The Realization of Smoothed Pseudo Wigner-Ville Distribution Based on LabVIEW Guoqing Liu 1, a, Xi Zhang 1, b 1, c, *

Transcription

1 Applied Mechanics and Materials Online: ISSN: , Vols , pp doi: / Trans Tech Publications, Switzerland The Realization of Smoothed Pseudo Wigner-Ville Distribution Based on LabVIEW Guoqing Liu 1, a, Xi Zhang 1, b 1, c, * and Qibing Lv 1 Welding Institute, College of Materials Science and Engineering, Southwest Jiaotong University, Chengdu, , Sichuan, P. R. China a xnjd_liuguoqing@163.com, b @qq.com, c xnjdlvqibing@163.com (corresponding author) Keywords: WVD, SPWVD, CWD, LabVIEW Abstract. The realization steps on LabVIEW of general SPWVD and one of special SPWVD named Choi-Williams distribution (CWD) were detailed in this paper. Through the comparative analysis, the series of simulation experiments clearly displayed the superiority of smoothed pseudo Wigner-Ville distribution, and also indicated that the programs of SPWVD and CWD based on LabVIEW could be applied in practical. Introduction In the past several decades, the study emphasis of Wigner-Ville distribution had focused on a hot problem that how to remove the cross-term interference and keep fine time-frequency resolution simultaneously. A series of methods were created to resolve the problem during that period [1], such as the smoothed pseudo Wigner-Ville distribution (SPWVD) and Choi-Williams distribution (CWD) which regarded as one of a special SPWVD in this paper. Smoothed pseudo Wigner-Ville distribution is the result of 2D convolution of the Wigner-Ville distribution. It could remove the cross-terms and keep fine time-frequency resolution simultaneously [2]. As the attractive advantages, the SPWVD should be widely applied in signal processing. But so far, there is no articles discussing how to realize the algorithm of SPWVD in practical programming, which has suppressed the applications of SPWVD. In order to promote the development of SPWVD in signal processing, the programs of general SPWVD and special SPWVD were developed based on high-level programming language LabVIEW in this paper. LabVIEW (Laboratory Virtual Instrument Engineering Workbench) is a graphical programming environment developed by National Instruments in 1986 [3]. LabVIEW is different from the previous high level programming language not based on text, but the newest data flow chart, and uses the terms, icons and concepts that the technicians, scientists and engineers are familiar with. So LabVIEW is one of tools that faces to the final users and widely used in data acquisition and signal processing. WVD and SPWVD WVD. For a signal (), the definition of Wigner-Ville distribution is the Fourier transform of instantaneous auto correlation, WVD (t,f) = st+ s t e dτ. (1) The product ( +/2) ( /2) is Hermitian symmetry in τ. For s(t) = s " (t)+s (t), the WVD of signal s(t) is WVD (t,f) = WVD # (t,f)+wvd $ (t,f)+2re&wvd #, $ (t,f)'. (2) All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, (ID: , Pennsylvania State University, University Park, USA-11/05/16,08:12:07)

2 1494 Measurement Technology and its Application Obviously, the WVD of the sum of two signals is not the sum of their corresponding WVD only. In addition to two auto-terms, contains one cross-term, WVD #, $ (t,f) = st+ s t e dτ. (3) Because the cross-term usually oscillates and its magnitude is twice as large as that of the auto-terms, it often obscures the useful patterns of the time- dependent spectra. The analytic signal z(t) can reduce the cross-terms introduced by negative frequency at the cost of some useful properties. In general, we calculate WVD of analytic signal z(t) instead of s(t), z(t) = s(t)+ih+s(t),. (4) SPWVD. Eq. 1 requires the range of τ is from minus infinity to plus infinity, which is not practical in real applications. Usually, a window h() in time-domain is imposed on it, i.e., PWVD / (t,f) = h(τ) zt+ z t e dτ, (5) which is the pseudo Wigner-Ville distribution (PWVD). Introducing a running window can lessen the cross-term interference at the cost of resolution. In order to further weaken the cross-term interference, we can impose a window 2(3) in frequency-domain based on PWVD to constitute SPWVD, the formula is SPWVD / (t,f) = h(τ) g(u) zt+ u z t udue dτ. (6) Procedures Discrete WVD calculation. The computer can only handle discrete data, so the continuous WVD must be discretized. If the length of sampled signal (7) is (28 +1), and the relevant analytic signal is :(7). The discrete Wigner-Ville distribution is [4]? DWVD / (n,k) = z(n+m) z (n m) e B@C/?, (7) in which, 0 7 8,0 F 8, 8 G 8, are the discrete variables of continuous variables t, f and τ. Referred to the subvi in LabVIEW,the WVD calculation is shown in Fig. 1. The time increment has been designed to reduce data calculation. Fig. 1. WVD calculation. Fig. 2. Imposed window function. Imposed window function. Imposed a discrete window function h(g) in time-domain based on Fig. 1, consisted the discrete pseudo Wigner-Ville distribution. The window h(g) is a real function with the length of (2H+1). DPWVD / (n,k) = 2 h(m) z(n+m) z (n m) e B@C/?. (8) The discrete variables of n, k and m correspond to the continuous variables t, f and τ respectively. As shown in Fig. 2 is the design of window function.

3 Applied Mechanics and Materials Vols Convolution. Apart from having imposed a window in time-domain, a window in frequency-domain must be imposed on WVD, so that we can get the SPWVD. According to properties of WVD, the WVD of two signals after convolution in time-domain equals to the product of the WVD of them in frequency-domain [5]. In consideration of the calculation of directly imposing window in frequency-domain is very large, the convolution algorithm in time-domain was come up with to replace the former. The convolution in LabVIEW is shown in Fig. 3. The discrete smoothed pseudo WVD is I Fig. 3. Convolution Q RAQ DSPWVD / (n,k) = 2 h(m) e JKLMN g(μ) z(n+m μ) z (n m μ), (9) where the discrete variables of n, k, m and µ correspond to the continuous variables t, f, τ and u respectively. The length of frequency window 2(S) is (2T +1). The Special SPWVD. If both of the following two criteria were satisfied, h(τ) = (4πατ ) # $,g(u) = e X$ BY $, (10) it comes into being the Choi-Williams distribution (CWD), which we regarded as the special SPWVD in this paper. CWD / (t,f) = (4πατ ) # $ e zt+ u z t+ u ex$ BY $ dudτ. (11) The parameter α controls the decay speed, the bigger the α, the more the cross-terms are suppressed. On the other hand, the bigger the α, the more the auto-terms are affected. Therefore, there is a trade-off for the selection of the parameter α. When α goes to zero, the CWD converges to the WVD. The LabVIEW design of time window h(τ) and frequency window g(u) in CWD are shown in Fig. 4. Experiments (a) h(τ) (b) g(u) Fig. 4. The design of time window and frequency window in CWD. In this section, the hopping signal was used to test the difference among WVD, PWVD, SPWVD and CWD. The analysis results of WVD, PWVD, SPWVD, and CWD are shown in Fig. 5. The following optimum parameters were chosen by a series of experiments: (1) Time increment was 4 in Fig. 5a of Wigner-Ville distribution. (2) Time window was Hanning with length of 129 in Fig. 5b of pseudo Wigner-Ville distribution.

4 1496 Measurement Technology and its Application (3) Time window and frequency window were Hanning with length of 129 and 65 respectively in Fig. 5c of smoothed pseudo Wigner-Ville distribution. (4) The α was 0.15 in Fig. 5d of Choi-Williams distribution, and the length of windows were chosen as the above. (a) Wigner-Ville distribution (b) Pseudo Wigner-Ville distribution (c) Smoothed pseudo Wigner-Ville distribution (d) Choi-Williams distribution Fig. 5. Comparative analysis of WVD, PWVD,SPWVD and CWD From the Fig. 5, we can see there are so many cross-terms in WVD clearly, and it proves Wigner-Ville distribution is unsuited for the practical application. As the imposed time window, the cross-terms in time-domain have been removed in pseudo Wigner-Ville distribution. However, it keeps the cross-terms in frequency-domain. The smoothed pseudo Wigner-Ville distribution further removes the cross-terms in frequency-domain by imposing frequency window. Thanks to the imposed special time window and frequency window, the Choi-Williams distribution which regarded as the special SPWVD in this paper, removes the cross-terms in time domain and frequency-domain as well. Moreover, Choi-Williams distribution can keep better time-frequency resolution than the general smoothed pseudo Wigner-Ville distribution. Conclusion The smoothed pseudo Wigner-Ville distribution was imposed time window and frequency window based on Wigner-Ville distribution, which could remove the cross-terms and keep fine time-frequency resolution. The experiments clearly displayed the superiority of smoothed pseudo Wigner-Ville distribution, and also indicated that the programs of SPWVD and CWD based on LabVIEW could be applied in practical. The programs of smoothed pseudo Wigner-Ville distribution in this paper fill the vacancy of SPWVD in LabVIEW. It will be helpful to promote the development of SPWVD in signal processing. References [1] ZHOU Hongxing, Dai Qionghai, in: SCIENCE IN CHINA (Series E) Vol. 31(2001), p [2] QIAN Shie: Time-Frequency and wavelet transforms (China Machine Press, Beijing 2005). [3] Cory L. Clark: LabVIEW digital signal processing and digital communications (McGraw-Hill Professional, New York 2005). [4] LIU Fang, YE Fei, in: Computer and Information Technology Vol. 15(2007), p. 28. [5] Hu Guangshu: Modern Signal Processing (Tsinghua University Press, Beijing 2004).

5 Measurement Technology and its Application / The Realization of Smoothed Pseudo Wigner-Ville Distribution Based on LabVIEW /