# The Gaussian Channel

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

## Parallel Gaussian channels

Parallel Gaussian channels are used to model bandlimited channels with a non-flat frequency response. We assume we have
*
k
*
Gaussian channels,

where

and the channels are independent. The total power used is constrained:

One question we might ask is: how do we distribute the power across the
*
k
*
channels to get maximum throughput.

We can find the maximum mutual information (the information channel capacity) as

Equality is obtained when the
*
X
*
s are independent normally distributed. We want to distribute the power available among the various channels, subject to not exceeding the power constraint:

with a side constraint (not shown) that . Differential w.r.t.

*P*

_{ j }to obtain

with equality only if all the constraints are inactive. After some fiddling, we obtain

(since λ is a constant). However, we must also have , so we must ensure that we don't violate that if . Thus, we let

where

and is chosen so that

Draw picture; explain "water filling.''