Correlation

Correlation $⋆$

φ_{s g} (τ) = s (τ) ⋆ g (τ) = \int s^{*} (t) g (t + τ) d t

$s^{*} (t)$ is the conjugate function of $s (t)$ . If $s (t)$ is real-valued, $s^{*} (t)$ is equal to $s (t)$ .

Auto-correlation function

φ_{s s} (τ) = s (τ) ⋆ s (τ)

due to the relation between correlation and convolution (see next section), it can be rewritten as:

φ_{s s} (τ) = s^{*} (- τ) * s (τ)

therefore, the auto-correlation is even:

φ_{s s} (τ) = φ (- τ)

with maximum at $τ = 0$ :

φ_{s s} (τ) \leq φ_{s s} (0) = \int | s (t) |^{2} d t

The convolution of auto-correlation is:

F T {φ_{s s} (τ)} = F T {s^{*} (- τ) * s (τ)} = S^{*} (f) S (f) = | S (f) |^{2}

(Wiener-Khinchin Theorem)
Application: The spectral power density is the Fourier transform of the auto-correlation function.

Just as a correlation function provides information about the temporal relationship between two quantities, so an autocorrelation function tells us about how a quantity at one time is related to itself evaluated at another time. For white noise, the stimulus autocorrelation function is 0 except for one time point $τ = 0$ .

Correlation ⋆

Auto-correlation function

Correlation $⋆$