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Random Processes Through Linear Systems

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Continuous  ::  Discrete  ::  White Noise

Continuous time systems

Recall: A signal $X_t$ through a linear system produces an output

\begin{displaymath}Y_t =\int_{-\infty}^\infty h(t,s) X_s ds.
\end{displaymath}

  • The system is causal if $h(t,s) = 0$ for $t < s$ . In this case,

    \begin{displaymath}Y_t = \int_{-\infty}^t h(t,s) X_s ds.
\end{displaymath}

  • The system is time invariant if $h(t,s) = h(t-s,0) \defeq
h(t-s)$ for all $t$ and $s$ . In this case, $Y_t$ is obtained by convolution:

    \begin{displaymath}Y_t = \int_{\infty}^t h(t-s) X_sds = h*X.
\end{displaymath}

    In this case, we can do analysis using Fourier transforms:

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

Suppose the input function is a random process instead of a deterministic signal. How can we characterize the output function?

Let

\begin{displaymath}Y_t = \int_{-\infty}^\infty h(t,s) X_s ds
\end{displaymath}

The integral is to be interpreted in in the mean-square sense. Properties:
  • $\int_{-\infty}^\infty h(t,s)X_s ds$ exists $\Leftrightarrow
\int_{-\infty}^\infty \int_{-\infty}^\infty h(t,s)h(t,q)
R_X(s,q) dsdq < \infty$ .

  • $\mu_Y(t) = \int_{-\infty}^\infty h(t,\tau) \mu_X(\tau)d\tau$ . That is, the mean of the output is the response of the mean of the input:

    \begin{displaymath}E[Y_t] = E\left[\int_{-\infty}^\infty h(t,\tau) X_\tau d1t\right]
= \int_{-\infty}^\infty h(t,\tau) E[X_t] d\tau
\end{displaymath}

  • $R_{XY}(t,s) = E[Y_tX_s] = \int_{-\infty}^\infty h(t,\tau)
R_X(\tau,s)\,d\tau$ . That is, the correlation between the input and the output is the response to the autocorrelation with respect to the 1st input.
  • $R_Y(t,s) = E[Y_t Y_s] = \int_{-\infty}^\infty
\int_{-\infty}^\infty h(t,\sigma) R_X(\sigma,\tau)
h(s,\tau) d\sigma d\tau$ .
  • If $\{X_t,t\in\Rbb\}$ is W.S.S. and $h$ is tine-invariant. Then $\{Y_t\}$ is also W.S.S., and $\{X_t\}$ and $\{Y_t\}$ are jointly W.S.S. and the following hold:
    1. $\mu_y = \mu_x \int_{-\infty}^\infty h(t) dt$
    2. $R_{YX}(\tau) = \int_{-\infty}^\infty
h(\tau-\sigma)R_X(\sigma)\,d\sigma = (h*R_X)(\tau)$ .
    3. $R_Y(\tau) = \int_{-\infty}^\infty
h(\sigma-\tau)R_{YX}(\sigma)d\sigma = (\hhat*R_{YX})(\tau) = (\hhat
* H * R_X)(\tau)$ , where $\hhat(t) = h(-t)$ .
    4. $S_{YX}(\omega) = H(\omega) S_X(\omega)$
    5. $S_Y(\omega) = H^*(\omega)H(\omega)S_X(\omega) = \vert H(\omega)\vert^2
S_X(\omega)$ . The quantity $\vert H(\omega)\vert^2$ is sometimes called the power transfer function.

\begin{example}
Suppose we have an RC circuit (C in parallel). Then
\begin{disp...
...au\vert} - e^{-\beta
\vert\tau\vert}
\right)
\end{displaymath}\par\end{example}
Copyright 2008, Todd Moon. 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_1.htm. This work is licensed under a Creative Commons License Creative Commons License