Auto/Cross Correlation

Correlation functions measure the statistical similarity between two signals (or a signal with itself) as a function of a time shift τ. They are fundamental tools for time delay estimation, source identification, and periodicity detection.

Definitions

Auto-Correlation

The auto-correlation of a signal x(t) measures its self-similarity at different time lags:

R_xx(τ) = lim_T→∞ (1/T) ∫₀^T x(t) · x(t + τ) dt

At τ = 0, the auto-correlation equals the mean-square value of the signal. Periodic components appear as periodic peaks in R_xx(τ), making it useful for detecting hidden periodicities in noisy signals.

Cross-Correlation

The cross-correlation between signals x(t) and y(t) quantifies their mutual similarity as a function of the relative time shift:

R_xy(τ) = lim_T→∞ (1/T) ∫₀^T x(t) · y(t + τ) dt

The peak of R_xy(τ) indicates the time delay at which the two signals are most similar, directly yielding the propagation delay between two measurement positions.

Normalized Correlation Coefficient

The normalized cross-correlation removes amplitude dependence, producing a dimensionless coefficient bounded between −1 and +1:

ρ_xy(τ) = R_xy(τ) / √(R_xx(0) · R_yy(0))

Value	Interpretation
ρ = +1	Perfectly correlated (identical signals, possibly delayed)
ρ = 0	Uncorrelated (statistically independent)
ρ = −1	Perfectly anti-correlated (inverted signal)

Time Delay Estimation

The primary application of cross-correlation is determining the propagation time between two sensors. If the peak of R_xy(τ) occurs at τ = τ₀, then the signal arrives at sensor y with a delay of τ₀ relative to sensor x. Combined with the known sensor spacing d, the propagation speed is:

c = d / τ₀

For source localization with multiple sensor pairs, triangulation based on the measured time delays provides the spatial coordinates of the noise source.

Parameters

Parameter	Options	Description
Channel X	Any signal channel	Reference signal for cross-correlation.
Channel Y	Any signal channel	Second signal. Set equal to X for auto-correlation.
Block Size	256 – 65536	Number of samples per analysis block.
Max Lag (τ_max)	0 – block length	Maximum time shift displayed. Limits the lag axis.
Normalization	On / Off	When enabled, produces the normalized coefficient ρ_xy.
Number of Averages	1 – unlimited	Averaging reduces noise floor and improves peak detection.

Variants

Correlation vs Time

Computes the correlation function for successive time blocks, allowing the time delay or periodicity to be tracked as it changes over time. The result is a 2D map with lag on the vertical axis and time on the horizontal axis.

Correlation vs RPM

Re-maps the time axis to rotational speed when a tachometer is available. This reveals how the dominant time delay or periodicity shifts with operating speed, which is useful for identifying speed-dependent transfer paths.

Practical Tips

Use a sufficiently long block size to accommodate the expected maximum time delay. If the block is too short, the true correlation peak may be truncated.
Bandpass filter the signals before cross-correlation to focus on the frequency range of interest and suppress out-of-band noise that degrades the peak sharpness.
For source localization, use at least three sensor pairs to resolve azimuth and distance unambiguously.
The cross-correlation can be computed efficiently in the frequency domain as the inverse FFT of the cross-spectrum: R_xy(τ) = IFFT{S_xy(f)}.

When both signals contain strong periodic components at the same frequency, the cross-correlation will show periodic peaks regardless of whether the sources are physically related. Use the coherence function to distinguish causally related signals from coincidental frequency matches.