Wigner-Ville Distribution

The WVD is a quadratic or bilinear distribution, meaning it correlates the signal with itself rather than with a predefined basis function. This self-correlation property yields the highest possible time-frequency resolution without the uncertainty trade-off inherent in linear transforms like the Short-Time Fourier Transform. For a signal x(t), the WVD is defined as the Fourier transform of the instantaneous autocorrelation function. This quadratic nature is what allows the WVD to perfectly localize linear chirps as delta functions in the time-frequency plane, but it also introduces the well-known cross-term interference artifact when analyzing multi-component signals.

The primary drawback of the WVD is the generation of cross-terms or interference terms. When analyzing a signal composed of multiple distinct components, the quadratic superposition creates spurious energy concentrations located midway between the true auto-terms in the time-frequency plane. These cross-terms are oscillatory, often negative, and can obscure the true signal structure. For RF fingerprinting, this is a critical consideration: while cross-terms can mask subtle hardware impairments, they also contain phase information that can be exploited. Mitigation strategies include applying smoothing kernels (leading to the Cohen class of distributions) or using the Cross Wigner-Ville Distribution to analyze signal pairs.

The WVD provides an optimal estimator for the instantaneous frequency of a monocomponent signal. The first-order moment of the WVD with respect to frequency yields the instantaneous frequency at each time instant. This property is invaluable for RF fingerprinting because hardware impairments like phase noise and voltage-controlled oscillator (VCO) non-linearity manifest as subtle, time-varying deviations from the ideal instantaneous frequency trajectory. By analyzing the WVD of a transmitter's steady-state or transient signal, one can extract a high-resolution signature of the local oscillator's instability, which is unique to each physical device and difficult to clone.

The WVD satisfies the marginal properties, a mathematically desirable trait for a joint distribution. Integrating the WVD over all frequencies yields the instantaneous power of the signal at each time instant: |x(t)|². Integrating over all time yields the energy spectral density: |X(f)|². This ensures the WVD is a true energy distribution, not just an arbitrary transform. For emitter identification, this means the energy concentration patterns in the time-frequency plane directly correspond to the physical power distribution of the transmitted waveform, allowing engineers to correlate specific energy signatures with hardware components like the power amplifier or pulse-shaping filter.

To suppress the oscillatory cross-terms while retaining the high resolution of the WVD, a 2D low-pass smoothing kernel can be applied. This generalizes the WVD into the Cohen class of distributions, which includes the Choi-Williams and Zhao-Atlas-Marks distributions. The kernel is designed in the ambiguity domain (the 2D Fourier transform of the WVD), where auto-terms are concentrated at the origin and cross-terms are displaced. By applying a low-pass filter in this domain, cross-terms are attenuated. For RF fingerprinting, selecting the optimal kernel involves a trade-off: aggressive smoothing removes cross-terms but also reduces the resolution of the transient and steady-state features critical for device identification.

Practical RF fingerprinting requires a discrete-time implementation of the WVD. The Discrete Wigner-Ville Distribution (DWVD) is computed from sampled I/Q data, but it suffers from aliasing unless the signal is sampled at twice the Nyquist rate or the analytic signal is used. The analytic signal, formed by adding the Hilbert transform as the imaginary part, suppresses negative frequencies and eliminates aliasing. For real-time embedded systems, efficient computation often uses the fast Fourier transform (FFT) on the discrete instantaneous autocorrelation matrix. This allows the DWVD to be deployed on FPGAs or SDRs for real-time time-frequency feature extraction from high-bandwidth emissions.