Shannon's Source Coding Theorem - Proof: Source Coding Theorem

Proof: Source Coding Theorem

Given is an i.i.d. source, its time series X₁, ..., X_n is i.i.d. with entropy H(X) in the discrete-valued case and differential entropy in the continuous-valued case. The Source coding theorem states that for any for any rate larger 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 .

Proof for achievability

Fix some . The typical set,, is defined as follows:

AEP shows that for large enough n, the probability that a sequence generated by the source lies in the typical set,, as defined approaches one. In particular there for large enough n, (See AEP for a proof):

The definition of typical sets implies that those sequences that lie in the typical set satisfy:

Note that:

• The probability of a sequence from being drawn from is greater than

• since the probability of the whole set is at most one.

• . For the proof, use the upper bound on the probability of each term in typical set and the lower bound on the probability of the whole set .

Since bits are enough to point to any string in this set.

The encoding algorithm: The encoder checks if the input sequence lies within the typical set; if yes, it outputs the index of the input sequence within the typical set; if not, the encoder outputs an arbitrary digit number. As long as the input sequence lies within the typical set (with probability at least ), the encoder doesn't make any error. So, the probability of error of the encoder is bounded above by

Proof for converse The converse is proved by showing that any set of size smaller than (in the sense of exponent) would cover a set of probability bounded away from 1.