Wigner-Ville Distribution Function for Time-Frequency Analysis

The Wigner-Ville Distribution (WVD) is a classical nonlinear time-frequency representation method in signal analysis that provides simultaneous high-resolution information in both time and frequency domains. Originally proposed by physicist Eugene Wigner in 1932, it was later introduced to signal processing by Jean Ville.

The fundamental principle of WVD involves computing the Fourier transform of the instantaneous autocorrelation function of a signal to obtain the time-frequency energy distribution. A key advantage is its superior resolution in the time-frequency plane compared to Short-Time Fourier Transform (STFT) and wavelet transform, making it particularly suitable for analyzing non-stationary signals such as radar, sonar, and biomedical signals. In implementation, the algorithm typically requires calculating the signal's auto-correlation function followed by Fourier transformation along the time delay dimension.

However, WVD suffers from a major drawback: when signals contain multiple components, it generates cross-terms (interference components) that can obscure genuine signal representations in the time-frequency plane. To suppress these artifacts, researchers have developed various improved methods including Smoothed Pseudo Wigner-Ville Distribution (SPWVD) and Choi-Williams distribution. Code implementations often incorporate smoothing kernels in both time and frequency domains to mitigate cross-term interference.

In practical applications, WVD computation usually processes the analytical form of signals (obtained through Hilbert transform to remove negative frequency components) to reduce cross-term interference. Discrete WVD implementation requires careful consideration of sampling rates and boundary effects to avoid aliasing and truncation errors. MATLAB implementations typically use the 'hilbert' function to obtain analytic signals before applying WVD algorithms.

Despite cross-term challenges, WVD remains a vital tool in time-frequency analysis, especially in high-resolution applications like instantaneous frequency estimation and signal component separation. Understanding its principles and limitations helps engineers select appropriate time-frequency analysis methods for specific signal processing tasks. Modern implementations often combine WVD with other techniques or use post-processing methods to enhance practical usability.