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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 $k < l$< l$" align="middle" border="0" height="29" width="38" />. 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 $\Leftrightarrow \sum_{l=-\infty}^\infty \sum_{m=-\infty}^\infty h_{k,l} R_X(l,m) h_{k,m} < \infty$< \infty$" align="middle" border="0" height="33" width="301" />.
  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 March 20, 2010, 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
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