Cross Spectrum (Cross Power Spectral Density)

The Cross Power Spectral Density (CPSD), denoted G_xy(f), describes the mutual spectral content shared between two signals x(t) and y(t). It is a complex-valued function whose magnitude indicates correlated power at each frequency and whose phase reveals the timing relationship between the two signals.

Definition

The cross-spectrum is defined as the averaged product of the Fourier transforms of the two signals:

G_xy(f) = (2 / T) · E[ X*(f) · Y(f) ]

where X(f) and Y(f) are the FFTs of signals x(t) and y(t), the asterisk (*) denotes complex conjugation, E[·] denotes ensemble averaging, and T is the block duration. The factor of 2 converts to a one-sided spectrum.

Magnitude and Phase

Magnitude

The magnitude |G_xy(f)| represents the correlated spectral power between the two channels at each frequency. It is always less than or equal to the geometric mean of the two auto-spectra (the Cauchy-Schwarz inequality):

|G_xy(f)| ≤ √(G_xx(f) · G_yy(f))

Phase

The phase ∠G_xy(f) represents the phase difference between the two signals at each frequency. A positive phase indicates that signal y leads signal x at that frequency, and vice versa. The phase is meaningful only where the coherence is sufficiently high.

Relationship to Coherence and Transfer Function

The cross-spectrum is the building block for both the coherence function and transfer function estimates:

Quantity	Formula
Coherence	γ²_xy = \|G_xy\|² / (G_xx · G_yy)
Transfer Function H₁	H₁(f) = G_xy(f) / G_xx(f)
Transfer Function H₂	H₂(f) = G_yy(f) / G_yx(f)

H₁ minimizes noise on the output channel, while H₂ minimizes noise on the input channel. The geometric mean of H₁ and H₂ provides the H_v estimator, which handles noise on both channels.

Parameters

Parameter	Options	Description
Input Channel (X)	Any signal channel	Reference signal (typically excitation/force).
Output Channel (Y)	Any signal channel	Response signal (typically accelerometer/microphone).
Spectrum Size	256 – 65536	FFT block size. Determines frequency resolution Δf = f_s/N.
Overlap	0% – 95%	Block overlap percentage for averaging.
Window	Hanning, Hamming, Flat Top, etc.	Window function applied to each block.
Number of Averages	1 – unlimited	Ensemble averages. More averages reduce random error.
Display	Magnitude, Phase, Real, Imaginary	Component of the complex cross-spectrum to display.

Variants

Cross Spectrum vs Time

Displays the evolution of G_xy(f) over successive time blocks as a waterfall or color map. Useful for tracking how the spectral coupling between two channels changes during transient events.

Cross Spectrum vs RPM

When a tachometer signal is available, the cross-spectrum is mapped to the rotational speed axis. This is essential in powertrain NVH to identify speed-dependent transfer paths and structural resonances excited by rotating components.

Practical Tips

Always inspect the coherence function alongside the cross-spectrum. At frequencies where coherence is low, the cross-spectrum magnitude and phase are unreliable.
The phase of G_xy becomes noisy at frequencies where the magnitude is near the noise floor. Use phase unwrapping cautiously and only in high-coherence regions.
For impact testing, use force and exponential windows to minimize leakage and improve the quality of the cross-spectral estimate.
The cross-spectrum is the Fourier transform of the cross-correlation function: G_xy(f) = FFT{R_xy(τ)}, providing a dual-domain perspective on the inter-channel relationship.

Unlike the auto-spectrum which is always real and non-negative, the cross-spectrum is generally complex. The real and imaginary parts carry distinct physical information: the real part (co-spectrum) represents in-phase correlated power, while the imaginary part (quad-spectrum) represents the 90-degree out-of-phase component.