Autocorrelation Function
The autocorrelation function R_xx(τ) = E[x(t)·x(t+τ)] measures how similar a signal is to a time-shifted version of itself. At zero lag it equals signal power. For periodic signals, autocorrelation is periodic. For white noise, it's a delta function at τ=0. Serial autocorrelation in time series data means successive observations are not independent — the Durbin-Watson test detects this in regression residuals (DW ≈ 2 means no autocorrelation).
Autocorrelation: Self-Similarity Over Time
Autocorrelation quantifies how predictable a signal's future is from its past. A highly autocorrelated signal like temperature changes slowly — knowing today's value tells you a lot about tomorrow's. White noise has zero autocorrelation — each sample is independent. Mathematically, R_xx(τ) = E[x(t)·x(t+τ)]. For ergodic signals, this equals the time average. The autocorrelation at zero lag R_xx(0) equals the signal's mean square value (power).
Key Formulas
Properties of the Autocorrelation Function
Autocorrelation is symmetric: R_xx(τ) = R_xx(−τ). Its maximum is at zero lag: |R_xx(τ)| ≤ R_xx(0). For periodic signals, autocorrelation is periodic with the same period — this is how pitch detection finds fundamental frequency of speech and music. For signals with DC offset, autocorrelation approaches the square of the mean as τ → ∞. The normalized autocorrelation ρ_xx(τ) = R_xx(τ)/R_xx(0) ranges from −1 to +1.
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Open CalculatorSerial Autocorrelation in Statistics
In time series, serial autocorrelation means successive observations are correlated — residuals that show autocorrelation indicate a model is missing systematic patterns. The Durbin-Watson statistic tests for first-order autocorrelation: DW ≈ 2(1 − ρ₁). DW near 2 means no autocorrelation; near 0 means strong positive; near 4 means strong negative. Ignoring serial autocorrelation leads to underestimated standard errors and unreliable hypothesis tests.
Power Spectral Density: Wiener-Khinchin Theorem
The Wiener-Khinchin theorem connects time and frequency domains: the power spectral density S_xx(f) is the Fourier transform of the autocorrelation R_xx(τ). White noise has flat PSD and delta autocorrelation. A low-pass signal has PSD that falls off at high frequencies and gradually decaying autocorrelation. This theorem bridges time-domain correlation measurements to frequency-domain energy distribution.
Applications: Pitch Detection, Radar, Signal Analysis
Pitch detection exploits periodicity of autocorrelation — the first peak after τ=0 occurs at the fundamental period. This method is robust to noise because autocorrelation suppresses uncorrelated noise. In radar, autocorrelation of received signal with transmitted waveform (matched filtering) maximizes detection. In CDMA telecommunications, spreading code autocorrelation properties determine system performance.
Related Topics in signal processing mathematics
Understanding autocorrelation function connects to several related concepts: autocorrelation formula, self correlation, what is autocorrelation, and serial autocorrelation. Each builds on the mathematical foundations covered in this guide.
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