Wigner-Ville distribution for time-frequency analysis

一、Wigner-Ville分布(WVD)

物理学家 Wigner 于1932年提出了 Wigner 分布,and apply it to the field of quantum mechanics.此后,Ville Introduce it to the field of signal processing,Thus developed into a representative time-frequency analysis technology——Wigner-Wille分布,简称WVD.

1.1 Wigner-Ville分布的原理

real-valued signalx(t) 的Wigner 分布定义为：

由于Wigner-WilleThe time-bandwidth product of the distribution is reached不确定性原理the lower bound given,Therefore, no time-frequency resolution of any joint time-frequency distribution can be surpassedWigner-Wille分布.

1.2 Wigner-VilleThe cross term of the distribution

Suppose there is a multicomponent signal of ：

则根据Wigner-WilleThe definition of distribution is obtained：

the first two in the above formula,This is what we often call signal autonomy(Autoterms),It results from the correlation between each signal's own components,而后两项,is the cross term(Crossterms),is caused by the interaction between different signal components.

Wigner-VilleThe distribution has the best signal time-frequency aggregation for sinusoidal and chirp signals.但是,For multi-component chirps,其WVDThe time-frequency resolution decreases,There will be cross terms in the time-frequency plane.图1(a)、1(b) represent single-component chirp and multi-component chirp, respectivelyWigner-Villedistributed time-frequency plane.

图1(a) Single-component chirp signal

图1(b) A chirp signal with two components

由图1(b)可见,Multicomponent chirps produce cross terms,This affects the resolution of time-frequency analysis.

1.3 伪Wigner-Ville分布（PWVD）

Wigner-VilleThe presence of cross terms in the distribution affects its time-frequency resolution,Not conducive to signal detection,因此,How to reduce the cross term becomesWigner-VilleThe main goal of distribution performance improvement.

Wigner-VilleA distribution is a characteristic of a signal that represents energy in terms of energy over the full time axis,但在实际工作中,All data are selected for analysis with limited length,This is equivalent to applying a window function that slides over the time axis to the original signal.因此,通过对变量加窗函数Can reduce the negative impact of cross terms.改进后的Wigner-WilleThe distribution is called pseudoWigner-Wille分布.

伪Wigner-WilleThe distribution is defined as ：

二、基于Wigner-VilleDistributed time-frequency peak filtering algorithm

Boashash 等人于2004The time-frequency peak filtering technique was proposed in 1999（Time-Frequency Peak Filtering,TFPF）,And applied to real-valued signal filtering.它的本质是基于Wigner-VilleAn instantaneous frequency estimate of the distribution.

Time-frequency peak filtering technology can recover non-stationary signals and multi-component signals with arbitrary instantaneous frequency rules without assuming waveforms,The effective signal is enhanced after time-frequency peak filtering,Random noise is suppressed.

瞬时频率（Instantaneous Frequency,IF）Defined as the time-varying position of the spectral peak of the signal,对于信号s(t),Its instantaneous frequency can be expressed as ：

2.1 Time-frequency peak filtering(TFPF)算法的原理

The noisy signal to be processed can be represented as ：

式中：s(t) 是有效信号;n(t) is additive random noise;The purpose of filtering is to remove noise from the signalx(t) to restore a valid signals(t). 

设z(t) 是x(t) 的解析信号,According to the definition of instantaneous frequency,z(t) 可表示为：

可采用WVDthe peak value to estimate the instantaneous frequencyIF：

2.2 Time-frequency peak filtering(TFPF)算法的实现步骤

第一步,Put the original observation signal x(t) Consider the form of the instantaneous frequency of an FM signal.将 x(t) The analytical signal is obtained by frequency modulation：

 第 二 步,Take the analytical signalz(t) 的伪Wigner-Wille分布（PWVD）的峰值,Instantaneous frequency estimation for the analytical signal,作为有效信号 s(t) 的估计值：

2.3 Factors that affect the time-frequency peak filtering results

从伪Wigner-WilleIt can be seen from the definition of distribution,The window function affects the time-frequency resolution,因此,The choice of window function type and size will affectTFPF算法的滤波效果.根据参考文献[2],The formula for calculating the length of the window function is as follows：

The type of window function can be selected according to the characteristics of the signal itself,Multiple experiments can be carried out.

三、MATLAB代码

clc;
clear;
close all;

%% 仿真信号
figure;
z=amgauss(160,90,40); 
subplot(3,1,1);plot(z);title('Commonly used functions that generate signals--Gaussian amplitude modulated signal');

z=fmconst(128,0.05,50);
subplot(3,1,2);plot(real(z));title('Commonly used functions that generate signals--A frequency modulated signal with a fixed frequency');

[z, f]=fmlin(128,0.05,0.3,50); 
subplot(3,1,3);plot(real(z));title('Commonly used functions that generate signals--线性调频信号');

%% Wigner-Ville时频分布图
sig=amgauss(160,90,40);
figure;
tfrwv(sig);

sig=fmconst(128,0.05,50);
figure;
tfrwv(sig);

sig = fmlin(128,0.1,0.4);
figure;
tfrwv(sig);

sig1 = fmlin(128,0.1,0.4);
sig2 = fmlin(128,0.2,0.5);
sig = sig1+sig2;
figure;
tfrwv(sig);

%% 伪Wigner-Ville时频分布图
sig = fmlin(128,0.1,0.4);
figure;
tfrpwv(sig);

sig1 = fmlin(128,0.1,0.4);
sig2 = fmlin(128,0.2,0.5);
sig = sig1+sig2;
figure;
tfrpwv(sig);

参考文献

[1] 金雷. Time-frequency peak filtering for random noise suppression in seismic exploration data[D]. 吉林长春：吉林大学,2006.

[2] Boashash B,Signal  enhancement  by  time-frequency  peak  Filtering[J],IEEE trans. Signal Processing,2004,52(4):929~937.

[3] Chen Yijun,程 浩,Gong Enpu,等. 基于 CEEMDAN A time-frequency peak filtering method for microseismic random noise suppression[J]. 石油地球物理勘探,2021,56(2)：234-241.

[4] Use of typical functions of the Time-Frequency Analysis Toolbox