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Typical Sequences

Module by: Behnaam Aazhang

Summary: A description of typical sequences.

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If the binary symmetric channel has crossover probability εε then if xx is transmitted then by the Law of Large Numbers the output yy is different from xx in nε n ε places if nn is very large.

d H xy≈nε d H x y n ε (1)
The number of sequences of length nn that are different from xx of length nn at nε n ε is
nnε=n!nε!n−nε! n n ε n n ε n n ε (2)

Example 1

x=000T x 0 0 0 and ε=13 ε 1 3 and nε=313 n ε 3 1 3 . The number of output sequences different from xx by one element: 3!1!2!=3×2×11×2=3 3 1 2 3 2 1 1 2 3 given by 101T 1 0 1 , 011T 0 1 1 , and 000T 0 0 0 .

Using Stirling's approximation

n!≈nnⅇ-n2πn n n n n 2 n (3)
we can approximate
nnε≈2n-εlog2ε−1−εlog21−ε=2n H b ε n n ε 2 n ε 2 ε 1 ε 2 1 ε 2 n H b ε (4)
where H b ε≡-εlog2ε−1−εlog21−ε H b ε ε 2 ε 1 ε 2 1 ε is the entropy of a binary memoryless source. For any xx there are 2n H b ε 2 n H b ε highly probable outputs that correspond to this input.

Consider the output vector YY as a very long random vector with entropy nHY n H Y . As discussed earlier, the number of typical sequences (or highly probably) is roughly 2nHY 2 n H Y . Therefore, 2n 2 n is the total number of binary sequences, 2nHY 2 n H Y is the number of typical sequences, and 2n H b ε 2 n H b ε is the number of elements in a group of possible outputs for one input vector. The maximum number of input sequences that produce nonoverlapping output sequences

M=2nHY2n H b ε=2nHY− H b ε M 2 n H Y 2 n H b ε 2 n H Y H b ε (5)

Figure 1
Figure 1 (Figure7-38.png)

The number of distinguishable input sequences of length nn is

2nHY− H b ε 2 n H Y H b ε (6)
The number of information bits that can be sent across the channel reliably per nn channel uses nHY− H b ε n H Y H b ε The maximum reliable transmission rate per channel use
R=log2Mn=nHY− H b εn=HY− H b ε R 2 M n n H Y H b ε n H Y H b ε (7)
The maximum rate can be increased by increasing HY H Y . Note that H b ε H b ε is only a function of the crossover probability and can not be minimized any further.

The entropy of the channel output is the entropy of a binary random variable. If the input is chosen to be uniformly distributed with p X 0= p X 1=12 p X 0 p X 1 1 2 .

Then

p Y 0=1−ε p X 0+ε p X 1=12 p Y 0 1 ε p X 0 ε p X 1 1 2 (8)
and
p Y 1=1−ε p X 1+ε p X 0=12 p Y 1 1 ε p X 1 ε p X 0 1 2 (9)
Then, HY H Y takes its maximum value of 1. Resulting in a maximum rate R=1− H b ε R 1 H b ε when p X 0= p X 1=12 p X 0 p X 1 1 2 . This result says that ordinarily one bit is transmitted across a BSC with reliability 1−ε 1 ε . If one needs to have probability of error to reach zero then one should reduce transmission of information to 1− H b ε 1 H b ε and add redundancy.

Recall that for Binary Symmetric Channels (BSC)

H Y | X = p x 0H Y | X = 0 + p x 1H Y | X = 1 = p x 0-(1−εlog21−ε−εlog2ε)+ p x 1-(1−εlog21−ε−εlog2ε)=-1−εlog21−ε−εlog2ε= H b ε H Y | X p x 0 H Y | X = 0 p x 1 H Y | X = 1 p x 0 1 ε 2 1 ε ε 2 ε p x 1 1 ε 2 1 ε ε 2 ε 1 ε 2 1 ε ε 2 ε H b ε (10)
Therefore, the maximum rate indeed was
R=HY−H Y | X =ℐX;Y R H Y H Y | X X Y (11)

Example 2

The maximum reliable rate for a BSC is 1− H b ε 1 H b ε . The rate is 1 when ε=0 ε 0 or ε=1 ε 1 . The rate is 0 when ε=12 ε 1 2

Figure 2
Figure 2 (Figure7-41.png)

This module provides background information necessary for an understanding of Shannon's Noisy Channel Coding Theorem. It is also closely related to material presented in Mutual Information.

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