6.3 Wigner-Ville distribution

The Wigner-Ville distribution is a powerful tool in advanced signal processing, offering high-resolution time-frequency analysis of non-stationary signals. It provides insights into signals with changing frequency content, making it valuable for applications in radar, sonar, and speech processing.

Despite its advantages, the Wigner-Ville distribution has limitations, including cross-terms that can obscure signal features. Various modifications and alternatives have been developed to address these issues, balancing resolution and interpretability for different signal analysis needs.

Definition of Wigner-Ville distribution

• The Wigner-Ville distribution (WVD) provides a joint time-frequency representation of a signal, allowing for the analysis of non-stationary signals whose frequency content varies over time
• It is a quadratic time-frequency distribution that belongs to the Cohen's class of distributions and offers high resolution in both time and frequency domains

Mathematical formulation

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• The Wigner-Ville distribution of a signal $x(t)$ is defined as: $W_x(t,f) = \int_{-\infty}^{\infty} x(t+\frac{\tau}{2}) x^*(t-\frac{\tau}{2}) e^{-j2\pi f \tau} d\tau$
• $x(t)$ is the signal, $x^*(t)$ is its complex conjugate, $t$ represents time, $f$ represents frequency, and $\tau$ is a time lag variable
• The WVD can be interpreted as the Fourier transform of the product $x(t+\frac{\tau}{2}) x^*(t-\frac{\tau}{2})$, which is known as the instantaneous autocorrelation function

Relation to time-frequency analysis

• The Wigner-Ville distribution maps a one-dimensional signal into a two-dimensional time-frequency representation
• It provides information about the energy distribution of the signal simultaneously in both time and frequency domains
• The WVD is particularly useful for analyzing signals with time-varying frequency content, such as chirps, frequency-modulated signals, and non-stationary signals encountered in various applications (radar, sonar, speech processing)

Properties of Wigner-Ville distribution

• The Wigner-Ville distribution exhibits several important properties that make it a valuable tool for time-frequency analysis in advanced signal processing applications

Time-frequency resolution

• The WVD offers high resolution in both time and frequency domains compared to other time-frequency representations (spectrogram)
• It does not suffer from the time-frequency resolution trade-off inherent in the uncertainty principle, as it is not limited by the fixed window size used in short-time Fourier transform (STFT) based methods
• The WVD can resolve signal components that are closely spaced in time or frequency, making it suitable for analyzing signals with rapidly changing frequency content or multiple closely spaced frequency components

Marginal properties

• The WVD satisfies the marginal properties, which relate the time-frequency distribution to the signal's time-domain and frequency-domain representations
• The time marginal property states that integrating the WVD over frequency yields the instantaneous power of the signal: $\int_{-\infty}^{\infty} W_x(t,f) df = |x(t)|^2$
• The frequency marginal property states that integrating the WVD over time yields the energy spectral density of the signal: $\int_{-\infty}^{\infty} W_x(t,f) dt = |X(f)|^2$, where $X(f)$ is the Fourier transform of $x(t)$

Moyal's formula

• Moyal's formula relates the inner product of two signals in the time-domain to the inner product of their Wigner-Ville distributions in the time-frequency domain
• It is given by: $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} W_x(t,f) W_y(t,f) dt df = |\langle x,y \rangle|^2$
• Moyal's formula highlights the preservation of signal energy in the WVD and its usefulness in signal detection and classification tasks

Convolution and multiplication

• The WVD satisfies the convolution property in the time-domain and the multiplication property in the frequency-domain
• The convolution property states that the WVD of the convolution of two signals is equal to the convolution of their individual WVDs along the time axis: $W_{x*y}(t,f) = \int_{-\infty}^{\infty} W_x(t-\tau,f) W_y(\tau,f) d\tau$
• The multiplication property states that the WVD of the product of two signals is equal to the convolution of their individual WVDs along the frequency axis: $W_{xy}(t,f) = \int_{-\infty}^{\infty} W_x(t,f-\nu) W_y(t,\nu) d\nu$
• These properties are useful in analyzing the time-frequency characteristics of signals that have undergone convolution or multiplication operations

Advantages of Wigner-Ville distribution

• The Wigner-Ville distribution offers several advantages over other time-frequency analysis techniques, making it a powerful tool in advanced signal processing applications

High resolution in time-frequency domain

• The WVD provides high resolution in both time and frequency domains, allowing for the accurate localization of signal components in the time-frequency plane
• Unlike the short-time Fourier transform (STFT) or wavelet transform, the WVD is not limited by the fixed window size or the choice of wavelet basis functions
• The high resolution of the WVD enables the analysis of signals with closely spaced frequency components or rapidly varying frequency content (chirps, frequency-hopping signals)

Ability to analyze non-stationary signals

• The WVD is particularly well-suited for analyzing non-stationary signals, where the frequency content of the signal changes over time
• Traditional frequency-domain techniques (Fourier transform) assume signal stationarity and fail to capture the time-varying nature of non-stationary signals
• The joint time-frequency representation provided by the WVD allows for the characterization of the instantaneous frequency and group delay of non-stationary signals, which is essential in applications (radar, sonar, speech processing)

Preservation of signal energy

• The WVD preserves the energy of the signal in the time-frequency domain, ensuring that the total energy of the signal is distributed across the time-frequency plane
• This property is a consequence of the marginal properties satisfied by the WVD, where the integration of the WVD over time or frequency yields the instantaneous power or energy spectral density of the signal, respectively
• The preservation of signal energy makes the WVD useful in energy-based signal detection, classification, and pattern recognition tasks

Disadvantages of Wigner-Ville distribution

• Despite its advantages, the Wigner-Ville distribution also has some limitations and drawbacks that should be considered when applying it to advanced signal processing tasks

Presence of cross-terms

• One of the main disadvantages of the WVD is the presence of cross-terms, which are artifacts that appear in the time-frequency representation between the actual signal components
• Cross-terms arise from the quadratic nature of the WVD and the interaction between different signal components
• The presence of cross-terms can obscure the true time-frequency structure of the signal and make the interpretation of the WVD challenging, especially for multi-component signals

Interpretation difficulties due to cross-terms

• The cross-terms in the WVD can lead to difficulties in interpreting the time-frequency representation of the signal
• Cross-terms can have significant amplitudes and may overlap with the actual signal components, making it difficult to distinguish between true signal features and artifacts
• The interpretation of the WVD requires careful analysis and may necessitate the use of additional techniques (filtering, masking) to suppress or mitigate the effects of cross-terms

Computational complexity

• The computation of the WVD involves the evaluation of the instantaneous autocorrelation function and the Fourier transform, which can be computationally intensive, especially for long signal durations or high sampling rates
• The computational complexity of the WVD grows quadratically with the signal length, making it more demanding than other time-frequency analysis techniques (STFT, wavelet transform)
• The high computational requirements of the WVD can limit its applicability in real-time or resource-constrained scenarios, necessitating the use of efficient algorithms or hardware implementations

Variants of Wigner-Ville distribution

• To address the limitations of the standard Wigner-Ville distribution, several variants have been proposed that aim to reduce cross-terms, improve readability, or adapt to specific signal characteristics

Pseudo Wigner-Ville distribution

• The pseudo Wigner-Ville distribution (PWVD) introduces a window function in the time-domain to smooth the instantaneous autocorrelation function before computing the Fourier transform
• The window function helps to suppress cross-terms and improve the readability of the time-frequency representation
• The PWVD offers a trade-off between cross-term suppression and time-frequency resolution, depending on the choice of the window function (Hamming, Hann, Gaussian)

Smoothed pseudo Wigner-Ville distribution

• The smoothed pseudo Wigner-Ville distribution (SPWVD) applies an additional frequency-domain smoothing window to the PWVD to further reduce cross-terms and enhance the clarity of the time-frequency representation
• The SPWVD allows for independent control over the time and frequency smoothing, providing flexibility in adapting to different signal characteristics and analysis requirements
• The choice of the time and frequency smoothing windows affects the trade-off between cross-term suppression and time-frequency resolution in the SPWVD

Cone-shaped kernel

• The cone-shaped kernel is a variant of the WVD that employs a cone-shaped weighting function in the time-frequency domain to reduce cross-terms
• The cone-shaped kernel emphasizes the regions close to the actual signal components and attenuates the regions where cross-terms are likely to occur
• The shape and parameters of the cone-shaped kernel can be adjusted to optimize the trade-off between cross-term suppression and time-frequency resolution for specific signal types and analysis goals

Choi-Williams distribution

• The Choi-Williams distribution (CWD) is a member of Cohen's class of time-frequency distributions that aims to reduce cross-terms while preserving the desirable properties of the WVD
• The CWD introduces an exponential kernel function in the time-lag domain, which suppresses cross-terms by attenuating the contributions from the regions away from the origin
• The CWD offers a balance between cross-term reduction and time-frequency resolution, controlled by a parameter that determines the spread of the exponential kernel

Applications of Wigner-Ville distribution

• The Wigner-Ville distribution finds applications in various fields of advanced signal processing, where the analysis of non-stationary signals and the joint time-frequency representation are crucial

Speech signal analysis

• The WVD is used in speech signal analysis to study the time-varying frequency content of speech signals, such as formants, pitch, and voice activity
• It helps in the characterization of speech production mechanisms, speaker identification, and speech recognition tasks
• The high resolution and ability to capture non-stationary behavior make the WVD suitable for analyzing the dynamic nature of speech signals

Radar signal processing

• In radar signal processing, the WVD is employed for the analysis and classification of radar signals, such as chirps, frequency-hopping signals, and micro-Doppler signatures
• The WVD enables the identification of target characteristics, the estimation of target motion parameters, and the detection of weak signals in the presence of noise and clutter
• The high resolution and cross-term suppression capabilities of the WVD variants are beneficial in radar applications

Sonar signal processing

• The WVD is applied in sonar signal processing for the analysis and classification of underwater acoustic signals, such as marine mammal vocalizations, ship noise, and seismic events
• It assists in the detection and localization of sound sources, the characterization of underwater acoustic environments, and the monitoring of marine life activities
• The ability of the WVD to handle non-stationary signals and provide high-resolution time-frequency representations is valuable in sonar applications

Biomedical signal analysis

• The WVD is used in biomedical signal analysis for the study of various physiological signals, such as electroencephalogram (EEG), electrocardiogram (ECG), and electromyogram (EMG)
• It helps in the identification of abnormalities, the extraction of relevant features, and the classification of different physiological states or conditions
• The WVD's capability to reveal time-varying frequency patterns and capture transient events is useful in detecting and characterizing biomedical signals

Comparison with other time-frequency distributions

• The Wigner-Ville distribution is one of several time-frequency analysis techniques available in advanced signal processing, and it is often compared with other distributions to highlight its strengths and limitations

Spectrogram vs Wigner-Ville distribution

• The spectrogram, based on the short-time Fourier transform (STFT), is a widely used time-frequency representation that provides a compromise between time and frequency resolution
• Compared to the WVD, the spectrogram has lower resolution in both time and frequency domains due to the fixed window size used in the STFT
• The spectrogram does not suffer from cross-terms, making it easier to interpret, but it may miss fine details and rapidly varying frequency components that the WVD can capture

Wavelet transform vs Wigner-Ville distribution

• The wavelet transform is another popular time-frequency analysis technique that uses a set of scaled and shifted wavelet basis functions to decompose the signal
• The wavelet transform provides a multi-resolution analysis, with better time resolution at high frequencies and better frequency resolution at low frequencies
• Compared to the WVD, the wavelet transform is less affected by cross-terms and offers a more intuitive interpretation of the time-frequency representation
• However, the wavelet transform's resolution is determined by the choice of the wavelet basis function and may not achieve the same high resolution as the WVD in some cases

Cohen's class of distributions

• The Wigner-Ville distribution is a member of Cohen's class of time-frequency distributions, which encompasses a wide range of distributions with different properties and kernel functions
• Other notable distributions in Cohen's class include the Choi-Williams distribution, the Born-Jordan distribution, and the Zhao-Atlas-Marks distribution
• Each distribution in Cohen's class offers a different trade-off between cross-term suppression, time-frequency resolution, and other desirable properties
• The choice of the appropriate distribution depends on the specific signal characteristics, the analysis goals, and the acceptable level of cross-terms and computational complexity

Implementation of Wigner-Ville distribution

• The implementation of the Wigner-Ville distribution involves several computational aspects and practical considerations for efficient and accurate calculation of the time-frequency representation

Discrete Wigner-Ville distribution

• In practice, signals are often represented as discrete-time sequences, and the WVD needs to be computed using discrete-time formulations
• The discrete Wigner-Ville distribution (DWVD) is the discrete-time counterpart of the continuous WVD and is defined as: $W_x[n,k] = \sum_{m=-\infty}^{\infty} x[n+m] x^*[n-m] e^{-j\frac{4\pi}{N}mk}$
• The DWVD is computed using discrete-time signals $x[n]$ and discrete frequency samples $k$, where $N$ is the number of frequency bins
• The DWVD inherits the properties and characteristics of the continuous WVD but is subject to aliasing and periodicity effects due to the discrete nature of the signals and the finite frequency resolution

Fast algorithms for computation

• The direct computation of the DWVD can be computationally expensive, especially for long signal lengths and high frequency resolutions
• Fast algorithms have been developed to efficiently compute the DWVD, exploiting the redundancy and symmetry properties of the distribution
• One common approach is the use of the fast Fourier transform (FFT) to calculate the discrete-time Fourier transforms required in the DWVD computation
• Other fast algorithms, such as the recursive DWVD and the time-frequency recursion method, have been proposed to further reduce the computational complexity and memory requirements

Software tools and libraries

• Several software tools and libraries are available for the computation and visualization of the Wigner-Ville distribution and its variants
• MATLAB provides built-in functions for computing the WVD, such as

  wvd()

  and

  tfrwv()

  , as part of its Signal Processing Toolbox
• Python libraries, such as SciPy and PyWavelets, offer implementations of the WVD and other time-frequency distributions
• Specialized software packages, like the Time-Frequency Toolbox (TFTB) for MATLAB and the Time-Frequency Analysis Toolbox (TFAT) for Python, provide comprehensive collections of time-frequency analysis tools, including the WVD and its variants
• These software tools and libraries facilitate the application of the WVD to real-world signal processing problems and enable the exploration and visualization of time-frequency representations