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# The Gaussian Channel

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

## Band-limited channels

We now come to the first time in the book where the information is actually carried by a time-waveform , instead of a random variable. We will consider transmission over a band-limited channel (such as a phone channel). A key result is the sampling theorem:

This is the classical Nyquist sampling theorem. However, Shannon's name is also attached to it, since he provided a proof and used it. A representation of the function f ( t ) is

where

From this theorem, we conclude (the dimensionality theorem) that a bandlimited function has only 2 W degrees of freedom per second.

For a signal which has "most'' of the energy in bandwidth W and "most'' of the energy in a time T , then there are about 2 WT degrees of freedom, and the time- and band-limited function can be represented using 2 WT orthogonal basis functions, known as the prolate spheroidal functions. We can view band- and time-limited functions as vectors in a 2 TW dimensional vector space.

Assume that the noise power-spectral density of the channel is N 0 /2. Then the noise power is ( N 0 /2)(2 W ) = N 0 W . Over the time interval of T seconds, the energy per sample (per channel use) is

Use this information in the capacity:

There are 2 W samples each second (channel uses), so the capacity is

or

This is the famous and key result of information theory.

As , we have to do a little calculus to find that

This is interesting: even with infinite bandwidth, the capacity is not infinite, but grows linearly with the power.

We cannot do better than capacity!

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_1.htm. This work is licensed under a Creative Commons License