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Monday, October 1, 2012

DSP facts

Autocorrelation
in general:
ϕyy[n,n+m]=E{y[n]y[n+m]}
if stationary
ϕyy[n,n+m]=ϕyy[m]=E{yn+myn}

Deterministic autocorrelation
chh[l]=k=h[k]h[l+k]=h[n]h[n]
Chh(ejω)=H(ejω)H(ejω)=|H(ejω)|2

Response of LTI system to random input
Φyy(ejω)=Chh(ejω)Φxx(ejω)

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]=12πππX(ejω)ejωndω
DTFT
X(ejω)=n=x[n]ejωn

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