# The Gaussian Channel

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

## Band-limited channels

We now come to the first time in the book where the information is
actually carried by a
*
time-waveform
*
, instead of a random
variable. We will consider transmission over a band-limited channel
(such as a phone channel). A key result is the sampling theorem:

This is the classical Nyquist sampling theorem. However, Shannon's name is also attached to it, since he provided a proof and used it. A representation of the function

*f*(

*t*) is

where

From this theorem, we conclude (the dimensionality theorem) that a

*bandlimited function has only 2*

*W*degrees of freedom per second.
For a signal which has "most'' of the energy in bandwidth
*
W
*
and
"most'' of the energy in a time
*
T
*
, then there are about 2
*
WT
*
degrees of freedom, and the time- and band-limited function can be
represented using 2
*
WT
*
orthogonal basis functions, known as the
*
prolate spheroidal
*
functions. We can view band- and time-limited
functions as vectors in a 2
*
TW
*
dimensional vector space.

Assume that the noise power-spectral density of the channel is
*
N
*
_{
0
}
/2. Then the noise power is
(
*
N
*
_{
0
}
/2)(2
*
W
*
) =
*
N
*
_{
0
}
*
W
*
. Over the
time interval of
*
T
*
seconds, the energy per sample (per channel use)
is

Use this information in the capacity:

There are 2

*W*samples each second (channel uses), so the capacity is

or

This is the famous and key result of information theory.

As
,
we have to do a little calculus to find that

This is interesting: even with infinite bandwidth, the capacity is not infinite, but grows linearly with the power.

We cannot do better than capacity!