Problem Set 3

Q5. Converse for channel coding theorem

intuitively, converse means what you can achieve using a certain of energy

(a). First prove that

      What we need: , , ,

                             , 

      Proof:

     Note: 1. since

              2. Chain Rule

              3. Directed Markov chain:

      Now prove that

      What we need:

     Proof: 

                                                                                               Fano's inequality

                                                                                                                   data processing theorem

                                                                                                                    Result of part A

Problem Set 4:

Zero-Error Data Compression

Q1. (DNA Encoding)

(a) For scheme 1. Error associated with it

                       2. encoding time complexity, exponential time to decode it. using the binary search, but you need to sort the sequence beforehead. considering he the DNA sequence, using the random scheme, you need a large codebook

                       note: complexity contains time and space complexity, for space complexity, it is codebook size.

(c) extend the tree to level , we get the tree as below:

Based on the observation, we get the descendents in level are

so

or and it is called Kraft Inequality

(d) from the observation of the tree above, we can conclude that prefix-free code leads to Kradt Inequality and vice versa