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# The Gaussian Channel

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Definitions   ::   Band-limited   ::   Kuhn-Tucker   ::   Parallel   ::   Colored Noise

## Channels with colored Gaussian noise

We will extend the results of the previous section now to channels with non-white Gaussian noise. Let K z be the covariance of the noise K x the covariance of the input, with the input constrained by

which is the same as

We can write

where

Now how do we choose K x to maximize K x + K z , subject to the power constraint? Let

then

where A = Q T K x Q . Observe that

So we want to maximize subject to . The key is to use an inequality, in this case Hadamard's inequality. Hadamard's inequality follows directly from the "conditioning reduces entropy'' theorem:

Let . Then

and

Substituting in and simplifying gives

with equality iff K is diagonal.

Getting back to our problem,

with equality iff A is diagonal. We have

(the power constraint), and . As before, we take

where is chosen so that

Now we want to generalize to a continuous time system. For a channel with AWGN and covariance matrix K Z ( n ) , the covariance is Toeplitz. If the channel noise process is stationary, then the covariance matrix is Toeplitz, and the eigenvalues of the covariance matrix tend to a limit as . The density of the eigenvalues on the real line tends to the power spectrum of the stochastic process. That is, if K ij = K i - j are the autocorrelation values and the power spectrum is

then

In this case, the water filling translates to water filling in the spectral domain. The capacity of the channel with noise spectrum N ( f ) can be shown to be

where is chosen so that

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 17). The Gaussian Channel. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Information_Theory/lecture11_4.htm. This work is licensed under a Creative Commons License