Introduction
The mean and autocorrelation functions of a stationary and second-order causal discrete process x(k) are, respectively, equal to:
\(m=E\left\{x\left(k\right)\right\}=\lim\limits_{N \to \infty} \frac {1} {N}\sum_{k=1}^{N}x\left(k\right)\) (13)
\(R_x\left(k\right)=E\left\{x\left(i\right)x\left(i-k\right)\right\}=\lim\limits_{N \to \infty} \frac {1} {N}\displaystyle\sum_{k=1}^{N} x\left(i\right)x\left(i-k\right)\) (14)
For a discrete realisation \(\left\{x\left(1\right),\ x\left(2\right),...,x\left(N-1\right)\right\}\) , of height N, we estimate this average by:
\(\hat{M}=\frac{1}{N}\sum_{k=1}^{N}x\left(k\right)\) (15)
\(\hat{M}\) is called the empirical mean.