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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,

 

 \begin{displaymath}
Y_j = X_j + Z_j,\qquad j=1,2,\ldots, k.
\end{displaymath}

 

where

 

 \begin{displaymath}
Z_j \sim \Nc(0,N_j)
\end{displaymath}

 

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

 

 \begin{displaymath}
E\sum_{j=1}^k X_j^2 \leq P.
\end{displaymath}

 

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

 

\begin{displaymath}\begin{aligned}
I(X_1,\ldots,X_k;Y_1,\ldots,Y_k) &= h(Y_1,\ld...
...i) \\
&\leq \sum_{i} \frac{1}{2} \log(1+P_i/N_i)
\end{aligned}\end{displaymath}

 

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:

 

 \begin{displaymath}
J(P_1,\ldots,P_k) = \sum_i \frac{1}{2}\log(1+\frac{P_i}{N_i}) + \lambda
\sum_{i=1}^k P_i
\end{displaymath}

 

with a side constraint (not shown) that $P_i \geq 0$ . Differential w.r.t. P j to obtain

 

 \begin{displaymath}
\frac{1}{P_j + N_j} + \lambda \geq 0.
\end{displaymath}

 

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

 

 \begin{displaymath}
P_j = \nu - N_j
\end{displaymath}

 

(since λ is a constant). However, we must also have $P_j \geq
0$ , so we must ensure that we don't violate that if $N_j > \nu$ . Thus, we let

 

 \begin{displaymath}
P_j = (\nu - N_j)^+
\end{displaymath}

 

where

 

 \begin{displaymath}
(x)^+ =
\begin{cases}
x & x \geq 0 \\
0 & x < 0
\end{cases}
\end{displaymath}

 

and nu is chosen so that

 

 \begin{displaymath}
\sum_{i=1}^n (\nu - N_i)^+ = P
\end{displaymath}

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 Creative Commons License