Cross-terms reduction in the Wigner–Ville distribution using tunable-Q wavelet transform

doi:10.1016/j.sigpro.2015.07.026

Signal Processing

Volume 120, March 2016, Pages 288-304

https://doi.org/10.1016/j.sigpro.2015.07.026 Get rights and content

Highlights

•
This paper presents a new method based on TQWT for cross terms reduction in WVD.
•
The proposed method has been studied on multi-component non-stationary signals.
•
The simulation results have been compared with the other existing methods.
•
The proposed method works well even in the presence of noise.

Abstract

This paper proposes a new method to reduce cross-terms in the Wigner–Ville distribution (WVD) using tunable-Q wavelet transform (TQWT). The suggested method exploits the advantages of sub-band filtering of filter-bank and also retaining the time-resolution property of the wavelet decomposition to achieve signal decomposition. Signal components in sub-bands obtained using TQWT are further separated in time-domain using time-domain energy distribution to eliminate inner-interference terms. Simulation results for multi-component non-stationary signals are presented in order to show the efficacy of the suggested method for cross-terms reduction in WVD. Results are compared with the existing methods based on the Fourier–Bessel (FB) series expansion and filter-bank based cross-terms reduction methods in WVD, in order to show the advantages over the compared methods.

Introduction

Non-stationary signals, for which model contains time-varying parameters, are commonly found in many areas like radar, communications, speech analysis and synthesis, biomedical, and mechanical engineering [1]. For such kind of signals, the Fourier transform based method is not suitable for spectral analysis. The time–frequency analysis based methods have been proposed for analysis of such signals [1], [2]. The analysis of non-stationary signals in time–frequency domain can contribute features for classification of non-stationary signals [3].

The short-time Fourier transform (STFT) provides time–frequency analysis of the non-stationary signals using moving window [4]. The use of window imposes a compromise between time-localization and frequency-localization [5]. Another widely employed time–frequency technique for analysis of the non-stationary signals is the Wigner–Ville distribution (WVD). In ideal case, the WVD has an infinite resolution in time-domain and frequency-domain due to the absence of averaging over any finite time duration [1].

It should be noted that in nature the WVD is quadratic and generates cross-terms for the multi-component non-stationary signals. The cross-terms in the WVD consist of outer-interference terms and inner-interference terms [1]. The outer-interference terms result if there are more than one mono-component, and inner-interference terms result due to nonlinear frequency modulated (FM) components [1]. The presence of cross-terms is a serious limitation of the WVD based method for time–frequency representation of non-stationary signals and sometimes these cross-terms can have significant magnitude in time–frequency plane which may mislead analysis interpretation [6].

In the literature, many methods have been proposed for the reduction of cross-terms. By using appropriate kernel functions, distributions with reduced interference terms were suggested which can suppress cross-terms and also preserve the mathematical properties of the WVD [1]. One such distribution using exponential kernel is known as the Choi–Williams distribution (CWD) or exponential distribution [7]. This distribution contains many mathematical properties of the WVD. It has been shown that kernels can be designed for reducing the interference from the WVD [8]. However, the fixed kernel based methods work only for a particular class of signals and do not provide cross-terms free time–frequency representation for non-stationary signals in general [9]. Significant distortions can take place in kernel based methods for cross-terms reduction in WVD [10]. Signal dependent kernel for reduction of cross-terms in WVD is proposed in [8]. A comparison of signal dependent time–frequency representation with fixed kernel time–frequency representations like WVD and CWD has been studied [11], [12]. In [13], a method is presented where the multiple smoothed pseudo-WVDs are used to reduce interference in the WVD. Similarly, an auto-term window method is presented in [14] to reduce the cross-terms in the WVD by appropriate selection of parameters like window function and threshold value. The auto-term window presented in [15] is used to suppress cross-terms in the WVD where it enhances the energy of auto-terms also.

New time–frequency distribution (TFD) based on the polynomial WVD and L class of the WVD has been developed specially for polynomial phase signals [16]. An instantaneous frequency (IF) estimation algorithm in noisy environment has been suggested based on the local singular value decomposition and the WVD [17]. It has been studied only for mono-component nonlinear frequency modulated (FM) signals. In [18], a method has been proposed where IF of mono-component nonlinear FM signal is estimated using pseudo-WVD along with adaptive window. A method based on shift invariant wavelet packet decomposition is developed for cross-terms reduction in WVD [19]. IF estimation has been studied for non-linear FM signals in the presence of noise using polynomial WVD in [20]. Statistical modeling and denoising methods were proposed to remove noise component and to estimate IF in the WVD and smoothed pseudo-WVD [21]. The use of two-dimensional signal processing techniques to reduce interference in WVD has been proposed in which fractional Fourier transform is used to isolate components, which are identified using image processing technique [22]. The use of morphological operators to remove cross-terms in the WVD has been proposed in [23]. In this method, the time–frequency image generated by thresholding spectrogram is used as the marker for performing λ-reconstruction. Only those components in the WVD of signal are retained which are also present in spectrogram. IF estimation for multi-component signals based on image processing techniques in time–frequency domain is proposed in [24]. Cross-terms reduction in the discrete WVD by applying non-linear filtering is suggested in [25].

It has been shown that if the signal has a single component without non-linear FM then the WVD will not have cross-terms [1]. Interference terms can be removed if the signal under analysis is decomposed into mono-component signals without non-linear FM. The uses of filter-bank and signal decomposition have been suggested before applying the WVD [10], [26]. A band-pass filter-bank is used to separate the signal in frequency-domain and the WVD is computed for each sub-band signal in [10]. Cross-terms can also result, if signal has components separable in time-domain. To avoid such type of cross-terms, the pseudo-WVD has been used in [27]. The Fourier–Bessel (FB) series expansion is used to separate the signal into components before computing the WVD [28]. This method proved effective for signals when components are separable in frequency-domain. When mono-component signals are well separated in time–frequency domain, then time-order representation based on the short-time FB series expansion has been proposed to separate the mono-component signals before computing the WVD in order to reduce cross-terms [29]. Similarly in [30], the FB transform has been used to separate the mono-component signals from the multi-component non-stationary signals before computing the WVD. The methods proposed in [28], [29], [30] require the identification of range of FB coefficients corresponding to each mono-component signals in order to separate the mono-component signals. In this paper, we propose a method to reduce cross-terms by segmenting signal both in frequency-domain and time-domain. In the proposed method, the tunable-Q wavelet transform (TQWT) [31] provides a frame work for sub-band filtering of signals, while providing time-localization. The TQWT is used to decompose the signal in frequency-domain and time-domain energy distribution is used to separate signal in time-domain to avoid windowing to achieve high concentrated energy distributions. The proposed method in this paper for cross-terms reduction in the WVD shows better performance even in the presence of noise.

The paper is organized as follows: a brief overview of the WVD followed by TQWT is presented in Section 2. The proposed method for cross-term reduction is presented in Section 3. In Section 4, the computational cost of the suggested method is provided. The Rényi entropy which has been used for performance evaluation of the proposed method is discussed in Section 5. Simulation results are shown in Section 6. Finally, conclusion is presented in Section 7.

Section snippets

Wigner–Ville distribution (WVD)

The WVD can be considered Fourier transform of the instantaneous autocorrelation function. Its mathematical expression in time-domain is as follows [1], [6], [32]: ${WVD}_{y} (t, ω) = \int_{- \infty}^{+ \infty} y (t + \frac{τ}{2}) y^{⁎} (t - \frac{τ}{2}) e^{- j ω τ} d τ$ where $y^{⁎} (t)$ represents the complex conjugate of y(t).

It can be seen from (1) that the WVD is bilinear in nature. Therefore, it deteriorates from the presence of cross-terms, if the signal under analysis is either multi-component signal or non-linear FM mono-component signal. The WVD has cross-terms

Cross-terms reduction using tunable-Q wavelet transform

It should be noted that there will be no cross-terms in the WVD if the signal under analysis consists of a single linear frequency modulated (LFM) mono-component signal. If a multi-component signal is decomposed into number of such signals, then it will be possible to have a cross-terms free time–frequency representation by summing up the WVD of the individual components. The problem lies in decomposition of a multi-component, non-linear FM signals into such LFM mono-component signals. This

Computational cost of the proposed method

The proposed method is based on TQWT for cross-terms reduction in WVD. The computational cost for a radix-2 TQWT of N point sequence is $O (rN \log_{2} N)$ [31] where r is the redundancy factor. In the proposed method, if the total number of TQWT blocks in TQWT array is M, then the computational cost for the TQWT part is $M \times O (rN \log_{2} N)$ . Then few number of sub-bands are selected from each TQWT block which are well localized in frequency-domain. Assuming an average of L number of sub-bands selected from

Performance evaluation

In order to quantitatively judge the performance of suggested time–frequency representation based on WVD, normalized Rényi entropy measure of TFD proposed in [35] has been used. The normalized Rényi entropy measure is computed using Rényi entropy [36]. The mathematical expression of Rényi entropy is given as [35] $R_{γ} = (1 / (1 - γ)) \log_{2} [\sum_{l}^{} \sum_{k} [C_{s}^{γ} (l, k)]]$ , where $C_{s} (l, k)$ is Cohen's class TFD [35] and γ is the order of information and chosen as 3 [37] for performance evaluation. The Rényi entropy is used

Simulation results

The proposed method for cross-terms reduction in WVD has been studied for four representative multi-component non-stationary signals and the same is compared with the WVD based time–frequency representation obtained using FB series expansion based method [28] or time-order based method [29], and filter-bank based method [10]. The brief description of these signals namely $y_{1} [n]$ , $y_{2} [n]$ , $y_{3} [n]$ , and $y_{4} [n]$ is as follows:

1.
$y_{1} [n]$ : The signal $y_{1} [n]$ is a two-component LFM non-stationary signal whose

Conclusion

In this work, a novel TQWT based technique is proposed for cross-terms reduction in the WVD of multi-component non-stationary signals. In the proposed method, the array of Q is used for decomposition of signal, which has generated different mother wavelets for different Qs. This is followed by decomposition of signal in time-domain using energy based segmentation. The time–frequency resolution of proposed method is better than other compared methods. This can also be observed by performance

Acknowledgment

The authors wish to thank Curtis Condon, Ken White, and Al Feng of the Beckman Institute of the University of Illinois for the bat data and for permission to use it in this paper.

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Cross-terms reduction in the Wigner–Ville distribution using tunable-Q wavelet transform

Highlights

Abstract

Introduction

Section snippets

Wigner–Ville distribution (WVD)

Cross-terms reduction using tunable-Q wavelet transform

Computational cost of the proposed method

Performance evaluation

Simulation results

Conclusion

Acknowledgment

Digit. Signal Process.

Signal Process.

Measurement

Measurement

IET Signal Process.

Signal Process.

Digit. Signal Process.

J. Vis. Commun. Image Represent.

Signal Process.

Signal Process.

Digit. Signal Process.

Expert Syst. Appl.

J. Frankl. Inst.

Signal Process.

Time–Frequency Signal Analysis and ProcessingA Comprehensive Reference

Time–frequency representation of digital signals and systems based on short-time Fourier analysis

IEEE Trans. Acoust. Speech Signal Process.

Time–Frequency Analysis

A comparison of the existence of "cross terms" in the Wigner distribution and the squared magnitude of the wavelet transform and the short-time Fourier transform

IEEE Trans. Signal Process.

Improved time–frequency representation of multicomponent signals using exponential kernels

IEEE Trans. Acoust. Speech Signal Process.