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Scribe Notes 7_2

Page history last edited by Harry 14 years, 11 months ago

 

1.converse of rate distortion theorem.

Definition:   Formula

                 Formula

                 Formula

the proof based on the following facts:

Formula

,which is a MC.

Formula 

(a) Formula is at most Formula

(b) data processing inequality

(c)definition of rate distortion function

(d)Jensen's inequality


 

2. Source&Channel separation theorem with distortion.

 

Let Formula be a finite alphabet i.i.d. source which is a

encoded as a sequence of n input symbols Formula of a discrete

memoryless channel with capacity C. The output of the channel Formula is

mapped onto the  reconstruction alphabet Formula . Let

Formulabe the

average distortion achieved by this combined source and channel

coding scheme. Then the distortion is achieved iff Formula

 

 


3. Particular rate-distortion prolbems

Bernoulli(p) soucre with squared-error distortion.

Formula

 

(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 Formula, how to find the minimun distance between them:

                    Formula

Algotithm: we would take any point Formula , and find the Formula  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. Formula

,where Formula.

 

 

3. Formula.

,which can be obtained directly by  the fact of (2)

4. Formula

 

,where A is the set of all joint distribution with marginal p(x) satisfy the distortion constraints.

 B is the set of product distributions Formula with arbitrary Formula 

 

 

 

 

 

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