Cross-Terms
Why Quadratic Distributions Generate Interference
The Wigner-Ville distribution is built from the instantaneous autocorrelation s(t + τ/2)·s*(t − τ/2), then Fourier transformed over the lag τ. That product is bilinear: it multiplies the signal by a shifted, conjugated copy of itself. When the signal contains a single component the product folds neatly onto that component and produces a perfectly localized auto-term. But the moment two components are present, the bilinear product also pairs each component with the other, and those mixed products do not vanish. They land halfway between the two real components and beat against each other, producing the oscillatory ridges engineers call cross-terms or interference terms.
The geometry is predictable and, in a sense, helpful. Two tones at f1 and f2 create a cross-term centered at the arithmetic midpoint (f1+f2)/2, oscillating in time at the difference frequency f1−f2. Two impulses at times t1 and t2 create a cross-term at (t1+t2)/2 oscillating in frequency at a rate proportional to t1−t2. The farther apart the components are, the faster the cross-term oscillates, which is precisely what lets a smoothing kernel separate interference from signal.
Counting matters when you read a real spectrogram of a wideband emitter. A signal with N distinct components produces N genuine auto-terms but N(N−1)/2 cross-terms. Six chirps in a radar pulse, for example, generate fifteen interference patterns, easily outnumbering and outweighing the real content. This combinatorial explosion is why raw Wigner-Ville displays of dense signals are often unreadable and why signal-processing chains almost always apply cross-term suppression before display.
Governing Relationships
Wx(t, f) = ∫ x(t + τ/2) · x*(t − τ/2) e−j2πfτ dτ
Cross-term of two components (a and b):
Wcross(t, f) ≈ 2 · |AaAb| · cos[2π((fa−fb)t − (ta−tb)f) + φ]
centered at t = (ta+tb)/2, f = (fa+fb)/2
Cohen-class smoothing (kernel Φ in the ambiguity plane):
Cx(t, f) = ∫∫∫ Φ(θ, τ) x(u + τ/2) x*(u − τ/2) ej2π(θu − θt − fτ) du dτ dθ
Where x(t) is the analytic signal, τ is lag, θ is Doppler, and Φ(θ,τ) is the kernel. Φ = 1 gives the raw Wigner-Ville distribution (no suppression); a kernel that passes the θτ origin and rolls off elsewhere attenuates the off-origin cross-terms. Choi-Williams uses Φ = exp(−θ2τ2/σ), with smaller σ giving stronger suppression.
How Distributions Trade Suppression for Resolution
| Distribution | Kernel Φ(θ,τ) | Cross-Term Suppression | Time-Freq Concentration | Marginals Preserved | Typical Use |
|---|---|---|---|---|---|
| Wigner-Ville | 1 | None | Optimal (ideal) | Both | Single-component / theory |
| Smoothed Pseudo WVD | Separable g(τ)·H(θ) | Adjustable, strong | High | Approx. | Multi-component radar / EW |
| Choi-Williams | exp(−θ2τ2/σ) | Strong, tunable via σ | High | Both | Crossing-component signals |
| Born-Jordan | sin(πθτ)/(πθτ) | Strong | Moderate | Both | General-purpose analysis |
| Spectrogram (STFT) | Ambiguity of window h | Complete | Limited by ΔtΔf ≥ 1/4π | Neither exact | Routine display / monitoring |
Frequently Asked Questions
Where do cross-terms appear in a Wigner-Ville distribution of two tones?
Two sinusoids at f1 and f2 produce two genuine ridges plus a spurious component centered exactly at (f1+f2)/2 that oscillates in time at the difference frequency f1−f2. Its peak amplitude is about twice that of either auto-term, so it visually dominates. Two impulses at t1, t2 behave the same way in the time direction. An N-component signal generates N(N−1)/2 such interference terms.
How do you suppress cross-terms without destroying time-frequency resolution?
Cross-terms are oscillatory and sit away from the origin of the ambiguity (Doppler-lag) plane, while auto-terms cluster near it. A 2-D lowpass kernel that passes the origin region and attenuates the rest, the Cohen class, suppresses them. The smoothed-pseudo-Wigner-Ville distribution uses separable time and lag windows (often 64 to 256 samples) to control the two directions independently. Choi-Williams uses exp(−θ2τ2/σ); smaller σ suppresses harder but blurs the auto-terms.
Why can cross-terms be negative when an energy density should be positive?
The Wigner-Ville distribution is a quadratic energy density that is not constrained to be non-negative; only its marginals are guaranteed (time integral gives the power spectral density, frequency integral gives instantaneous power). Cross-terms oscillate symmetrically about zero, so on integration their positive and negative lobes cancel and the marginals stay correct. That cancellation is exactly why cross-terms look alarming yet do not corrupt total energy.
Do cross-terms also appear in the spectrogram and STFT?
The spectrogram is a member of the Cohen class whose kernel is the ambiguity function of its analysis window, which heavily smooths the plane. That smoothing removes cross-terms almost entirely, so spectrograms are effectively interference-free. The cost is resolution: the window length forces the time-bandwidth product Δt·Δf ≥ 1/(4π), so a spectrogram can never localize components as sharply as a raw Wigner-Ville distribution. It trades the cross-term problem for a blur problem.