Cohen's Class Distribution

All distributions in Cohen's Class are generated by taking the 2D Fourier transform of the product of a kernel function and the ambiguity function. This unified formulation allows any member to be expressed as a smoothed version of the Wigner-Ville Distribution (WVD). The kernel acts as a 2D low-pass filter in the ambiguity domain, attenuating cross-terms that are typically located away from the origin.

The primary purpose of the kernel is to suppress cross-term interference without destroying auto-terms. The kernel is designed in the ambiguity domain (Doppler-delay plane) where signal components are concentrated at the origin and cross-terms are displaced. A properly designed kernel, such as the Choi-Williams exponential kernel, passes the origin while attenuating distant contributions, yielding a high-resolution representation with minimal artifacts.

To be physically interpretable as a time-frequency energy density, the distribution must satisfy the time and frequency marginals. This requires the kernel to equal 1 along the Doppler axis (for the time marginal) and along the delay axis (for the frequency marginal). The Born-Jordan and Choi-Williams distributions are designed to satisfy these marginal conditions, ensuring the integral over frequency yields the instantaneous power and the integral over time yields the spectral energy density.

All members of Cohen's Class are covariant to time and frequency shifts. If the input signal is shifted in time or modulated in frequency, the resulting time-frequency distribution is shifted by the exact same amount. This property is guaranteed by the kernel's independence from time and frequency, ensuring that the analysis does not depend on the absolute location of the signal in the time-frequency plane.

A subset of Cohen's Class known as Reduced Interference Distributions uses low-pass kernels specifically shaped to suppress cross-terms while preserving the time and frequency marginals. Examples include the Choi-Williams Distribution (exponential kernel) and the Born-Jordan Distribution (sinc kernel). These are standard choices when analyzing multi-component signals where WVD cross-terms would obscure the true signal structure.

The spectrogram (squared magnitude of the STFT) is a member of Cohen's Class. Its kernel is the ambiguity function of the analysis window. This reveals that the spectrogram's poor joint resolution is a direct consequence of its kernel being a fixed smoothing function determined by the window, rather than a signal-adaptive filter. This connection unifies linear and quadratic time-frequency analysis under a single mathematical framework.

A primary application of Cohen's class is the design of kernels to suppress cross-term interference that plagues the Wigner-Ville Distribution (WVD). In RF fingerprinting, a raw signal often contains multiple components (e.g., a carrier and modulated data). Distributions like the Choi-Williams Distribution (CWD) use an exponential kernel to filter these artifacts in the ambiguity domain, revealing the true auto-terms of individual signal components without obscuring the subtle hardware impairments used for identification.

The turn-on transient of a transmitter is a rich source of unique hardware signatures. Cohen's class distributions, particularly those with high time-frequency resolution, are used to analyze these brief, non-stationary events. By applying a kernel that balances resolution and cross-term suppression, analysts can visualize the precise spectral evolution of a transient:

• Frequency Settling Time: The time it takes for an oscillator to lock to its carrier frequency.
• Amplitude Ramp Profile: The unique shape of the power envelope during startup. This joint-domain view provides a distinct, unclonable fingerprint.

Even during steady-state transmission, subtle hardware impairments like I/Q imbalance and phase noise create unique, cyclostationary signatures. A custom Cohen's class kernel can be designed to emphasize these specific modulation artifacts. For example, a kernel can be shaped to highlight the periodic energy patterns in the time-frequency plane that correspond to the symbol rate, allowing a neural network to learn a device-specific distortion profile that is robust to the transmitted data content.

Raw Cohen's class distributions serve as powerful 2D input features for Convolutional Neural Networks (CNNs). Instead of manually engineering features, a time-frequency image is generated using a kernel optimized for a specific emitter type. The CNN then learns the discriminative patterns within this representation. This approach is highly effective for open set recognition, where the model must distinguish between known emitters and identify unknown ones based on the structural anomalies visible in the Cohen's class representation.

Multipath propagation distorts the time-frequency representation of a signal, potentially masking the hardware fingerprint. Cohen's class provides a framework for channel-robust feature learning. By applying a smoothing kernel in the ambiguity domain, the representation can be made more invariant to the time-dispersive effects of the channel while preserving the fine-scale spectral features caused by transmitter impairments. This allows for reliable device authentication even in complex, dynamic environments.

In dense electromagnetic environments, a receiver captures a mixture of signals from multiple emitters. Cohen's class distributions, when combined with techniques like Hough transforms on the time-frequency plane, can separate these components. By designing a kernel that sharpens the ridges of individual signals, analysts can isolate the instantaneous frequency trajectory of each emitter. This decomposed view allows for parallel fingerprinting of multiple devices from a single, overlapping capture.