Subject: Speech Processing

Topic: Speech Analysis in Time Domain

Difficulty: Low

Auto-correlation Function: If {x(t)} is a stationary random process {x(t)} then E[x(t).X(t+z)] which is a function of Z is called auto-correlation function of {x(t)} and denoted by R$_{xx}$(Z), R$_x$(Z) or R(Z). Thus $$ R_{xx} (Z) = E[x(t).E(t+Z)]$$

Properties of Auto-Correlation Function R(Z):

(i) The mean square value of a random process can be obtained from the auto-correlation function R(Z).

(ii) R(Z) is even function Z.

(iii) R(Z) is maximum at Z = 0 e.e. |R(Z)| $ \leq $ R(0). In other words, this means the maximum value of R(Z) is attained at Z = 0.

(iv) If R(Z) is the auto-correlation of a stationary random process {x(t)} with no periodic components and with non-zeros means then $$ \lim_{z \to \infty} R(Z) = [E(x)]^2 $$

The short time auto-correlation function is usually computed using the following equation: $$ R_n(k) = \sum_{m-w}^{w} W[-(m-n)]x(m+k)W[-(m-n+k)] $$