hw41 - Mathematical methods in communication 2nd Semester...

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Mathematical methods in communication 2nd Semester 2009 Homework Set #4 Channel and Source coding 1. Lossless source coding with side information. Consider the lossless source coding with side information that is avail- able at the encoder and decoder, where the source X and the side information Y are i.i.d. P X,Y ( x, y ). X n f ( X n , Y n ) ∈ { 1 , 2 , ..., 2 nR } Encoder Decoder Y n Y n ˆ X n ( f ( X n , Y n ) , Y n ) Figure 1: Lossless source coding with side information at the encoder and decoder. Show that a code with rate R < H ( X | Y ) can not be achievable, and interpret the result. Hint: Let T defines f ( X n , Y n ). Consider nR H ( T ) H ( T | Y n ) , (1) and use similar steps, including Fano’s inequality, as we used in the class to prove the converse where side information was not available. 2. Preprocessing the output. One is given a communication channel with transition probabilities p ( y | x ) and channel capacity C = max p ( x ) I ( X ; Y ) . A helpful statisti- cian preprocesses the output by forming ˜ Y = g ( Y ) , yielding a channel p y | x ). He claims that this will strictly improve the capacity.

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