The Gaussian Channel
Parallel Gaussian channels
Parallel Gaussian channels are used to model bandlimited channels with a non-flat frequency response. We assume we have
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
and is chosen so that
Draw picture; explain "water filling.''