1.converse of rate distortion theorem.
Definition:
the proof based on the following facts:
,which is a MC.
(a) is at most
(b) data processing inequality
(c)definition of rate distortion function
(d)Jensen's inequality
2. Source&Channel separation theorem with distortion.
Let be a finite alphabet i.i.d. source which is a
encoded as a sequence of n input symbols of a discrete
memoryless channel with capacity C. The output of the channel is
mapped onto the reconstruction alphabet . Let
be the
average distortion achieved by this combined source and channel
coding scheme. Then the distortion is achieved iff
3. Particular rate-distortion prolbems
Bernoulli(p) soucre with squared-error distortion.
(it is a template to compute R(D))
4. Compute rate-distortion functions
1. A simple Case.
Given two convex sets A and B in , how to find the minimun distance between them:
Algotithm: we would take any point , and find the that is closest to it. Then fix this y and find the closest point in A. Repeating this process to minimum distance dereases at each stage.
2.
,where .
3. .
,which can be obtained directly by the fact of (2)
4.
,where A is the set of all joint distribution with marginal p(x) satisfy the distortion constraints.
B is the set of product distributions with arbitrary
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