**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

## Comments (1)

## Mak said

at 10:38 am on Feb 23, 2009

Oh... D:\贴图法? :-)

The notes contain the links:

file:///D:/DOCUME~1/POPPIN~1/LOCALS~1/Temp/moz-screenshot.jpg

file:///D:/DOCUME~1/POPPIN~1/LOCALS~1/Temp/moz-screenshot-1.jpg

I guess these can be removed?

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