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Random Processes Through Linear Systems

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Continuous  ::  Discrete  ::  White Noise

White Noise

We have seen that for a discrete-time signal we can create a ``white'' noise with the properties

\begin{displaymath}\mu_X(k) = 0 \qquad\qquad R_X(k,l) = \frac{N_0}{2} \delta_{k,l}.
\end{displaymath}

For continuous time random processes, we say that it is white noise if

\begin{displaymath}\mu_X(t) = 0
\end{displaymath}


\begin{displaymath}R_X(t,s) = \frac{N_0}{2} \delta(t-s).
\end{displaymath}

This is thus W.S.S. However, since we are dealing with the Dirac $\delta$ function, we do not have a second order random process. Because of this, such a process cannot be said to exist in a physical sense. Nevertheless, it is a very important and practical model for use in conjunction with linear systems.

Using a white noise process is fine as long as it is input to a linear system (which will integrate the process, thus smoothing it out).

Note that if the process is Gaussian and white, then the output is also Gaussian. In this case, knowing the first and second distributions (e.g., mean and correlation functions) is sufficient to entirely characterize the f.d.d.s of the joint processes.

We mentioned earlier the Wiener random process. Let us consider one now having with zero mean:

\begin{displaymath}R_W(t,s) = \sigma^2 \min(t,s) + \mu^2 t s = \sigma^2 \min(t,s)
\end{displaymath}

We can approximate the white random process using differences,

\begin{displaymath}\{W_{t+\Delta t} - W_t\}.
\end{displaymath}

In the limit, we can take the derivative of the Wiener process to obtain a white noise process.

What is the autocorrelation function of the derivative of the Wiener process?

\begin{displaymath}\partiald{R_W(t,s)}{t} =
\begin{cases}
\sigma^2 & t < s \\ 0 & t > s.
\end{cases}\end{displaymath}


\begin{displaymath}\partialw{R_W(t,s)}{s}{t} = \sigma^2 \delta(t-s).
\end{displaymath}

So we get a W.S.S. white random noise process. Strictly speaking, however, the derivative does not exist in the m.s. sense.

We always employ white noise r.p. in the context of an integration operation (e.g., running through a system).

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_3.htm. This work is licensed under a Creative Commons License Creative Commons License