derive some new and important properties of theWDL The relationships between WDL and other time-frequency analysis tools are also discussed for instance theLCTAF STFT and WT The research on these theories ofthe WDL lays the foundation for its further application andenriches theoretical systems of the LCT and WVD A fastand precise algorithm based on the generalized WDL forQFM signal parameter estimation is proposed too The newalgorithm only needs 1D maximizations so it does not haveheavy computational burden and is efficient Also the newalgorithm is accurate and has a low SNR threshold becauseof its moderate order of nonlinearity


100 200 300 400 500 600 700 800 900 10000

u

100

200

300

400

500

600

700

800

900

1000

0

t


3))

Figure 4 119905-119906 distribution of QFM signal in GWDL domain

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was partially supported by the National NaturalScience Foundation of China (Grant no 11201482) andthe science and technology innovation platform of Beijingeducation committee (Grant no PXM2013-014225-000051)

References

[1] D Gabor ldquoTheory of communicationrdquo Journal of the IEE vol93 no 26 pp 429ndash457 1946

[2] L Stankovic ldquoAuto-term representation by the reduced inter-ference distributions a procedure for kernel designrdquo IEEETransactions on Signal Processing vol 44 no 6 pp 1557ndash15631996

[3] M-J Bastiaans T Alieva and L Stankovic ldquoOn rotated time-frequency kernelsrdquo IEEE Signal Processing Letters vol 9 no 11pp 378ndash381 2002

[4] B Boashash Time Frequency Analysis Elsevier Science Ams-terdam The Netherlands 2003

[5] M Moshinsky and C Quesne ldquoLinear canonical transforma-tions and their unitary representations rdquo Journal of Mathemat-ical Physics vol 12 pp 1772ndash1780 1971

[6] J Collins and A Stuart ldquoLens-system diffraction integralwritten in terms of matrix opticsrdquo JOSA vol 60 no 9 pp 1168ndash1177 1970

[7] R Tao B Deng and Y Wang ldquoResearch progress of thefractional Fourier transform in signal processingrdquo Science inChina F Information Sciences vol 49 no 1 pp 1ndash25 2006

[8] S-C Pei and J-J Ding ldquoRelations between fractional opera-tions and time-frequency distributions and their applicationsrdquoIEEE Transactions on Signal Processing vol 49 no 8 pp 1638ndash1655 2001

[9] B Barshan M-A Kutay and H M Ozaktas ldquoOptimal filteringwith linear canonical transformationsrdquoOptics Communicationsvol 135 no 1ndash3 pp 32ndash36 1997

[10] S Barbarossa and V Petrone ldquoAnalysis of polynomial-phasesignals by the integrated generalized ambiguity functionrdquo IEEETransactions on Signal Processing vol 45 no 2 pp 316ndash3271997

[11] T-J Abatzoglou ldquoFastmaximurm likelihood joint estimation offrequency and frequency raterdquo IEEE Transactions on Aerospaceand Electronic Systems vol 22 no 6 pp 708ndash715 1986

[12] H-K Kwok and D L Jones ldquoImproved instantaneous fre-quency estimation using an adaptive short-time Fourier trans-formrdquo IEEE Transactions on Signal Processing vol 48 no 10 pp2964ndash2972 2000

[13] B Boashash and P OrsquoShea ldquoPolynomial Wigner-Ville distri-butions and their relationship to time-varying higher orderspectrardquo IEEE Transactions on Signal Processing vol 42 no 1pp 216ndash220 1994

[14] Y Wang and Y-C Jiang ldquoISAR imaging of a ship targetusing product high-order matched-phase transformrdquo IEEEGeoscience and Remote Sensing Letters vol 6 no 4 pp 658ndash661 2009

[15] R Tao Y-E Song Z-J Wang and Y Wang ldquoAmbiguityfunction based on the linear canonical transformrdquo IET SignalProcessing vol 6 no 6 pp 568ndash576 2012

[16] R-F Bai B-Z Li and Q-Y Cheng ldquoWigner-Ville distributionassociated with the linear canonical transformrdquo Journal ofAppliedMathematics vol 2012 Article ID 740161 14 pages 2012

[17] T A C M Claasen and W F G Mecklenbrauker ldquoTheWigner distributionmdasha tool for time-frequency signal analysisI Continuous-time signalsrdquo Philips Journal of Research vol 35no 3 pp 217ndash250 1980

[18] T A C M Claasen and W F G Mecklenbrauker ldquoTheWigner distributionmdasha tool for time-frequency signal analysisII Discrete-time signalsrdquo Philips Journal of Research vol 35 no4-5 pp 276ndash300 1980

[19] T A C M Claasen andW F G Mecklenbrauker ldquoTheWignerdistributionmdasha tool for time-frequency signal analysis IIIRelations with other time-frequency signal transformationsrdquoPhilips Journal of Research vol 35 no 6 pp 372ndash389 1980

[20] R Tao B-Z Li Y Wang and G K Aggrey ldquoOn samplingof band-limited signals associated with the linear canonicaltransformrdquo IEEE Transactions on Signal Processing vol 56 no11 pp 5454ndash5464 2008

[21] B Deng R Tao and Y Wang ldquoConvolution theorems for thelinear canonical transform and their applicationsrdquo Science inChina F Information Sciences vol 49 no 5 pp 592ndash603 2006

[22] X Guanlei W Xiaotong and X Xiaogang ldquoUncertaintyinequalities for linear canonical transformrdquo IET Signal Process-ing vol 3 no 5 pp 392ndash402 2009

[23] J J Healy and J T Sheridan ldquoSampling and discretization of thelinear canonical transformrdquo Signal Processing vol 89 no 4 pp641ndash648 2009

[24] S-C Pei and J-J Ding ldquoEigenfunctions of linear canonicaltransformrdquo IEEE Transactions on Signal Processing vol 50 no1 pp 11ndash26 2002

[25] XGuanleiWXiaotong andXXiaogang ldquoGeneralizedHilberttransform and its properties in 2D LCT domainrdquo Signal Process-ing vol 89 no 7 pp 1395ndash1402 2009

[26] L Qi R Tao S Y Zhou and Y Wang ldquoDetection and param-eter estimation of multicomponent LFM signal based on thefractional Fourier transformrdquo Science in China F InformationSciences vol 47 no 2 pp 184ndash198 2004


[27] P Wang H Li I Djurovic and B Himed ldquoIntegrated cubicphase function for linear FM signal analysisrdquo IEEE Transactionson Aerospace and Electronic Systems vol 46 no 3 pp 963ndash9772010

[28] P OrsquoShea and R-A Wiltshire ldquoA new class of multilinear func-tions for polynomial phase signal analysisrdquo IEEE Transactionson Signal Processing vol 57 no 6 pp 2096ndash2109 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


order of nonlinearity (sixth-order to be exact) and this leadsto high signal-to-noise ratio (SNR) threshold Thereforemethods to estimate the QFM signal parameters quickly andaccurately are still an important issue to be solved

In [15 16] Tao et al and Bai et al have defined theWigner-Ville distribution based on linear canonical trans-form (WDL) separately For the WDL Bai has derived someproperties and used them to detect the linear frequencymodulated (LFM) signal The WDL is a new and importantsignal processing tool but they have not discussed theWDL in depth enough In this paper we deduce some newproperties of WDL and investigate the relationship betweenWDL and other transforms We also derive WDLs of somecommon signals In order to estimate QFM signal param-eters we define a new kind of Wigner-Ville distributionmdashthe generalized Wigner-Ville distribution based on the linearcanonical transform (GWDL) The GWDL algorithm justneeds 1D maximization so the amount of calculations issmaller compared to ML method and LCTAF method Andthe new algorithm only needs fourth-order nonlinearity soit has lower SNR threshold than the PHMT algorithm andPWVDs algorithm The simulation results are provided tosupport the theoretical results

The remainder of this paper is organized as followsSection 2 reviews the preliminaries about the WVD and theLCT In Section 3 some new properties of WDL are deducedand the relationship between WDL and other transforms isinvestigated The GWDL is defined and its application toQFM signal parameter estimation is illustrated in Section 4Finally Section 5 gives the conclusion

2 Preliminary

21 The Winger-Ville Distribution The instantaneous auto-correlation function 119896119891(119905 120591) of signal 119891(119905) is

119896119891 (119905 120591) = 119891(119905 +120591

2)119891

lowast(119905 minus

120591

2) (1)

The WVD of 119891(119905) is defined as the Fourier transform of119896119891(119905 120591) for 120591

119882119891 (119905 120596) = int

+infin

minusinfin119896119891 (119905 120591) 119890

minus119895120596120591119889120591

= int

+infin


120591

2)119891

lowast(119905 minus

120591

2) 119890

minus119895120596120591119889120591

(2)

Another definition of WVD is

119882119865 (119905 120596) =1

2120587int

+infin


V2)119865

lowast(120596 minus

V2) 119890

119895V119905119889V (3)

where 119865(120596) is the Fourier transform of 119891(119905)The WVD has many important properties such as con-

jugation symmetry property time marginal property andenergy distribution property For more results about theWVD one can refer to [17ndash19]

22 Linear Canonical Transform (LCT) The LCT of a signal119891(119905) with parameter matrix 119860 = (119886 119887 119888 119889) is defined as

119865119860 (119906) = 119871119860 [119891] (119906)

=

int

+infin

minusinfin119891 (119905)119870119860 (119906 119905) 119889119905 119887 = 0

radic119889119890(1198951198881198892)1199062

119891 (119889119906) 119887 = 0

(4)

where the kernel function119870119860(119906 119905) is

119870119860 (119906 119905) =1

radic1198952120587119887exp(119895 119889

21198871199062minus 119895

119906119905

119887+ 119895

119886

21198871199052) (5)

and parameters 119886 119887 119888 119889 isin 119877 and satisfy 119886119889minus119887119888 = 1The LCThas additive property

119871 (1198862119887211988821198892) [119871 (119886

1119887111988811198891) [119891 (119905)]] = 119871 (119890119891119892ℎ) [119891 (119905)] (6)

where ( 119890 119891119892 ℎ ) = (11988621198872

11988821198892

) (11988611198871

11988811198891

) and reversible property

119871119860 [119871119860minus1 [119891 (119905)]] = 119891 (119905) (7)

Other properties of LCT such as sampling and dis-cretization uncertainty principles product and convolutiontheorems and Hilbert Transform are discussed in detail in[20ndash23]

When the parameter matrix119860 is with some special casesthe LCT reduces to FT FRFT Fresnel transform and scalingoperation [24] So the LCT is the generalization of thesetransforms From (4) we can see that when the parameter119887 = 0 the LCT is a scaling transform operation multiplying alinear frequency modulation signal so we suppose that 119887 = 0

in the following discussion

3 The Wigner-Ville Distribution Based onLinear Canonical Transform

In [16] the authors have defined the WDL according tothe actual needs but they have missed some importantproperties In this section we deduce some new propertiesof WDL and investigate the relationship between WDL andother transforms

31 The Definition of WDL

Definition 1 Keeping the instantaneous autocorrelationfunction 119896119891(119905 120591) unchanged and replacing the kernel func-tion 119890minus119895119908120591 of FT by the kernel function119870119860(119906 120591) of LCT in theWVD definition the WDL is defined as [15]

WD119860119891 (119905 119906) = int+infin

minusinfin119896119891 (119905 120591)119870119860 (119906 120591) 119889120591 (8)

where the 119896119891(119905 120591) is given by (1) and119870119860(119906 120591) is shown by (5)

Obviously when 119860 = (0 1 minus1 0) the LCT reduces to FTAccordingly the WDL reduces to classical WVD

WD119860119891 (119905 119906) = radic1

1198952120587119882119891 (119905 119906) (9)



WD119860119891 (119905 119906) = radic119890minus119895120572119882120572119891 (119905 119906) (10)





WD119860119891119892 (119905 119906) = int+infin


120591

2) 119892

lowast(119905 minus

120591

2)119870119860 (119906 120591) 119889120591

(11)




119891 (minus119905 119906)]lowast

(12)


is

WD119860119891(119905)119890119895120596119905

2 (119905 119906) = 1198901198952119889120596119905119906minus119895211988711988912059621205912WD119860119891 (119905 119906 minus 2119887120596119905) (13)


WD119860119891(119905minus1199050)119890

119895120596119905 (119905 119906) = 119890119895(119906119889120596minus11988711988912059622)


TheWDL of signal 119892(119905) = 119891(119905)1198901198951205961119905+11989512059621199052

is

WD119860119892(119905) (119905 119906) = 119890119895(119906119889(120596

1+21205962119905)minus119887119889(120596

1+21205962119905)22)


Especially when 119891(119905) = 1 119892(119905) = 1198901198951205961119905+11989512059621199052


WD119860119892(119905) (119905 119906)

=

radic2120587119887

119895119890119895(1198892119887)1199062

120575 (21205962119887119905 + 1205961119887 minus 119906) 119886 = 0

radic1

119886119890119895(1198892119887)1199062minus119895((119906minus120596

1119887minus2119887120596

2119905)22119886119887)

119886 = 0

(16)


int

+infin


119889119905 = radic120587

119874119890(1198752119874+119876)

(17)


WD119860119892(119905) (119905 119906) = int

+infin


120591

2) 119892

lowast(119905 +

120591

2)119870119860 (119906 120591) 119889120591

= radic1

1198952120587119887119890119895(1198892119887)1199062

times int

+infin


119889120591

(18)


WD119860119892(119905) (119905 119906) = radic1

1198952120587119887119890119895(1198892119887)1199062

int

+infin


119889120591

= radic2120587119887

119895119890119895(1198892119887)1199062

120575 (21205962119905119887 + 1205961119887 minus 119906)

(19)


WD119860119892(119905) (119905 119906) = radic1

119886119890119895(1198892119887)1199062minus119895((2119887120596

2119905+1205961119887minus119906)22119886119887)

(20)

Then we obtain (16)


WD119860ℎ (119905 119906) = radic1

|120582|WD119861119891 (120582119905 119906) (21)

where 119861 = (119886120582 119887120582 119888120582 119889120582)




AF119860119891 (120591 119906) = int+infin

minusinfin119896119891 (119905 120591)119870119860 (119906 119905) 119889119905 (22)


119882119863119860119891 (119905 119906) = ∬

+infin


119860119891 (120591 V) 119870119860119860minus1 (120591 V 119906 119905) 119889120591 119889V (23)




119878119908119891 (119905 120596) = int

+infin

minusinfin119891 (119906) 119892

lowast(119906 minus 119905) 119890

minus119895120596119906119889119906 (24)


119882119863119860119891 (119905 119906)

= radic1

1198952120587119887∬

+infin


sdot 119878119882119891 (120591

2V minus 119886120591119887

)

times 119870119860119860minus1 (120591 V 119906 119905) 119889120591 119889V

(25)

where 119878119882119891 (120591 (2V minus 119886120591)2119887) = int

+infin


1015840)[119891(119905

1015840minus 120591)

119890minus119895(1198862119887)(1199051015840minus120591)

2


1198891199051015840


AF119860119891 (120591 119906) = radic1

1198952120587119887119890119895((1198892119887)1199062+1199061205912119887minus311988612059128119887)

sdot 119878119882119891 (120591

2119906 minus 119886120591

119887)

(26)



WT119891 (119886 119887) =1

radic119886int

+infin

minusinfin119891 (119905) 120595


119886)119889119905 (27)



119882119863119860119891 (119905 119906) = radic

1

1198952120587119887

times∬

+infin


sdotWT119891 (1 120591)

times 119870119860119860minus1 (120591 V 119906 119905) 119889120591 119889V(28)



AF119860119891 (120591 119906) = radic1

1198952120587119887119890119895((1198892119887)1199062minus1199061205912119887+11988612059128119887)

sdotWT119891 (1 120591)

(29)





119895(1198861119905+11988621199052+11988631199053) where


WD119860119891 (119905 119906) = radic1

119895212058711988711986020119890119895(1198892119887)1199062

times int

+infin


312059134]

119889120591

(30)





(1198951198882119886) 1199062


119890119895120582119905

radic2120587119887

119895119890119895(1198892119887) 1199062

120575(119906 minus 120582119887) 119886 = 0

radic1

119886119890119895(1198892119887)1199062minus119895((119906minus1205961119887)

22119886119887) 119886 = 0

11989011989512058311990522

radic2120587119887

119895119890119895(1198892119887)1199062

120575(119906 minus 2120583119887119905) 119886 = 0

radic1

119886119890119895(1198892119887)1199062minus119895((119906minus2119887120583119905)22119886119887)

119886 = 0

11989011989512058311990522+119895120582119905+119895120574

radic2120587119887

119895119890119895(1198892119887)1199062

120575(119906 minus 120582119887 minus 2120583119887119905) 119886 = 0

radic1


119886 = 0

119890minus11990522

radic2120587119887

119895119890119895(1198892119887)1199062

120575(119895119887119905 minus 119906) 119886 = 0

radic1

119886119890(1198951198882119886)1199062minus(119905119906119886)+119895(11990521198872119886)

119886 = 0




follows

GWD119860119891 (119905 119906) = int+infin

minusinfin1198961015840119891 (119905 120591)119870119860 (119906 120591) 119889120591 (31)


119895(1198861119905+11988621199052+11988631199053) into 1198961015840119891(119905 120591) we obtain

1198961015840119891(119905 120591) = 119860

40119890119895(4(1198861119905+11988631199053)+3119886

31199051205912) So the phase of 1198961015840119891(119905 120591) is



GWD119860119891 (119905 119906)

=

11986040radic

2120587119887

119895119890119895((1198892119887)1199062+4(119886

1119905+11988631199053))120575 (119906) 119886 + 61198871198863119905 = 0

11986040

radic119886 + 61198871198863119905

times119890119895((1198892119887)1199062+4(119886

1119905+11988631199053)minus1199062(2119886119887+241198872119905119886

3)) 119886 + 61198871198863119905 = 0

(32)


31199051205912) into the


GWD119860119891 (119905 119906) =11986040

radic1198952120587119887119890119895((1198892119887)1199062+4(119886

1119905+11988631199053))

times int

+infin


3119905)1205912119889120591

(33)



1198863 = minus119886

6119887 argmax119905 |GWDL (119905 0)| (34)


3







1198862 =argmaxΩ |CPF (119899 Ω)|

2 (35)



1198861 = argmax120596

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119873minus1)2

sum

119899=minus(119873minus1)2

119891 (119905) 119890minus119895(11988621198992+11988631198993)minus119895120596119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

(36)


1198600 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873

(119873minus1)2

sum

119899=minus(119873minus1)2

119891 (119905) 119890minus119895(1198861119899+11988621198992+11988631198993)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(37)


10038161003816100381610038161003816119886119894Δ

11989410038161003816100381610038161003816le

120587

119894((119873 minus 1) 2)119894minus1 119894 = 1 2 3 (38)



300

250

200

150

100

50

01000

500

0

t

0200

400600

8001000

u

Am

plitu

de


3))

500

t200

400600

800

u







300

250

200

150

100

100 200 300 400 500 600 700 800 900 1000

50

0

0

t

Am

plitu

de


3))


300

250

200

150

100

100 200 300 400 500 600 700 800 900 1000

50

0

0

u

Am

plitu


2+ 001t

3))


5 Conclusions



100 200 300 400 500 600 700 800 900 10000

u

100

200

300

400

500

600

700

800

900

1000

0

t


3))




Acknowledgment


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











WD119860119891 (119905 119906) = radic119890minus119895120572119882120572119891 (119905 119906) (10)





WD119860119891119892 (119905 119906) = int+infin


120591

2) 119892

lowast(119905 minus

120591

2)119870119860 (119906 120591) 119889120591

(11)




119891 (minus119905 119906)]lowast

(12)


is

WD119860119891(119905)119890119895120596119905

2 (119905 119906) = 1198901198952119889120596119905119906minus119895211988711988912059621205912WD119860119891 (119905 119906 minus 2119887120596119905) (13)


WD119860119891(119905minus1199050)119890

119895120596119905 (119905 119906) = 119890119895(119906119889120596minus11988711988912059622)


TheWDL of signal 119892(119905) = 119891(119905)1198901198951205961119905+11989512059621199052

is

WD119860119892(119905) (119905 119906) = 119890119895(119906119889(120596

1+21205962119905)minus119887119889(120596

1+21205962119905)22)


Especially when 119891(119905) = 1 119892(119905) = 1198901198951205961119905+11989512059621199052


WD119860119892(119905) (119905 119906)

=

radic2120587119887

119895119890119895(1198892119887)1199062

120575 (21205962119887119905 + 1205961119887 minus 119906) 119886 = 0

radic1

119886119890119895(1198892119887)1199062minus119895((119906minus120596

1119887minus2119887120596

2119905)22119886119887)

119886 = 0

(16)


int

+infin


119889119905 = radic120587

119874119890(1198752119874+119876)

(17)


WD119860119892(119905) (119905 119906) = int

+infin


120591

2) 119892

lowast(119905 +

120591

2)119870119860 (119906 120591) 119889120591

= radic1

1198952120587119887119890119895(1198892119887)1199062

times int

+infin


119889120591

(18)


WD119860119892(119905) (119905 119906) = radic1

1198952120587119887119890119895(1198892119887)1199062

int

+infin


119889120591

= radic2120587119887

119895119890119895(1198892119887)1199062

120575 (21205962119905119887 + 1205961119887 minus 119906)

(19)


WD119860119892(119905) (119905 119906) = radic1

119886119890119895(1198892119887)1199062minus119895((2119887120596

2119905+1205961119887minus119906)22119886119887)

(20)

Then we obtain (16)


WD119860ℎ (119905 119906) = radic1

|120582|WD119861119891 (120582119905 119906) (21)

where 119861 = (119886120582 119887120582 119888120582 119889120582)




AF119860119891 (120591 119906) = int+infin

minusinfin119896119891 (119905 120591)119870119860 (119906 119905) 119889119905 (22)


119882119863119860119891 (119905 119906) = ∬

+infin


119860119891 (120591 V) 119870119860119860minus1 (120591 V 119906 119905) 119889120591 119889V (23)




119878119908119891 (119905 120596) = int

+infin

minusinfin119891 (119906) 119892

lowast(119906 minus 119905) 119890

minus119895120596119906119889119906 (24)


119882119863119860119891 (119905 119906)

= radic1

1198952120587119887∬

+infin


sdot 119878119882119891 (120591

2V minus 119886120591119887

)

times 119870119860119860minus1 (120591 V 119906 119905) 119889120591 119889V

(25)

where 119878119882119891 (120591 (2V minus 119886120591)2119887) = int

+infin


1015840)[119891(119905

1015840minus 120591)

119890minus119895(1198862119887)(1199051015840minus120591)

2


1198891199051015840


AF119860119891 (120591 119906) = radic1

1198952120587119887119890119895((1198892119887)1199062+1199061205912119887minus311988612059128119887)

sdot 119878119882119891 (120591

2119906 minus 119886120591

119887)

(26)



WT119891 (119886 119887) =1

radic119886int

+infin

minusinfin119891 (119905) 120595


119886)119889119905 (27)



119882119863119860119891 (119905 119906) = radic

1

1198952120587119887

times∬

+infin


sdotWT119891 (1 120591)

times 119870119860119860minus1 (120591 V 119906 119905) 119889120591 119889V(28)



AF119860119891 (120591 119906) = radic1

1198952120587119887119890119895((1198892119887)1199062minus1199061205912119887+11988612059128119887)

sdotWT119891 (1 120591)

(29)





119895(1198861119905+11988621199052+11988631199053) where


WD119860119891 (119905 119906) = radic1

119895212058711988711986020119890119895(1198892119887)1199062

times int

+infin


312059134]

119889120591

(30)





(1198951198882119886) 1199062


119890119895120582119905

radic2120587119887

119895119890119895(1198892119887) 1199062

120575(119906 minus 120582119887) 119886 = 0

radic1

119886119890119895(1198892119887)1199062minus119895((119906minus1205961119887)

22119886119887) 119886 = 0

11989011989512058311990522

radic2120587119887

119895119890119895(1198892119887)1199062

120575(119906 minus 2120583119887119905) 119886 = 0

radic1

119886119890119895(1198892119887)1199062minus119895((119906minus2119887120583119905)22119886119887)

119886 = 0

11989011989512058311990522+119895120582119905+119895120574

radic2120587119887

119895119890119895(1198892119887)1199062

120575(119906 minus 120582119887 minus 2120583119887119905) 119886 = 0

radic1


119886 = 0

119890minus11990522

radic2120587119887

119895119890119895(1198892119887)1199062

120575(119895119887119905 minus 119906) 119886 = 0

radic1

119886119890(1198951198882119886)1199062minus(119905119906119886)+119895(11990521198872119886)

119886 = 0




follows

GWD119860119891 (119905 119906) = int+infin

minusinfin1198961015840119891 (119905 120591)119870119860 (119906 120591) 119889120591 (31)


119895(1198861119905+11988621199052+11988631199053) into 1198961015840119891(119905 120591) we obtain

1198961015840119891(119905 120591) = 119860

40119890119895(4(1198861119905+11988631199053)+3119886

31199051205912) So the phase of 1198961015840119891(119905 120591) is



GWD119860119891 (119905 119906)

=

11986040radic

2120587119887

119895119890119895((1198892119887)1199062+4(119886

1119905+11988631199053))120575 (119906) 119886 + 61198871198863119905 = 0

11986040

radic119886 + 61198871198863119905

times119890119895((1198892119887)1199062+4(119886

1119905+11988631199053)minus1199062(2119886119887+241198872119905119886

3)) 119886 + 61198871198863119905 = 0

(32)


31199051205912) into the


GWD119860119891 (119905 119906) =11986040

radic1198952120587119887119890119895((1198892119887)1199062+4(119886

1119905+11988631199053))

times int

+infin


3119905)1205912119889120591

(33)



1198863 = minus119886

6119887 argmax119905 |GWDL (119905 0)| (34)


3







1198862 =argmaxΩ |CPF (119899 Ω)|

2 (35)



1198861 = argmax120596

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119873minus1)2

sum

119899=minus(119873minus1)2

119891 (119905) 119890minus119895(11988621198992+11988631198993)minus119895120596119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

(36)


1198600 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873

(119873minus1)2

sum

119899=minus(119873minus1)2

119891 (119905) 119890minus119895(1198861119899+11988621198992+11988631198993)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(37)


10038161003816100381610038161003816119886119894Δ

11989410038161003816100381610038161003816le

120587

119894((119873 minus 1) 2)119894minus1 119894 = 1 2 3 (38)



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AF119860119891 (120591 119906) = int+infin

minusinfin119896119891 (119905 120591)119870119860 (119906 119905) 119889119905 (22)


119882119863119860119891 (119905 119906) = ∬

+infin


119860119891 (120591 V) 119870119860119860minus1 (120591 V 119906 119905) 119889120591 119889V (23)




119878119908119891 (119905 120596) = int

+infin

minusinfin119891 (119906) 119892

lowast(119906 minus 119905) 119890

minus119895120596119906119889119906 (24)


119882119863119860119891 (119905 119906)

= radic1

1198952120587119887∬

+infin


sdot 119878119882119891 (120591

2V minus 119886120591119887

)

times 119870119860119860minus1 (120591 V 119906 119905) 119889120591 119889V

(25)

where 119878119882119891 (120591 (2V minus 119886120591)2119887) = int

+infin


1015840)[119891(119905

1015840minus 120591)

119890minus119895(1198862119887)(1199051015840minus120591)

2


1198891199051015840


AF119860119891 (120591 119906) = radic1

1198952120587119887119890119895((1198892119887)1199062+1199061205912119887minus311988612059128119887)

sdot 119878119882119891 (120591

2119906 minus 119886120591

119887)

(26)



WT119891 (119886 119887) =1

radic119886int

+infin

minusinfin119891 (119905) 120595


119886)119889119905 (27)



119882119863119860119891 (119905 119906) = radic

1

1198952120587119887

times∬

+infin


sdotWT119891 (1 120591)

times 119870119860119860minus1 (120591 V 119906 119905) 119889120591 119889V(28)



AF119860119891 (120591 119906) = radic1

1198952120587119887119890119895((1198892119887)1199062minus1199061205912119887+11988612059128119887)

sdotWT119891 (1 120591)

(29)





119895(1198861119905+11988621199052+11988631199053) where


WD119860119891 (119905 119906) = radic1

119895212058711988711986020119890119895(1198892119887)1199062

times int

+infin


312059134]

119889120591

(30)





(1198951198882119886) 1199062


119890119895120582119905

radic2120587119887

119895119890119895(1198892119887) 1199062

120575(119906 minus 120582119887) 119886 = 0

radic1

119886119890119895(1198892119887)1199062minus119895((119906minus1205961119887)

22119886119887) 119886 = 0

11989011989512058311990522

radic2120587119887

119895119890119895(1198892119887)1199062

120575(119906 minus 2120583119887119905) 119886 = 0

radic1

119886119890119895(1198892119887)1199062minus119895((119906minus2119887120583119905)22119886119887)

119886 = 0

11989011989512058311990522+119895120582119905+119895120574

radic2120587119887

119895119890119895(1198892119887)1199062

120575(119906 minus 120582119887 minus 2120583119887119905) 119886 = 0

radic1


119886 = 0

119890minus11990522

radic2120587119887

119895119890119895(1198892119887)1199062

120575(119895119887119905 minus 119906) 119886 = 0

radic1

119886119890(1198951198882119886)1199062minus(119905119906119886)+119895(11990521198872119886)

119886 = 0




follows

GWD119860119891 (119905 119906) = int+infin

minusinfin1198961015840119891 (119905 120591)119870119860 (119906 120591) 119889120591 (31)


119895(1198861119905+11988621199052+11988631199053) into 1198961015840119891(119905 120591) we obtain

1198961015840119891(119905 120591) = 119860

40119890119895(4(1198861119905+11988631199053)+3119886

31199051205912) So the phase of 1198961015840119891(119905 120591) is



GWD119860119891 (119905 119906)

=

11986040radic

2120587119887

119895119890119895((1198892119887)1199062+4(119886

1119905+11988631199053))120575 (119906) 119886 + 61198871198863119905 = 0

11986040

radic119886 + 61198871198863119905

times119890119895((1198892119887)1199062+4(119886

1119905+11988631199053)minus1199062(2119886119887+241198872119905119886

3)) 119886 + 61198871198863119905 = 0

(32)


31199051205912) into the


GWD119860119891 (119905 119906) =11986040

radic1198952120587119887119890119895((1198892119887)1199062+4(119886

1119905+11988631199053))

times int

+infin


3119905)1205912119889120591

(33)



1198863 = minus119886

6119887 argmax119905 |GWDL (119905 0)| (34)


3







1198862 =argmaxΩ |CPF (119899 Ω)|

2 (35)



1198861 = argmax120596

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119873minus1)2

sum

119899=minus(119873minus1)2

119891 (119905) 119890minus119895(11988621198992+11988631198993)minus119895120596119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

(36)


1198600 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873

(119873minus1)2

sum

119899=minus(119873minus1)2

119891 (119905) 119890minus119895(1198861119899+11988621198992+11988631198993)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(37)


10038161003816100381610038161003816119886119894Δ

11989410038161003816100381610038161003816le

120587

119894((119873 minus 1) 2)119894minus1 119894 = 1 2 3 (38)



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Algebra












(1198951198882119886) 1199062


119890119895120582119905

radic2120587119887

119895119890119895(1198892119887) 1199062

120575(119906 minus 120582119887) 119886 = 0

radic1

119886119890119895(1198892119887)1199062minus119895((119906minus1205961119887)

22119886119887) 119886 = 0

11989011989512058311990522

radic2120587119887

119895119890119895(1198892119887)1199062

120575(119906 minus 2120583119887119905) 119886 = 0

radic1

119886119890119895(1198892119887)1199062minus119895((119906minus2119887120583119905)22119886119887)

119886 = 0

11989011989512058311990522+119895120582119905+119895120574

radic2120587119887

119895119890119895(1198892119887)1199062

120575(119906 minus 120582119887 minus 2120583119887119905) 119886 = 0

radic1


119886 = 0

119890minus11990522

radic2120587119887

119895119890119895(1198892119887)1199062

120575(119895119887119905 minus 119906) 119886 = 0

radic1

119886119890(1198951198882119886)1199062minus(119905119906119886)+119895(11990521198872119886)

119886 = 0




follows

GWD119860119891 (119905 119906) = int+infin

minusinfin1198961015840119891 (119905 120591)119870119860 (119906 120591) 119889120591 (31)


119895(1198861119905+11988621199052+11988631199053) into 1198961015840119891(119905 120591) we obtain

1198961015840119891(119905 120591) = 119860

40119890119895(4(1198861119905+11988631199053)+3119886

31199051205912) So the phase of 1198961015840119891(119905 120591) is



GWD119860119891 (119905 119906)

=

11986040radic

2120587119887

119895119890119895((1198892119887)1199062+4(119886

1119905+11988631199053))120575 (119906) 119886 + 61198871198863119905 = 0

11986040

radic119886 + 61198871198863119905

times119890119895((1198892119887)1199062+4(119886

1119905+11988631199053)minus1199062(2119886119887+241198872119905119886

3)) 119886 + 61198871198863119905 = 0

(32)


31199051205912) into the


GWD119860119891 (119905 119906) =11986040

radic1198952120587119887119890119895((1198892119887)1199062+4(119886

1119905+11988631199053))

times int

+infin


3119905)1205912119889120591

(33)



1198863 = minus119886

6119887 argmax119905 |GWDL (119905 0)| (34)


3







1198862 =argmaxΩ |CPF (119899 Ω)|

2 (35)



1198861 = argmax120596

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119873minus1)2

sum

119899=minus(119873minus1)2

119891 (119905) 119890minus119895(11988621198992+11988631198993)minus119895120596119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

(36)


1198600 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873

(119873minus1)2

sum

119899=minus(119873minus1)2

119891 (119905) 119890minus119895(1198861119899+11988621198992+11988631198993)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(37)


10038161003816100381610038161003816119886119894Δ

11989410038161003816100381610038161003816le

120587

119894((119873 minus 1) 2)119894minus1 119894 = 1 2 3 (38)



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References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article The Wigner-Ville Distribution Based on ...Wigner-Ville distribution based on linear...

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Transcript of Research Article The Wigner-Ville Distribution Based on ...Wigner-Ville distribution based on linear...