Rate Distortion Theory February 20, 2006

Source coding aims to reduce redundancy in information to improve storage/transmission efficiency, while channel coding aims to increase redundancy to improve reliable transmission over a noisy channel. The source coding theorem:

Given a discrete memoryless source of entropy, H(ζ), the average code-word length L for any distortionless source encoding scheme, is bounded as L ≥ H(ζ)

In a system, source symbols with alphabet X~p(X=xi) go through an encoder and become representation symbols with alphabet Y~p(Y=yj)

The single letter distortion measure, d(xi, yj) is defined as the cost in representing xi by yj. Define


Single Letter Distortion Measure

p(yj| xi) is said to be “D-admissible” iff d≤D for some D. The set of D-admissible p(yj| xi) is denoted by PD = {p(yj | xi) : d ≤ D}

[Mutual Information]
The mutual information, I(X; Y) is given by


Mutual Information

[Rate Distortion Measure]
A rate distortion function, R(D) (and measured in bits) is defined as the smallest coding rate possible for which the average distortion, d is guaranteed not to exceed D.




subject to the constraint



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