Wigner-Ville Distribution Associated with the Linear Canonical Transform

The linear canonical transform is shown to be one of the most powerful tools for nonstationary signal processing. Based on the properties of the linear canonical transform and the classical Wigner-Ville transform, this paper investigates theWigner-Ville distribution in the linear canonical transform domain. Firstly, unlike the classical Wigner-Ville transform, a new definition of WignerVille distribution associated with the linear canonical transform is given. Then, the main properties of the newly defined Wigner-Ville transform are investigated in detail. Finally, the applications of the newly defined Wigner-Ville transform in the linear-frequency-modulated signal detection are proposed, and the simulation results are also given to verify the derived theory.


Introduction
With the development of the modern signal processing technology for the nonstationary signal processing, a series of novel signal analysis theories and processing tools have been put forward to meet the requirements of modern signal processing, for example, the shorttime Fourier transform 1 , the wavelet transform WT 2 , the ambiguity function AF 3 , the Wigner-Ville distribution WVD 4 , the the fractional Fourier transform FRFT , and the linear canonical transform LCT 5-7 .Recently, more and more results 8, 9 show that the LCT is one of the most powerful signal processing tools; it receives much interests in signal processing community and has been applied in many fields, such as the time-frequency analysis 10 , the filter design 11 , the pattern recognition 6 , encryption, and watermarking 12 .For more results associated with the LCT, one can refer to 5-7 .
The linear-frequency-modulated LFM signal is one of the most important nonstationary signals, which is widely used in communications, radar, and sonar system 13-17 .The detection and parameter estimation of LFM signal are important in signal processing where A a b c d is the parameter matrix of LCT satisfying ad − bc 1, that is, det A 1. The inverse transform of the LCT ILCT is given by an LCT having parameter From the definition of LCT, we can see that, when b 0, the LCT of a signal is essentially a chirp multiplication and it is of no particular interest to our object.Therefore, without loss of generality, we set b > 0 in the following sections of the paper.
It is shown in 5-7 that the FT, FRFT, chirp, and scaling operations are all the special cases of the LCT.Therefore, the LCT can be used to solve some problems that cannot be solved well by these operations 20 .The well-known theories and concepts in the classical Fourier transform domain are generalized to the LCT domain by different researchers.The uniform and nonuniform sampling theories are well studied in the LCT domain and showed that we can obtain the better results compared to the classical ones in the Fourier domain 8, 9, 21-23 .The other concepts, for example, the WVD 10 , the convolution and product theories 24, 25 , the uncertainty principle 26 , the spectral analysis 27 , and the eigenfunctions 28 , are also proposed and investigated in the LCT domain.The discrete methods and the fast computation of the LCT are investigated in detail in 29-31 .

The Wigner-Ville Distribution (WVD)
The instantaneous autocorrelation function of a signal f t is defined as 1 and the classical WVD of f t is defined as the FT of The WVD is one of the most powerful time-frequency analysis tools and has a series of good properties, the main properties of the WVD are listed as follows.

The Previous Results about WVD Associated with LCT
With the developments of the FRFT and the LCT, Almeida in 32 and Lohmann in 33 investigate the relationship between the WVD and the FRFT; they show that the WVD of the FRFTed signal can be seen as a rotation of the classical WVD in the time-frequency plane.Along this direction, Pei and Ding discuss the relationship between the classical WVD and the WVD associated with the LCT 10 .In their definition, suppose the LCT of a signal f t with parameter A is denoted as F A u L A f t u ; then the WVD associated with the LCT is defined as It is shown in 10, 32, 33 that this definition of the WVD associated with the LCT can be seen as the rotation or affine transform of the LCTed signal in the time-frequency plane.If the classical WVD of a signal f t is denotes as W f t, w and the newly defined WVD associated in 2.9 is denotes as W F A u, v , we have the following result 10 : where Unlike the WVD definition in 2.9 associated with the LCT, we propose a new kind of definition for WVD in the LCT domain and the potential applications in the LFM signal detection are also proposed in the following sections.

The New Definition of WVD
Based on the properties of the LCT and the form of the classical WVD definition associated with the Fourier transform, we give a new definition of WVD by the LCT of instantaneous autocorrelation function R f t, τ .In other words, we take place of the kernel of FT with the kernel of LCT to get a new kind of WVD associated with the LCT as follows.
Definition 3.1.Suppose the kernel of the LCT with parameter A is K A t, u ; then the WVD of a signal f t associated with the LCT is defined as In order to make different from the existing results about the WVD, we denote the WVD associated with the LCT for parameter A a, b; c, d by W f A t, u and simplified as the WDL of f t .
The LCT of a signal f t can be looked as the affine transform of the signal in the time-frequency plane; so the WDL of a signal can be interpreted as the affine transform of the instantaneous autocorrelation function R f t, τ of this signal in the time-frequency plane.Some of the important properties are investigated in the following subsection.

The Properties
Suppose the WDL of a signal f t is denoted as W f A t, u , then the following important properties of WDL can be obtained.
1 Conjugation symmetry property: the WDL of 4 Inverse property: the signal f t can be expressed by the WDL of f t as: Proof.From the definition of WDL for a signal f t , we know By the inverse transform of the LCT, we obtain the instantaneous autocorrelation function R f t, τ as follows: Letting τ/2 t, 3.7 will reduce to and the final result can be obtained by letting 2t s

3.10
Proof.From the definition of the WDL, we obtain

3.11
Let μ t − τ/2; then the above equation reduces to the final result:

6
The relationship between the classical WVD and WDL from the definition of LCT, it is easy to verify that when the parameter A reduces to A 0, 1; −1, 0 , the WDL reduces to the classical WVD.In this sense, the WDL can be seen as the generalization of the classical WVD to the LCT domain:

Applications of the WDL
The newly defined WDL is applied in the LFM signal detection in this section, the oneand two-component LFM signals are analyzed with the WDL in the LCT domain, and the simulation results are also proposed to verify the derived results.

One-Component LFM
If the LFM signal is modeled as f t e j w 0 t mt 2 /2 ; w 0 , m represent the initial frequency and frequency rate of f t , respectively.From the definition of the WDL, the WDL of f t is

4.1
We can see from this equation that if we choose the especial parameter, the WDL of f t will produce an impulse in t, u plane.From this fact, we propose the following algorithm for the detection and estimation of the of LFM signal by WDL.
Step 1. Compute the WDL of a signal.
Step 2. Search for the peak values in the time-frequency plane, then estimate the instantaneous frequency.
Step 3. Apply the least-squares ap proximation to the instantaneous frequency and obtain the final estimation value.The diagram of the LFM signal detection can be summarized in Figure 1.

Bicomponent Signal
When the processing signal is modeled as a bicomponent finite-length signal as follows.
this signal can be expressed as f t f 1 t f 2 t , and the WDL of f t can be represented by the WDL of f 1 t and f 2 t as follows: The first two terms represent the autoterms of the signal, whereas the rest is the crossterm.If the parame a, b, c, d are chosen to be special numbers, the graph of WDL for signal f t will be composed of the WDL of f 1 t and f 2 t , respectively.

The WDL of One-Component LFM
The simulations are performed to verify the derived results; a finite-length LFM signal as follows is chosen:  2, and the projection of W f A t, u onto time-frequency plane is plotted in Figure 3.
We can see from Figures 2 and 3 that the WDL of f t has the energy accumulation property.Energy is accumulated in a straight line of the plane t, u , which is the same as discussed before.

The Parameter Estimation of One LFM
Suppose the signal f t is added with the white Gaussian noise; then it can be modeled as f t e j w 0 t m 0 t 2 /2 n t , |t| < T 2 .the initial parameters are set as w 0 10, m 0 0.8, and the length of signal T 40.The magnitude of the WDL of f t and the contour picture of the above signal is plotted in Figures 4 and 5, respectively.

4.5
Applying the parameter estimation algorithm as shown in Figure 1, search for the peak value in the time-frequency plane of WDL, we can obtain the instantaneous frequency as shown in Figure 6.Applying least-squares approximation to the instantaneous frequency and obain the ultimate instantaneous frequency estimation value m 0 0.808, w 0 9.8918.

Comparison with the Classical WVD
In order to compare the WVD with the WDL, we investigate the performance of peak value estimating method of them for the signal f t added with noise.The contour picture of WVD and WDL of f t with SNR −5 dB is plotted in Figures 7 and 8, respectively.From Figures 7 and 8, we can obtain better results by the WDL under the low SNR circumstance as we discussed before.

Conclusion
Based on the LCT and the classical WVD theory, this paper proposes a new kind of definition of WVD associated with the LCT, namely WDL, which can be seen as the generalization of classical WVD to the LCT domain.Its main properties are derived in detail, and the applications of the WDL in the detection the parameters of the LFM signals are investigated.The simulations are also performed to verify the derived results.The future works will be the   applications of the newly defined WDL in the nonstationary signal processing and the study of the marginal properties for Cohen's class along this direction.
. The parameters a, b, c, d are the real numbers satisfying ad − bc 1.

Figure 1 :
Figure 1: Detection algorithm diagram of instantaneous frequency.

Figure 2 :
Figure 2: The WDL of f t .

Figure 3 :Figure 4 :
Figure 3: The contour picture of WDL of f t .

Figure 5 :
Figure 5: The contour of WVD of f t with SNR 5 dB.

Figure 6 :
Figure 6: Search for the peak value in the time-frequency plane of WDL.

Figure 7 :
Figure 7: The contour picture of WVD of f t SNR −5 dB.

Figure 8 :
Figure 8: The contour picture of WDL of f t with SNR −5 dB.