The Gaussian Channel
The Gaussian channel
Suppose we send information over a channel that is subjected to
additive white Gaussian noise. Then the output is
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
is constrained so that
Let is consider the probability of error for binary transmission.
Suppose that we can send either
channel. The receiver looks at the received signal amplitude and
determines the signal transmitted using a threshold test. Then
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 X is Gaussian distributed . (How do we make the input distribution look Gaussian?)
For a codeword of length
the received vector (in
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
The received vectors a limited
in energy by
, 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 .