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Mutual Information

Module by: Behnaam Aazhang

Summary: A description of mutual information between two random variables with examples.

Recall that

HXY=-∑x∑ypXYxylogpXYxy H X Y x x y y p X Y x y p X Y x y (1)
HY+H X | Y =HX+H Y | X H Y H X | Y H X H Y | X (2)

Definition 1: Mutual Information
The mutual information between two discrete random variables is denoted by ℐX;Y X Y and defined as
ℐX;Y=HX-H X | Y X Y H X H X | Y (3)
Mutual information is a useful concept to measure the amount of information shared between input and output of noisy channels.

In our previous discussions it became clear that when the channel is noisy there may not be reliable communications. Therefore, the limiting factor could very well be reliability when one considers noisy channels. Claude E. Shannon in 1948 changed this paradigm and stated a theorem that presents the rate (speed of communication) as the limiting factor as opposed to reliability.

Example 1

Consider a discrete memoryless channel with four possible inputs and outputs.

Figure 1
Figure 1 (Figure7-32.png)

Every time the channel is used, one of the four symbols will be transmitted. Therefore, 2 bits are sent per channel use. The system, however, is very unreliable. For example, if "a" is received, the receiver can not determine, reliably, if "a" was transmitted or "d". However, if the transmitter and receiver agree to only use symbols "a" and "c" and never use "b" and "d", then the transmission will always be reliable, but 1 bit is sent per channel use. Therefore, the rate of transmission was the limiting factor and not reliability.

This is the essence of Shannon's noisy channel coding theorem, i.e., using only those inputs whose corresponding outputs are disjoint (e.g., far apart). The concept is appealing, but does not seem possible with binary channels since the input is either zero or one. It may work if one considers a vector of binary inputs referred to as the extension channel.

X input vector= X 1 X 2 ⋮ X n ∈ X ¯ n=01n X input vector X 1 X 2 ⋮ X n X ¯ n 0 1 n

Y output vector= Y 1 Y 2 ⋮ Y n ∈ Y ¯ n=01n Y output vector Y 1 Y 2 ⋮ Y n Y ¯ n 0 1 n

Figure 2
Figure 2 (Figure7-34.png)

This module provides a description of the basic information necessary to understand Shannon's Noisy Channel Coding Theorem. However, for additional information on typical sequences, please refer to Typical Sequences.

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