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

$\begin{displaymath} f(t) = \sum_n f(\frac{n}{2W}) \sinc(t-\frac{n}{2W}) \end{displaymath}$

where

$\begin{displaymath} \sinc(t) = \frac{\sin(2 \pi Wt)}{2\pi W t} \end{displaymath}$

From this theorem, we conclude (the dimensionality theorem) that a bandlimited function has only 2W 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 2WT degrees of freedom, and the time- and band-limited function can be represented using 2WT orthogonal basis functions, known as the prolate spheroidal functions. We can view band- and time-limited functions as vectors in a 2TW dimensional vector space.

Assume that the noise power-spectral density of the channel is N₀/2. Then the noise power is (N₀/2)(2W) = N₀ W. Over the time interval of T seconds, the energy per sample (per channel use) is

$\begin{displaymath} \frac{PT}{2WT} = \frac{P}{2W}. \end{displaymath}$

Use this information in the capacity:

$\begin{displaymath}\begin{aligned} C &= \frac{1}{2}\log(1+\frac{P}{N}) \text{ bi... ...(1+\frac{P}{N_0 W}) \text{ bits per channel use}. \end{aligned}\end{displaymath}$

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

$\begin{displaymath} C = (2W) \frac{1}{2}\log(1+\frac{P}{N_0 W}) \text{ bits/second} \end{displaymath}$

$\begin{displaymath} \boxed{C = W \log(1+\frac{P}{N_0 W})} \end{displaymath}$

This is the famous and key result of information theory.

As $W\rightarrow \infty$ , we have to do a little calculus to find that

$\begin{displaymath} C = \frac{P}{N_0} \log_2 e \text{ bits per second}. \end{displaymath}$

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

$\begin{example} For a phone channel, take $W=3300$\ Hz. If the SNR is $P/N_0 W ... ...= 21972 \text{ bits/second}. \end{displaymath}(The book is dated.) \end{example}$
We cannot do better than capacity!

Copyright 2008, by the Contributing Authors. Cite/attribute Resource. admin. (2006, May 17). The Gaussian Channel. Retrieved November 23, 2009, 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.

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	Info The Gaussian Channel Document Actions 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 $\begin{displaymath} f(t) = \sum_n f(\frac{n}{2W}) \sinc(t-\frac{n}{2W}) \end{displaymath}$ where $\begin{displaymath} \sinc(t) = \frac{\sin(2 \pi Wt)}{2\pi W t} \end{displaymath}$ From this theorem, we conclude (the dimensionality theorem) that a bandlimited function has only 2W 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 2WT degrees of freedom, and the time- and band-limited function can be represented using 2WT orthogonal basis functions, known as the prolate spheroidal functions. We can view band- and time-limited functions as vectors in a 2TW dimensional vector space. Assume that the noise power-spectral density of the channel is N₀/2. Then the noise power is (N₀/2)(2W) = N₀ W. Over the time interval of T seconds, the energy per sample (per channel use) is $\begin{displaymath} \frac{PT}{2WT} = \frac{P}{2W}. \end{displaymath}$ Use this information in the capacity: $\begin{displaymath}\begin{aligned} C &= \frac{1}{2}\log(1+\frac{P}{N}) \text{ bi... ...(1+\frac{P}{N_0 W}) \text{ bits per channel use}. \end{aligned}\end{displaymath}$ There are 2W samples each second (channel uses), so the capacity is $\begin{displaymath} C = (2W) \frac{1}{2}\log(1+\frac{P}{N_0 W}) \text{ bits/second} \end{displaymath}$ or $\begin{displaymath} \boxed{C = W \log(1+\frac{P}{N_0 W})} \end{displaymath}$ This is the famous and key result of information theory. As $W\rightarrow \infty$ , we have to do a little calculus to find that $\begin{displaymath} C = \frac{P}{N_0} \log_2 e \text{ bits per second}. \end{displaymath}$ This is interesting: even with infinite bandwidth, the capacity is not infinite, but grows linearly with the power. $\begin{example} For a phone channel, take $W=3300$\ Hz. If the SNR is $P/N_0 W ... ...= 21972 \text{ bits/second}. \end{displaymath}(The book is dated.) \end{example}$ We cannot do better than capacity! Copyright 2008, by the Contributing Authors. Cite/attribute Resource. admin. (2006, May 17). The Gaussian Channel. Retrieved November 23, 2009, 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.	Reuse Course Download this course

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Band-limited channels