Cross-Term Interference

The fundamental mechanism behind cross-term interference is the failure of the superposition principle in quadratic distributions. When the Wigner-Ville Distribution (WVD) processes a two-component signal s(t) = s₁(t) + s₂(t), it generates not only the desired auto-terms W₁₁ and W₂₂ but also cross-terms W₁₂ and W₂₁. These cross-terms appear at the midpoint in time and frequency between the two real components, creating oscillatory structures where no actual signal energy exists. This bilinear artifact is mathematically expressed as:

W(t,f) = W₁₁(t,f) + W₂₂(t,f) + 2Re[W₁₂(t,f)]

• The cross-term magnitude is proportional to the product of the two signal components' amplitudes
• Interference oscillates at a frequency equal to the difference frequency between components
• The phenomenon is inherent to all quadratic Cohen's Class distributions

Cross-terms exhibit a characteristic oscillatory pattern that distinguishes them from genuine auto-terms. Unlike the smooth, positive energy distribution of real signal components, cross-terms alternate between positive and negative values in both time and frequency directions. This oscillatory nature creates a ripple or checkerboard pattern in the time-frequency plane. Key structural properties include:

• Zero net energy: The oscillations integrate to zero over the time-frequency plane
• Frequency of oscillation: Directly proportional to the distance between the interacting components in the joint domain
• Amplitude modulation: Cross-term magnitude decreases as the separation between real components increases
• Phase coupling: The interference pattern preserves the relative phase relationship between the original signal components

This alternating polarity is a primary visual cue for identifying and distinguishing cross-terms from legitimate signal features during analysis.

Cross-term interference can be mitigated through kernel function design in Cohen's Class distributions. The general formulation applies a two-dimensional kernel Φ(θ,τ) to the ambiguity function before the final Fourier transform:

C(t,f) = ∫∫∫ Φ(θ,τ) A(θ,τ) e^{-j2π(θt+τf)} dθ dτ

Different kernels provide varying trade-offs between cross-term suppression and auto-term resolution:

• Choi-Williams Distribution: Uses an exponential kernel Φ(θ,τ) = exp(-θ²τ²/σ) that attenuates cross-terms located away from the origin in the ambiguity domain while preserving auto-terms near the origin
• Born-Jordan Distribution: Employs a sinc-shaped kernel that provides uniform cross-term reduction
• Zhao-Atlas-Marks Distribution: Uses a cone-shaped kernel for directional smoothing
• Smoothed Pseudo WVD: Applies independent time and frequency smoothing windows for separable filtering

Each kernel represents a specific compromise between the Heisenberg-Gabor uncertainty principle and interference reduction.

The number of cross-terms grows quadratically with the number of signal components, making interference management critical for complex signals. For a signal with N distinct components, the WVD generates:

• N auto-terms: Representing genuine signal energy
• N(N-1)/2 cross-term pairs: Representing spurious interference between every possible component pair

This combinatorial explosion means:

• A 3-component signal produces 3 auto-terms and 3 cross-term pairs
• A 5-component signal produces 5 auto-terms and 10 cross-term pairs
• A 10-component signal produces 10 auto-terms and 45 cross-term pairs

In dense electromagnetic environments with numerous emitters, cross-term interference can completely obscure the true time-frequency structure. This scaling behavior is the primary motivation for developing reassignment methods and sparse decomposition techniques that avoid quadratic processing altogether.

The ambiguity function provides the most intuitive framework for understanding cross-term geometry. In the Doppler-lag domain (θ,τ), auto-terms are concentrated near the origin, while cross-terms are displaced away from the origin by an amount proportional to the time-frequency separation of the interacting components:

• Auto-terms: Located at the ambiguity domain origin (θ=0, τ=0), representing the signal's self-correlation
• Cross-terms: Located at coordinates (Δf, Δt), where Δf is the frequency separation and Δt is the time separation between the two real components

This spatial separation in the ambiguity domain is the foundation for kernel-based filtering. By applying a low-pass kernel that preserves energy near the origin while attenuating distant regions, cross-terms can be suppressed without destroying auto-term information. The Choi-Williams exponential kernel exploits this property by providing frequency-dependent attenuation that increases with distance from the origin.

Cross-term interference poses significant challenges for RF fingerprinting and emitter identification systems that rely on time-frequency representations for feature extraction. The artifacts can:

• Mask transient signatures: Cross-terms from strong signal components can obscure the weak turn-on transients critical for device identification
• Create false features: Spurious oscillatory patterns may be misinterpreted as hardware impairment signatures by downstream classifiers
• Degrade modulation classification: Interference between subcarriers in OFDM signals complicates automatic modulation recognition
• Reduce open-set detection accuracy: Unknown emitter detection relies on clean time-frequency representations to identify novel signatures

Mitigation strategies in RF applications include:

• Using reassignment methods like the Synchrosqueezing Transform to concentrate energy along true instantaneous frequency ridges
• Applying signal decomposition (EMD, VMD) before time-frequency analysis to separate components
• Employing deep learning models trained to distinguish cross-term artifacts from genuine hardware impairments
• Preferring linear transforms (STFT, CWT) for initial screening despite their resolution limitations

The turn-on transient of a power amplifier is a goldmine for RF fingerprinting, containing unique hardware-specific distortions. However, these transients are inherently multi-component and non-stationary. When analyzed with the Wigner-Ville Distribution (WVD), cross-term interference generates spurious oscillatory energy precisely in the time-frequency region between the transient's spectral components. This artifact can completely mask the subtle, device-specific amplitude and phase trajectories that a neural network needs to learn, rendering the high-resolution representation useless for identification.

In steady-state analysis, a transmitted signal is often a composite of the modulated carrier and non-linear distortion products like spectral regrowth from a saturated amplifier. The WVD of this signal will contain cross-terms between the main signal lobe and the adjacent channel leakage. These interference terms appear as oscillating artifacts in the adjacent channels, which can be falsely interpreted by a deep learning model as a unique spectral feature. This leads to models that fingerprint the signal's modulation format rather than the hardware impairment, destroying robustness to protocol changes.

Mitigating cross-term interference requires applying a smoothing kernel, moving from the WVD to a Cohen's Class distribution like the Choi-Williams Distribution (CWD). This is a fundamental engineering trade-off:

• No Kernel (WVD): Maximum time-frequency resolution, but severe cross-terms for multi-component signals.
• Exponential Kernel (CWD): Effectively suppresses cross-terms between components with different frequencies, but smears the auto-terms, reducing the precision of instantaneous frequency estimates. Selecting the optimal kernel is a hyperparameter problem that directly impacts the separability of device clusters in the feature space.

Cross-term interference forces a critical architectural decision in deep learning signal identification. A model cannot simply ingest a raw WVD as an image. The interference creates non-physical patterns that act as adversarial noise, causing the model to latch onto spurious correlations. The standard mitigation is to pre-process the signal with a cross-term attenuated distribution (like a spectrogram or CWD) before feeding it to a Convolutional Neural Network (CNN). This preprocessing step, while necessary, discards the super-resolution benefits of quadratic distributions, limiting the model's ability to resolve closely spaced hardware impairments.

When generating synthetic RF impairment datasets to train fingerprinting models, the choice of time-frequency representation is critical. If a simulator generates a clean, multi-component signal and you apply a WVD, the resulting cross-terms are a mathematical artifact of the analysis, not the device. A model trained on these artifacts will fail catastrophically in the real world. Validation requires comparing the time-frequency representation of simulated signals against real captured waveforms using a cross-term suppressed distribution to ensure the model is learning true hardware physics, not transform mathematics.

A powerful strategy to avoid cross-term interference entirely is to decompose the signal before time-frequency analysis. Techniques like Variational Mode Decomposition (VMD) or Empirical Mode Decomposition (EMD) separate a multi-component signal into its constituent Intrinsic Mode Functions (IMFs). Each IMF is a mono-component signal. Applying the WVD to each IMF individually yields a cross-term-free, high-resolution time-frequency representation. The resulting set of WVDs can then be stacked as separate channels in a multi-channel CNN, providing the benefits of quadratic resolution without the interference penalty.

The quadratic time-frequency distribution that provides the highest possible joint resolution but is the primary source of cross-term interference. The WVD is computed as the Fourier transform of the signal's instantaneous autocorrelation function. For a multi-component signal, the bilinear nature of this calculation generates oscillatory cross-terms located midway between every pair of true signal components. These artifacts are not physically present in the signal but are a mathematical consequence of the quadratic superposition principle.

A prominent member of Cohen's class that uses an exponential kernel in the ambiguity domain to attenuate cross-term interference while preserving auto-term resolution. The kernel has the form Φ(θ,τ) = exp(-θ²τ²/σ), where σ is a tunable parameter:

• Small σ: aggressive cross-term suppression but reduced auto-term resolution
• Large σ: approaches the WVD with higher resolution but more interference
• The CWD is particularly effective because cross-terms tend to be located away from the origin in the ambiguity domain, where the exponential kernel provides strong attenuation.

The two-dimensional Fourier transform of the Wigner-Ville Distribution, representing the signal in the Doppler-delay domain. The ambiguity function is the natural domain for analyzing cross-term geometry:

• Auto-terms: Concentrated near the origin (zero Doppler, zero delay)
• Cross-terms: Located at coordinates corresponding to the time-frequency separation between interacting components This separation property makes the ambiguity domain the ideal space for designing kernel filters that pass auto-terms while rejecting cross-terms.

A post-processing technique that sharpens time-frequency representations by relocating computed energy from its geometric coordinates to the center of gravity of the signal's energy distribution. For the WVD, reassignment can partially mitigate cross-term visibility by concentrating oscillatory energy onto the true instantaneous frequency ridges. However, reassignment does not eliminate cross-terms—it redistributes them. The method is most effective when combined with a smoothed distribution like the spectrogram before reassignment is applied.

The fundamental distinction that determines whether cross-term interference exists:

• Linear transforms (STFT, CWT, S-Transform): Satisfy the superposition principle. The transform of a sum equals the sum of transforms. No cross-terms are generated.
• Quadratic transforms (WVD, Cohen's class): The energy distribution of a sum includes cross-terms because the calculation involves a product of the signal with itself. Cross-terms are inherent. Choosing between linear and quadratic representations is the first design decision when cross-term interference is a concern.