## Section: Scientific Foundations

### Rate-distortion theory

Coding and joint source channel coding rely on fundamental concepts of information theory, such as notions of entropy, memoryless or correlated sources, of channel capacity, or on rate-distortion performance bounds. Compression algorithms are defined to be as close as possible to the optimal rate-distortion bound, R(D) , for a given signal. The source coding theorem establishes performance bounds for lossless and lossy coding. In lossless coding, the lower rate bound is given by the entropy of the source. In lossy coding, the bound is given by the rate-distortion function R(D) . This function R(D) gives the minimum quantity of information needed to represent a given signal under the constraint of a given distortion. The rate-distortion bound is usually called OPTA (Optimum Performance Theoretically Attainable ). It is usually difficult to find close-form expressions for the function R(D) , except for specific cases such as Gaussian sources. For real signals, this function is defined as the convex-hull of all feasible (rate, distortion) points. The problem of finding the rate-distortion function on this convex hull then becomes a rate-distortion minimization problem which, by using a Lagrangian formulation, can be expressed as

The Lagrangian cost function J is derivated with respect to the different optimisation parameters, e.g. with respect to coding parameters such as quantization factors. The parameter is then tuned in order to find the targeted rate-distortion point. When the problem is to optimise the end-to-end Quality of Service (QoS) of a communication system, the rate-distortion metrics must in addition take into account channel properties and channel coding. Joint source-channel coding optimisation allows to improve the tradeoff between compression efficiency and robustness to channel noise.

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