Choi-Williams Distribution (CWD)

The defining feature of the CWD is its exponential kernel in the ambiguity domain, parameterized by a smoothing factor (σ). This kernel acts as a low-pass filter that attenuates cross-terms located far from the origin while preserving auto-terms centered near the origin.

• Kernel form: Φ(θ, τ) = exp(-θ²τ² / σ)
• The parameter σ controls the trade-off between cross-term suppression and time-frequency resolution
• As σ → ∞, the CWD converges to the Wigner-Ville Distribution (WVD)
• As σ → 0, the distribution becomes increasingly smoothed, approaching a spectrogram-like representation

Unlike the Wigner-Ville Distribution, which generates oscillatory cross-terms for every pair of signal components, the CWD strategically suppresses these artifacts by exploiting their location in the ambiguity domain.

• Auto-terms are concentrated near the origin of the ambiguity function
• Cross-terms are displaced away from the origin, proportional to the time-frequency separation of the interacting components
• The exponential kernel attenuates components farther from the origin, effectively filtering cross-terms while passing auto-terms
• This makes the CWD particularly effective for analyzing multi-component signals common in RF fingerprinting

The CWD achieves a practical balance between the ideal resolution of the WVD and the interference-free readability of linear transforms like the Short-Time Fourier Transform (STFT).

• Maintains bilinear time-frequency resolution for auto-terms, avoiding the uncertainty principle trade-off inherent in linear transforms
• Preserves sharp localization for linear frequency-modulated (chirp) signals
• The kernel design ensures the distribution satisfies desirable mathematical properties including time and frequency marginal conditions
• Real-valued output with correct energy interpretation when the kernel is properly normalized

The CWD belongs to Cohen's class of distributions, a unified framework where any quadratic time-frequency representation can be generated by applying a two-dimensional kernel to the ambiguity function.

• General form: C(t, f) = ∫∫∫ x(u + τ/2) x*(u - τ/2) Φ(θ, τ) e^{-j2π(θt + fτ - θu)} du dτ dθ
• The kernel Φ(θ, τ) uniquely defines each distribution within the class
• The CWD kernel is product-separable in θ and τ, enabling efficient implementation
• This formulation allows direct comparison with other distributions like the Born-Jordan or Zhao-Atlas-Marks distributions

The CWD is widely used in Specific Emitter Identification (SEI) because it reveals subtle transient and steady-state features that are obscured in spectrograms or raw time-domain analysis.

• Captures transient signal behavior during power amplifier turn-on and turn-off sequences with high precision
• Reveals unintentional modulation artifacts caused by hardware impairments like I/Q imbalance and oscillator phase noise
• The reduced cross-term interference allows clear visualization of closely spaced signal components in dense electromagnetic environments
• Often used as a pre-processing step before feeding time-frequency images into convolutional neural networks (CNNs) for deep learning-based emitter classification

The CWD is computationally more intensive than linear transforms but offers significant advantages for offline analysis and feature extraction pipelines.

• Direct implementation requires O(N³) operations for an N-point signal, though fast algorithms reduce this to O(N² log N)
• The smoothing parameter σ must be tuned to the specific signal characteristics; typical values range from 0.1 to 10
• Modern GPU-accelerated implementations enable near real-time processing for moderate signal lengths
• Often implemented via a filtered ambiguity function approach: compute the ambiguity function, apply the exponential kernel mask, then perform a 2D Fourier transform

The CWD excels at resolving the rapid, non-stationary turn-on and turn-off transients of a transmitter. Unlike linear transforms, it captures the instantaneous frequency trajectory of the power amplifier's stabilization phase with high precision.

• Resolves closely spaced transient components in time and frequency
• Reveals unique damping characteristics of oscillator startup
• Provides a stable feature space for fingerprinting short-duration bursts

A primary advantage of the CWD is its exponential kernel in the ambiguity domain, which effectively suppresses oscillatory cross-term artifacts that plague the Wigner-Ville Distribution.

• Preserves auto-term resolution for individual signal components
• Prevents false feature generation from multi-component interference
• Enables clean analysis of signals with multiple modulation sub-carriers

The CWD visualizes the dynamic non-linear behavior of power amplifiers under varying drive levels. It captures the spectral regrowth and memory effects that manifest as unique, time-varying frequency excursions during a transmission burst.

• Identifies device-specific AM/AM and AM/PM distortion patterns
• Tracks the temporal evolution of harmonic content
• Distinguishes between static non-linearity and thermal memory effects

The distribution provides a high-resolution view of oscillator phase noise spread around the carrier. By analyzing the time-frequency broadening, one can extract unique phase noise signatures that are intrinsic to the synthesizer's phase-locked loop design.

• Resolves close-in phase noise skirts from the main carrier
• Captures spurious tones generated by reference clock leakage
• Provides a robust feature invariant to modulation data

The CWD's exponential kernel contains a scaling parameter (σ) that controls the trade-off between cross-term suppression and auto-term resolution. Tuning this parameter is critical for maximizing fingerprinting accuracy.

• A small σ maximizes resolution but retains some cross-terms
• A large σ aggressively suppresses interference but smooths auto-terms
• Optimal value is often determined empirically per device class

The CWD can reveal the time-varying nature of I/Q modulator impairments, such as gain imbalance and quadrature skew. It shows how the unwanted negative-frequency image component fluctuates relative to the desired signal over the burst duration.

• Tracks dynamic DC offset drift during transmission
• Visualizes the frequency-dependent nature of quadrature errors
• Separates static constellation warping from transient imbalance effects

The Wigner-Ville Distribution is the foundational quadratic distribution from which the CWD is derived. It calculates the Fourier transform of the signal's instantaneous autocorrelation function, providing the highest possible joint time-frequency resolution for mono-component signals. However, its bilinear nature generates severe cross-term interference—spurious oscillatory artifacts that appear midway between any two genuine signal components. The CWD directly addresses this limitation by applying an exponential kernel to suppress these artifacts while preserving auto-term resolution.

The CWD is a specific member of Cohen's Class, a unified framework for generating quadratic time-frequency distributions. Every distribution in this class is defined by a two-dimensional kernel function applied in the ambiguity domain. Key members include:

• Wigner-Ville: Kernel = 1 (no filtering)
• Choi-Williams: Kernel = exponential function
• Born-Jordan: Kernel = sinc function
• Spectrogram: Kernel = ambiguity function of the window The kernel design represents a fundamental trade-off between cross-term suppression and auto-term concentration.

The Spectrogram is the squared magnitude of the Short-Time Fourier Transform and remains the most widely used time-frequency representation. Unlike the CWD, it is a linear distribution that avoids cross-term interference entirely by analyzing the signal through a sliding window. However, this comes at the cost of the Heisenberg-Gabor uncertainty principle: the fixed window imposes an unavoidable trade-off between time and frequency resolution. The CWD achieves superior resolution for multi-component signals where the spectrogram's windowing would blur closely spaced components.

Cross-term interference is the primary artifact that the CWD is engineered to suppress. In quadratic distributions, the interaction between any two signal components generates a third, oscillatory term located at the midpoint of their time-frequency coordinates. These artifacts can be stronger than the actual signal components and severely obscure interpretation. The CWD's exponential kernel acts as a low-pass filter in the ambiguity domain, attenuating cross-terms that appear far from the origin while preserving auto-terms concentrated near the origin. The kernel parameter σ controls the suppression aggressiveness.

The Ambiguity Function is the dual-domain representation where the CWD's kernel is designed and applied. It maps a signal into a two-dimensional space of Doppler shift (ν) and time delay (τ). In this domain, auto-terms are concentrated near the origin (0,0), while cross-terms are displaced away from the origin. The CWD's exponential kernel, exp(-ν²τ²/σ), is a two-dimensional low-pass filter that attenuates energy far from the origin. This elegant separation in the ambiguity domain is the mathematical mechanism enabling the CWD's selective interference suppression.

Reassignment is a post-processing technique that can be applied to the CWD to further sharpen its representation. It relocates the computed energy from the geometric center of the analysis window to the center of gravity of the signal's energy distribution. This creates nearly ideal concentration for chirp-like signals. However, reassignment is a non-invertible operation—the original signal cannot be reconstructed from the reassigned distribution. For RF fingerprinting applications requiring feature extraction rather than signal reconstruction, reassigned CWD often provides the cleanest visualization of transient and steady-state signatures.