Correlation

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Correlation between $f$ and $g$ , 2 piecewise continuous functions absolutely integrable over $(-\infty, \infty)$ :

(f \star g)(t) = \int_{-\infty}^{\infty} f(\lambda) g(\lambda - t) \, \text{d}\lambda

Relation with Convolution

(f \star g)(t) = (-f * g)(t) = (f * {-g})(t)

Types

Cross-Correlation

Correlation of 2 different signals $f$ and $g$ . Measures the similarity between them as a function of time lag $t$ . Not commutative in general:

(f \star g)(t) \neq (g \star f)(t)

Autocorrelation

Correlation of a signal with itself:

R_{ff}(t) = (f \star f)(t) = \int_{-\infty}^{\infty} f(\lambda) f(\lambda - t) \, \text{d}\lambda

Always symmetric: $R_{ff}(t) = R_{ff}(-t)$ .

Applications

Signal detection
Finding a known pattern within a noisy signal by sliding it over the signal.
Time-delay estimation
Determining the lag between 2 related signals from different sensors.
Power spectral density
Autocorrelation and PSD are a Fourier transform pair (Wiener-Khinchin theorem).

Written by

Sahithyan K.

September 13, 2026 1 min read

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