# hw7s - ECE 178 HW#7 Solutions Anindya Sarkar Emre Sargin Q1...

This preview shows pages 1–5. Sign up to view the full content.

ECE 178: HW #7 Solutions Q1 . a) H ( X ) = 4 × 1 / 32 × log 2 (32)+2 × 1 / 16 × log 2 (16)+8 × 1 / 64 × log 2 (64)+1 / 8 × log 2 (8)+1 / 2 × log 2 (2) H ( X ) = 22 / 8bits b) Entropy maximizing distribution is the uniform distribution: p i = 1 / 16 , i = 1 ,..., 16 c) Proof. H ( X ) = X i p i log 2 ( 1 p i ) |{z} Jensen log 2 ( X i p i 1 p i ) = log 2 ( |X| ) Equality holds only if p i = 1 / |X| ∀ i , which completes the proof. 1

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

View Full Document
Q2 . Huﬀman code can be constructed as follows: Message Probability a 1 2 15 a 2 2 15 a 3 3 15 a 4 3 15 a 5 5 15 Message Probability a 3 3 15 a 4 3 15 a 5 5 15 a 6 = { a 1 ,a 2 } 4 15 Message Probability a 7 = { a 3 ,a 4 } 6 15 a 5 5 15 a 6 = { a 1 ,a 2 } 4 15 Message Probability a 7 = { a 3 ,a 4 } 6 15 a 8 = { a 5 ,a 6 } 9 15 Message Probability Codeword a 1 2 15 100 a 2 2 15 101 a 3 3 15 00 a 4 3 15 01 a 5 5 15 11 2
Q3 . Message is in between (0 . 931878 , 0 . 931905) 0 a 1 2/15 a 2 4/15 a 3 7/15 a 4 10/5 a 5 1 0.000000 0.133333 0.266667 0.466667 0.666667 1.000000 0.666667 0.711111 0.755556 0.822222 0.888889 1.000000 0.888889 0.903704 0.918519 0.940741 0.962963 1.000000 0.918519 0.921481 0.924444 0.928889 0.933333 0.940741 0.928889 0.929481 0.930074 0.930963 0.931852 0.933333 0.931852 0.932049 0.932247 0.932543 0.932840 0.933333 0.931852 0.931878 0.931905 0.931944 0.931984 0.932049 3

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

View Full Document
Q4 . Figure 1: A lossy open-loop predictive coding model: (a) encoder and (b) decoder e n = f n - ˆ f n , where in HW problem ˆ f n = f n - 1 In the decoder side, the bit-sequence (of the prediction error) is decoded to obtain the error terms (let ˙ e n denote the quantized version of e n , as in Fig. 1). The reconstructed output ˙ f n is obtained by adding the quantized error to the predicted value ˆ f n . ˙ f
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 06/12/2009 for the course ECE 178 taught by Professor Manjunath during the Winter '08 term at UCSB.

### Page1 / 11

hw7s - ECE 178 HW#7 Solutions Anindya Sarkar Emre Sargin Q1...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online
The provided text is a set of solutions to homework problems for an ECE 178 course, which appears to cover topics in information theory and coding. It includes solutions to four questions, with mathematical calculations and explanations.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# hw7s - ECE 178 HW#7 Solutions Anindya Sarkar Emre Sargin Q1...

This preview shows pages 1–5. Sign up to view the full content.

ECE 178: HW #7 Solutions Q1 . a) H ( X ) = 4 × 1 / 32 × log 2 (32)+2 × 1 / 16 × log 2 (16)+8 × 1 / 64 × log 2 (64)+1 / 8 × log 2 (8)+1 / 2 × log 2 (2) H ( X ) = 22 / 8bits b) Entropy maximizing distribution is the uniform distribution: p i = 1 / 16 , i = 1 ,..., 16 c) Proof. H ( X ) = X i p i log 2 ( 1 p i ) |{z} Jensen log 2 ( X i p i 1 p i ) = log 2 ( |X| ) Equality holds only if p i = 1 / |X| ∀ i , which completes the proof. 1

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

View Full Document
Q2 . Huﬀman code can be constructed as follows: Message Probability a 1 2 15 a 2 2 15 a 3 3 15 a 4 3 15 a 5 5 15 Message Probability a 3 3 15 a 4 3 15 a 5 5 15 a 6 = { a 1 ,a 2 } 4 15 Message Probability a 7 = { a 3 ,a 4 } 6 15 a 5 5 15 a 6 = { a 1 ,a 2 } 4 15 Message Probability a 7 = { a 3 ,a 4 } 6 15 a 8 = { a 5 ,a 6 } 9 15 Message Probability Codeword a 1 2 15 100 a 2 2 15 101 a 3 3 15 00 a 4 3 15 01 a 5 5 15 11 2
Q3 . Message is in between (0 . 931878 , 0 . 931905) 0 a 1 2/15 a 2 4/15 a 3 7/15 a 4 10/5 a 5 1 0.000000 0.133333 0.266667 0.466667 0.666667 1.000000 0.666667 0.711111 0.755556 0.822222 0.888889 1.000000 0.888889 0.903704 0.918519 0.940741 0.962963 1.000000 0.918519 0.921481 0.924444 0.928889 0.933333 0.940741 0.928889 0.929481 0.930074 0.930963 0.931852 0.933333 0.931852 0.932049 0.932247 0.932543 0.932840 0.933333 0.931852 0.931878 0.931905 0.931944 0.931984 0.932049 3

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

View Full Document
Q4 . Figure 1: A lossy open-loop predictive coding model: (a) encoder and (b) decoder e n = f n - ˆ f n , where in HW problem ˆ f n = f n - 1 In the decoder side, the bit-sequence (of the prediction error) is decoded to obtain the error terms (let ˙ e n denote the quantized version of e n , as in Fig. 1). The reconstructed output ˙ f n is obtained by adding the quantized error to the predicted value ˆ f n . ˙ f
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 06/12/2009 for the course ECE 178 taught by Professor Manjunath during the Winter '08 term at UCSB.

### Page1 / 11

hw7s - ECE 178 HW#7 Solutions Anindya Sarkar Emre Sargin Q1...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online
user
The provided text is a set of solutions to homework problems for an ECE 178 course, which appears to cover topics in information theory and coding. It includes solutions to four questions, with mathematical calculations and explanations.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS