Correlation

1 min read Last updated Sat Jun 27 2026 08:46:52 GMT+0000 (Coordinated Universal Time)

Open in ChatGPT Open in Claude

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)

Was this helpful?

Previous Convolution

Next Solving ODEs