The Wigner-Ville Transform as an Information Theoretic Tool in Radio-frequency Signal Analysis

TL;DR

This work reframes the Wigner-Ville transform as an information-theoretic tool for classical signal analysis by treating its time-frequency output as a quasi-distribution and applying Tsallis entropy (order $\alpha=2$) to define normalized information measures $I_2$ and $S_2$, along with localized densities. It derives fundamental properties, discusses cross-term effects, and demonstrates how these information measures enable sensitive, modulation-agnostic detection, localization, and estimation of information volume in RF signals under noisy and cluttered backgrounds. In RF sensing experiments, WVT-based detectors outperform traditional energy-based methods by substantial margins (often $>10$ dB) for broad-band and transient signals, with robustness to interference and without requiring extensive training. The results advocate broader applications of information-sensitive WVT methods across time-frequency analysis tasks and potentially for training landscapes in AI/ML systems.

Abstract

This paper presents novel interpretations to the field of classical signal processing of the Wigner-Ville transform as an information measurement tool. The transform's utility in detecting and localizing information-laden signals amidst noisy and cluttered backgrounds, and further providing measure of their information volumes, are detailed herein using Tsallis' entropy and information and related functionals. Example use cases in radio frequency communications are given, where Wigner-Ville-based detection measures can be seen to provide significant sensitivity advantage, for some shown contexts greater than 15~dB advantage, over energy-based measures and without extensive training routines. Such an advantage is particularly significant for applications which have limitations on observation resources including time/space integration pressures and transient and/or feeble signals, where Wigner-Ville-based methods would improve sensing effectiveness by multiple orders of magnitude. The potential for advancement of several such applications is discussed.

Figures (11)

• Figure 1: An example RF signal and sample resulting Wigner-Ville-type Cohen's class transformations. Only the non-negative-frequency half-plane is shown, though each of the transforms is symmetric about zero. The signal is composed of an additive superposition of white Gaussian noise and a BPSK transmission with root-raised-cosine filter with shape parameter RRC=0.5, containing a message of random symbols, injected with a (log) signal-to-noise ratio (SNR) of 10. Alternatives to the WVT are referenced in Appendix \ref{['appx:wigner_alternates']}. The length of the waveform is $N = 2^{12} = 4,096$ samples, $N_s = 100$ is the number of symbols. (A) Time domain representation of the waveform components, vertically offset, with the I-Q representation of the BPSK message at top, individual noise and transmission components ad middle, and superposed waveform at bottom. (B) The spectrogram of the waveform with FFT windows of size $N_{FFT} = 2^7 = 128$. (C) The modulus squared of the Gabor time-frequency representation using smoothing length of $L_s = 60$. (D) The full WVT of the waveform. (E) The pseudo-Wigner transform with Gaussian smoothing kernel of width $\sigma = 60$, (F) Polynomial Wigner with smoothing length $\sigma = 60$, $q = 4$, $b = [2,0,0,0,-2]$, and $c = [1,0,0,0,-1]$. All intensities are plotted in linear units.
• Figure 2: Example properties of WVT, information and entropy densities on a noise-less waveform containing two QPSK signals with identical messages (RRC = 0.35, $R_{\text{Symb}} = 100$ Symb/s) collected at a rate of 10,000. (A) The waveform spectrogram with $N_{FFT} = 128$. (B) The full WVT of the waveform. (C) The information density $i_2(t,\nu)$ of the waveform. (D) The projected information density $I_{2,\nu}$ (red) and entropy density $S_{2,\nu}$ (Blue) of the waveform. All intensities are plotted in linear units.
• Figure 3: Global detection rate for common modulation types injected over AWGN. The detection threshold in each metric was computed in the no-injection limit using a probability of false alarm $p_{FA} = 0.05$, and was reached using $N_{\text{trials}} = p_{FA}^{-2} = 400$. The detection statistics at each point along SNR are also generated by $N_{\text{trials}}$. The pseudo-WVT-based measures use a width parameter of $\sigma=100$.
• Figure 4: Localized PSD, WVT frequency spectrum $\tilde{\rho}_{\nu}$, and WVT-squared frequency spectrum $WVT^{2}_{\nu}$, the last two being the building blocks of TE and TI. Each measure's spectra is normalized to the AWGN background for better visualization of the emerging signatures. The spectra are computed using the same AWGN background to highlight the changes induced by injections of increasing intensity. The injections use a symbol rate of $R_{\text{Symb}} = 100$ Symb/s.
• Figure 5: Relative entropy volume for example single-injection signals over AWGN. The relative measurements are made between the carrier mode and a nearby region of the same width. MFSK injections figures are largely unsettled due to lack of a well-defined prescribed center frequency. Note the significantly higher SNR required for convergence of the TE/TI volume.