In information theory, the error exponent of a channel code or source code over the block length of the code is the rate at which the error probability decays exponentially with the block length of the code. Formally, it is defined as the limiting ratio of the negative logarithm of the error probability to the block length of the code for large block lengths. For example, if the probability of error of a decoder drops as , where is the block length, the error exponent is . In this example, approaches for large . Many of the information-theoretic theorems are of asymptotic nature, for example, the channel coding theorem states that for any rate less than the channel capacity, the probability of the error of the channel code can be made to go to zero as the block length goes to infinity. In practical situations, there are limitations to the delay of the communication and the block length must be finite. Therefore, it is important to study how the probability of error drops as the block length go to infinity.
Error exponent in channel coding
For time-invariant DMC's
The channel coding theorem states that for any ε > 0 and for any rate less than the channel capacity, there is an encoding and decoding scheme that can be used to ensure that the probability of block error is less than ε > 0 for sufficiently long message block X. Also, for any rate greater than the channel capacity, the probability of block error at the receiver goes to one as the block length goes to infinity.
Assuming a channel coding setup as follows: the channel can transmit any of messages, by transmitting the corresponding codeword (which is of length n). Each component in the codebook is drawn i.i.d. according to some probability distribution with probability mass function Q. At the decoding end, maximum likelihood decoding is done.
Let be the th random codeword in the codebook, where goes from to . Suppose the first message is selected, so codeword is transmitted. Given that is received, the probability that the codeword is incorrectly detected as is:
The function has upper bound
Since there are a total of M messages, and the entries in the codebook are i.i.d., the probability that is confused with any other message is times the above expression. Using the union bound, the probability of confusing with any message is bounded by:
for any . Averaging over all combinations of :
Choosing and combining the two sums over in the above formula:
Using the independence nature of the elements of the codeword, and the discrete memoryless nature of the channel:
Using the fact that each element of codeword is identically distributed and thus stationary:
Replacing M by 2nR and defining
probability of error becomes
Q and should be chosen so that the bound is tighest. Thus, the error exponent can be defined as
Error exponent in source coding
For time invariant discrete memoryless sources
The source coding theorem states that for any and any discrete-time i.i.d. source such as and for any rate less than the entropy of the source, there is large enough and an encoder that takes i.i.d. repetition of the source, , and maps it to binary bits such that the source symbols are recoverable from the binary bits with probability at least .
Let be the total number of possible messages. Next map each of the possible source output sequences to one of the messages randomly using a uniform distribution and independently from everything else. When a source is generated the corresponding message is then transmitted to the destination. The message gets decoded to one of the possible source strings. In order to minimize the probability of error the decoder will decode to the source sequence that maximizes , where denotes the event that message was transmitted. This rule is equivalent to finding the source sequence among the set of source sequences that map to message that maximizes . This reduction follows from the fact that the messages were assigned randomly and independently of everything else.
Thus, as an example of when an error occurs, supposed that the source sequence was mapped to message as was the source sequence . If was generated at the source, but then an error occurs.
Let denote the event that the source sequence was generated at the source, so that Then the probability of error can be broken down as Thus, attention can be focused on finding an upper bound to the .
Let denote the event that the source sequence was mapped to the same message as the source sequence and that . Thus, letting denote the event that the two source sequences and map to the same message, we have that
and using the fact that and is independent of everything else have that
A simple upper bound for the term on the left can be established as
for some arbitrary real number This upper bound can be verified by noting that either equals or because the probabilities of a given input sequence are completely deterministic. Thus, if then so that the inequality holds in that case. The inequality holds in the other case as well because
for all possible source strings. Thus, combining everything and introducing some , have that
Where the inequalities follow from a variation on the Union Bound. Finally applying this upper bound to the summation for have that:
Where the sum can now be taken over all because that will only increase the bound. Ultimately yielding that
Now for simplicity let so that Substituting this new value of into the above bound on the probability of error and using the fact that is just a dummy variable in the sum gives the following as an upper bound on the probability of error:
- and each of the components of are independent. Thus, simplifying the above equation yields
The term in the exponent should be maximized over in order to achieve the highest upper bound on the probability of error.
Letting see that the error exponent for the source coding case is:
R. Gallager, Information Theory and Reliable Communication, Wiley 1968