ece6605-hwk3

ece6605-hwk3 - X = { 1 , 2 , 3 , 4 } and that the...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
GEORGIA INSTITUTE OF TECHNOLOGY School of Electrical and Computer Engineering ECE 6605 Information Theory Assigned: Monday, Sep. 27, 2010 Due: Atlanta and Savannah students: Wed, Oct. 6, 2010 Due: Video students: Wed, Oct. 13, 2010 Problem Set #3 Problem 1-2: Solve the questions 4.12 and 4.13 from Chapter 4 of the textbook (the second edition). Problem 3: Let assume the time is discretized to time slots or epochs. At each time epoch we throw a fair (unbiased) coin. Let X be the waiting time (i.e., the number of time epochs) for the first heads to appear in successive flips of a fair coin. Thus, the distribution of the waiting time is given by Pr { X = i } = (1 / 2) i . Let S n be the waiting time for the n th head to appear. Thus, we have: S 0 = 0 S n +1 = S n + X n +1 for n = 0 , 1 , 2 ,..., where X 1 ,X 2 ,X 3 ,... are iid according to the above distribution. (a) (10 points) Determine H ( S 1 ,S 2 ,...,S n ). (b) (10 points) Does the process S n have an entropy rate? If so, what is it? If not, why not? Problem 4: Suppose that
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: X = { 1 , 2 , 3 , 4 } and that the probabilities of the four possible outcomes are p = { 1 2 , 1 8 , 1 4 , 1 8 } (a) (2 points) Determine H ( X ). (b) (2 points) Let q = { 1 8 , 1 4 , 1 2 , 1 8 } be probabilities associated with a random variable Y also dened on the set { 1 , 2 , 3 , 4 } . Compute H ( Y ). (c) (2 points) Find the relative entropy between p and q , (i.e., D ( p k q ). Also nd D ( q k p ). (d) (6 points) Find a Human code for X . (e) (2 points) Find the expected codeword length for the Human code. 1 (f) (6 points) Now suppose that q had been the true distribution, but the Human code was designed using p as in part d. Find the expected codeword length. What is the cost for not using the true distribution q to design the code? Problem 5: Solve the questions 4.28 from Chapter 4 of the textbook (the second edition). 2...
View Full Document

Page1 / 2

ece6605-hwk3 - X = { 1 , 2 , 3 , 4 } and that the...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online