The Gaussian Channel
Definitions :: Bandlimited :: KuhnTucker :: Parallel :: Colored Noise
Channels with colored Gaussian noise
We will extend the results of the previous section now to channels with nonwhite 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