DSP facts

Autocorrelation
in general:
\[
\phi_{yy}[n, n+m] = \text{E}\{y[n]y[n+m]\}
\]
if stationary
\[
\phi_{yy}[n, n+m] =  \phi_{yy}[m] = \text{E}\{\mathbf{y}_{n+m}\mathbf{y}_n^*\}
\]

Deterministic autocorrelation

\[
c_{hh}[l] = \sum_{k=-\infty}^{\infty}h[k]h[l+k] = h[n]*h[-n]
\]
\[
C_{hh}(e^{j\omega}) = H(e^{j\omega})H^*(e^{j\omega}) = |H(e^{j\omega})|^2
\]

Response of LTI system to random input
\[
\Phi_{yy}(e^{j\omega})=C_{hh}(e^{j\omega})\Phi_{xx}(e^{j\omega})
\]

Mean
$m_{\mathbf{x}_n}} = \text{E}\{\mathbf{x}_n\}$
if stationary $m_{\mathbf{x}_n}}=m_x \quad \text{for all } n$
Mean-squared (average power)
$\text{E}\{\mathbf{x}_n\mathbf{x}_n^*\} = \text{E}\{|\mathbf{x}_n|^2\} $
if stationary $\text{E}\{\mathbf{x}_n\mathbf{x}_n^*\} = \text{E}\{\mathbf{x}[n+m]\mathbf{x}[n]\}|_{m=0} = \phi_{xx}[0]$

Inverse DTFT
\[
x[n]=\frac{1}{2\pi} \int_{-\pi}^{\pi} X(e^{j\omega})e^{j\omega n}d\omega
\]
DTFT
\[
X(e^{j\omega}) = \sum_{n=-\infty}^{\infty}x[n]e^{j\omega n}
\]