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Scribe Notes 7-1

Page history last edited by Lingxiao XIA 13 years, 1 month ago

Scalar quantization

A source generates an element Formula from the set {0,1,2,3} with uniform probability. The source encoder needs to describe the source to the decoder, but is allowed to use only a single bit Formula. The decoder's task is to reconstruct the source as the symbol Formula (which must also be in the set {0.1.2.3}) while minimizing the expected distortion. The distortion between two symbols here is defined as the mean-squared error (MSE) Formulabetween Formula and Formula. Propose a scheme with a distortion that is as small as possible.

 

We can just  encode both 0 and 1 to '0' and both 2 and 3 to '1', and decode '0' as 0 and '1' as 2, thus the expected distortion computed by expected MSE is

Formula

 

 

Vector quantization

Now, suppose the source generates two elements Formula i.i.d. from the same source. The encoder encodes this as two bits Formula, and the decoder decodes these as Formula. Propose a scheme with an MSE distortion Formulathat is as small as possible.

 

Since Formula is drawn from {0,1,2,3}, we can denote Formula as 16 different symbols shown in the image below, with the first bit for Formula and the last bit for Formula, and we have divided the 16 different symbols into four different groups, with one representitive (13, 32, 01, 20) for each of the groups. We call the different groups different assignment rigions/Voronoi Cells, and the different representitives different reproduction points.

 

 

And the expected MSE is:

Formula

 

 

 

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