# The Gaussian Channel

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

## The Gaussian channel

Suppose we send information over a channel that is subjected to
additive white Gaussian noise. Then the output is

*Y*

_{ i }=

*X*

_{ i }+

*Z*

_{ i }

where

*Y*

_{ i }is the channel output,

*X*

_{ i }is the channel input, and

*Z*

_{ i }is zero-mean Gaussian with variance

*N*: . This is different from channel models we saw before, in that the output can take on a continuum of values. This is also a good model for a variety of practical communication channels.

We will assume that there is a constraint on the input power. If we
have an input codeword
,
we will assume that the
**
average power
**
is constrained so that

Let is consider the probability of error for binary transmission.
Suppose that we can send either
or
over the
channel. The receiver looks at the received signal amplitude and
determines the signal transmitted using a threshold test. Then

where

or

We can compute this as follows:

since

*E Y*

^{ 2 }=

*P*+

*N*and the Gaussian is the maximum-entropy distribution for a given variance. So

bits per channel use. The maximum is obtained

*when*. (How do we make the input distribution look Gaussian?)

*X*is Gaussian distributed

**
Geometric plausibility
**
For a codeword of length
*
n
*
,
the received vector (in
*
n
*
space) is normally distributed with mean
equal to the true codeword. With high probability, the received
vector is contained in sphere about the mean of radius
.
Why? Because with high probability, the
vector falls within one standard deviation away from the mean in each
direction, and the total distance away is the Euclidean sum:

This is the square of the expected distance within which we expect to fall. If we assign everything within this sphere to the given codeword, we misdetect only if we fall outside this codeword.

Other codewords will have other spheres, each with radius
approximately
.
The received vectors a limited
in energy by
*
P
*
, so they all must lie in a sphere of radius
.
The number of (approximately) nonintersecting
decoding spheres is therefore

The volume of a sphere of radius

*r*in

*n*space is proportional to

*r*

^{ n }. Substituting in this fact we get

The converse is that rate
*
R
*
>
*
C
*
are not achievable, or, equivalently,
that if
then it must be that
.