**S***calar quantization*

**A source generates an element 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 . The decoder's task is to reconstruct the source as the symbol (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) between and . 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

**Vector quantization**

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

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

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