更改

A message W is transmitted through a noisy channel by using encoding and decoding functions. An encoder maps W into a pre-defined sequence of channel symbols of length n. In its most basic model, the channel distorts each of these symbols independently of the others. The output of the channel –the received sequence– is fed into a decoder which maps the sequence into an estimate of the message. In this setting, the probability of error is defined as:

A message W is transmitted through a noisy channel by using encoding and decoding functions. An encoder maps W into a pre-defined sequence of channel symbols of length n. In its most basic model, the channel distorts each of these symbols independently of the others. The output of the channel –the received sequence– is fed into a decoder which maps the sequence into an estimate of the message. In this setting, the probability of error is defined as:

::<math> P_e = \text{Pr}\left\{ \hat{W} \neq W \right\}. </math>

::<math> P_e = \text{Pr}\left\{ \hat{W} \neq W \right\}. </math>

1. For every discrete memoryless channel, the channel capacity is defined in terms of the mutual information <math>I(X; Y)</math>,

1. For every discrete memoryless channel, the channel capacity is defined in terms of the mutual information <math>I(X; Y)</math>,

1.对于每一个离散的无记忆信道，信道容量是根据互信息 i (x; y) </math > 来定义的,

1.对于每一个离散的无记忆信道，信道容量是根据互信息I(x; y)来定义的,

has the following property.  For any <math>\epsilon>0</math> and <math>R<C</math>, for large enough <math>N</math>, there exists a code of length <math>N</math> and rate <math>\geq R</math> and a decoding algorithm, such that the maximal probability of block error is <math>\leq \epsilon</math>.

has the following property.  For any <math>\epsilon>0</math> and <math>R<C</math>, for large enough <math>N</math>, there exists a code of length <math>N</math> and rate <math>\geq R</math> and a decoding algorithm, such that the maximal probability of block error is <math>\leq \epsilon</math>.

具有以下属性。对于任何 < math > epsilon > 0 </math > 和 < math > r < c </math > ，对于足够大的 < math > n </math > ，存在一个长度为 < math > n </math > 和速率 < math > geq r </math > 的代码和一个解码算法，使得块错误的最大概率为 < math > leq epq </math > 。

具有以下属性。对于任何ε>0 和 R<C ，对于足够大的N ，存在一个长度为 N 和速率R的代码和一个解码算法，使得块错误的最大概率为ε 。

2. If a probability of bit error <math>p_b</math> is acceptable, rates up to <math>R(p_b)</math> are achievable, where

2. If a probability of bit error <math>p_b</math> is acceptable, rates up to <math>R(p_b)</math> are achievable, where

2.如果位错概率 < math > p _ b </math > 是可以接受的，那么达到 < math > r (p _ b) </math > 的速率是可以实现的

2.如果位错概率pb是可以接受的，那么达到R(pb)的速率是可以实现的

3. For any <math>p_b</math>, rates greater than <math>R(p_b)</math> are not achievable.

3. For any <math>p_b</math>, rates greater than <math>R(p_b)</math> are not achievable.

3.对于任何 < math > p _ b </math > ，比率大于 < math > r (p _ b) </math > 是无法实现的。

3.对于任何pb ，比率大于R(pb)是无法实现的。

== Outline of proof ==

== Outline of proof证明概述 ==

As with the several other major results in information theory, the proof of the noisy channel coding theorem includes an achievability result and a matching converse result.  These two components serve to bound, in this case, the set of possible rates at which one can communicate over a noisy channel, and matching serves to show that these bounds are tight bounds.

As with the several other major results in information theory, the proof of the noisy channel coding theorem includes an achievability result and a matching converse result.  These two components serve to bound, in this case, the set of possible rates at which one can communicate over a noisy channel, and matching serves to show that these bounds are tight bounds.

As with the several other major results in information theory, the proof of the noisy channel coding theorem includes an achievability result and a matching converse result.  These two components serve to bound, in this case, the set of possible rates at which one can communicate over a noisy channel, and matching serves to show that these bounds are tight bounds.

As with the several other major results in information theory, the proof of the noisy channel coding theorem includes an achievability result and a matching converse result.  These two components serve to bound, in this case, the set of possible rates at which one can communicate over a noisy channel, and matching serves to show that these bounds are tight bounds.

The following outlines are only one set of many different styles available for study in information theory texts.

The following outlines are only one set of many different styles available for study in information theory texts.

===Achievability for discrete memoryless channels===

===Achievability for discrete memoryless channels离散无记忆信道的可达性===

This particular proof of achievability follows the style of proofs that make use of the asymptotic equipartition property (AEP).  Another style can be found in information theory texts using error exponents.

This particular proof of achievability follows the style of proofs that make use of the asymptotic equipartition property (AEP).  Another style can be found in information theory texts using error exponents.

Both types of proofs make use of a random coding argument where the codebook used across a channel is randomly constructed - this serves to make the analysis simpler while still proving the existence of a code satisfying a desired low probability of error at any data rate below the channel capacity.

Both types of proofs make use of a random coding argument where the codebook used across a channel is randomly constructed - this serves to make the analysis simpler while still proving the existence of a code satisfying a desired low probability of error at any data rate below the channel capacity.