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# The Gaussian Channel

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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.''

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 17). The Gaussian Channel. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Information_Theory/lecture11_3.htm. This work is licensed under a Creative Commons License