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Random Processes Through Linear Systems

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Continuous  ::  Discrete  ::  White Noise

Discrete-time filters

Let $T = \Zbb$ . The output of a discrete-time system is

\begin{displaymath}Y_k = \sum_{l=-\infty}^\infty h_{k,l} X_l
\end{displaymath}

Causal: $h_{k.l} = 0$ for $k < l$ . Time-invariant: $h_{k,l} =
h_{k-l,0} = h_{k-l}$ for all $k,l \in \Zbb$ . Then

\begin{displaymath}Y_k =\sum_{l=-\infty}^\infty h_{k-l} X_l = (h*X)_k.
\end{displaymath}

We can transform in the time-invariant case using a Z-transform,

\begin{displaymath}Y(z) = H(z)X(z),
\end{displaymath}

or

\begin{displaymath}Y_k =frac{1}{2\pi i} \oint Y(z) z^{k-1} dz.
\end{displaymath}

We also deal with the discrete-time Fourier transform, obtained by evaluating on the unit circle $z = e^{i\omega}$ . We write

\begin{displaymath}\left. H(z)\right\vert _{z = e^{i \omega}} = H(\omega).
\end{displaymath}

(This is an abuse of notation, but makes the notation consistent with continuous time.)

For a random process, we still have

\begin{displaymath}Y_k =\sum_{l=-\infty}^\infty h_{k-l} X_l = (h*X)_k,
\end{displaymath}

but we interpret this in a m.s. sense:

\begin{displaymath}\lim_{m,n\rightarrow \infty} E[(Y_k - \sum_{l=-m}^n h_{k,l} X_l)^2]
= 0.
\end{displaymath}

Properties:

  1. $Y_k$ exists $\Leftrightarrow \sum_{l=-\infty}^\infty
\sum_{m=-\infty}^\infty h_{k,l} R_X(l,m) h_{k,m} < \infty$ .
  2. $\mu_Y(k) = \sum_{l=-\infty}^\infty h_{k,l} \mu_x(l)$ .
  3. $R_{YX}(k,n) = \sum_{l=-\infty}^\infty h_{k,l}R_X(l,n)$
  4. $R_Y(k,n) = \sum_{l=-\infty}^\infty h_{n,l} R_{YX}(k,l) =
\sum_{l}\sum_m h_{n,l} h_{k,m}R_X(m,l)$
  5. If $\{X_k\}$ is W.S.S. and $\{h_{k,l}\}$ is time-invariant, then $\{Y_k\}$ is W.S.S. and $\{X_k\}$ , $\{Y_k\}$ are jointly W.S.S. Then:
    1. $\mu_y = \mu_x \sum_{k} h_k$
    2. $R_{YX}(k) = (h * R_X)(k)$
    3. $R_Y(k) = (\hhat * X * h)(k)$ , where $\hhat(k) = h(-k)$ .
    These properties can be expressed in the Z-transform domain:
    1. $\mu_y = \left. \mu_x H(z)\right\vert _{z=1}$ .
    2. $S_{YX}(z) = \sum_k R_{YX}(k) z^{-k} = H(Z) S_X(z)$
    3. $S_Y(z) = \sum_k R_{YX}(z)z^{-k} = H(z^{-1})H(z) S_X(z)$ .
    These can be further expressed on the unit circle as:
    1. $\mu_y = \left. \mu_x H(\omega)\right\vert _{\omega=0}$ .
    2. $S_{YX}(\omega) = H(\omega) S_X(\omega)$
    3. $S_Y(\omega) = \vert H(\omega)\vert^2 S_X(\omega)$ .

\begin{example}
Consider the feedback system with
\begin{displaymath}Y_k = X_k ...
...Y(\omega) = \frac{N_0}{2} \vert H(\omega)\vert^2.
\end{displaymath}\end{example}
Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, June 07). Random Processes Through Linear Systems. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lecture8_2.htm. This work is licensed under a Creative Commons License Creative Commons License