2S2122 Activity 3.1

Problem 1 - The Complete Channel Model (2 pts.)

Answer comprehensively and do the following:

1. Draw the complete channel model. (0.2 pts)

2. What is the source? (0.2 pts)

3. What is the channel? (0.2 pts)

4. What is the receiver? (0.2 pts)

5. What are the encoder and decoder? (0.2 pts)

6. What are codebooks? (0.2 pts)

7. What is coding efficiency? (0.2 pts)

8. What is the maximum capacity of a channel? (0.4 pts)

9. What are symbol rates? (0.4 pts)

10. Where is Waldo? (if we like your answer you get 0.2 pts extra)

Problem 2 - Review of BSC (2 pts.)

1. Draw the BSC channel with correct annotations of important parameters $p$ and $\epsilon$. (0.1 pts) 2. Fill up the probability table in terms of the important parameters: (0.1 pts each entry)

Probability Term	Function
$P(s=1)$
$P(s=0)$
$P(r=1|s=0)$
$P(r=0|s=1)$
$P(r=0,s=0)$
$P(r=0,s=1)$
$P(r=1,s=0)$
$P(r=1,s=1)$
$P(r=1)$
$P(r=0)$

3. Fill up the table below with the measures of information. Answer only in terms of $p$ and $\epsilon$ parameters (and of course constants). When necessary you can use "let $q$ be ...". Also, answer with equations that are programming friendly. Answering the simplified version merits half points for that part. (0.1 pts per term)

Information Measure	Function
$H(S)$
$H(R)$
$H(R|S)$
$I(R|S)$

4. Explain why $I(R,S)=0$ when $\epsilon =0.5$ regardless of varying the source probability distribution $p$. (0.2 pts.) 5. Explain what does negative information mean and when can it happen? (0.2 pts.) 6. Explain why $I(R,S)<0$ means information from the receiver can short-circuit back to the source. (0.1 pts.)

Problem 3 - CRAZY Channels

Problem 3.a Binary Erasure Channel (2 pts.)

A Binary Erasure Channel (BEC) is shown below:

Binary Erasure Channel

BEC is an interesting channel because it has 3 different outputs. The received values $0$, $1$, and $*$. This channel assumes that we can erase or throw unwanted values into the trash bin. It's worth investigating! 😁 Show your complete solutions and box your final answers.

1. Fill up the table below but show your solutions. Solve for each probability first then summarize your results in the table. Answers should be in terms of $p$ and $\gamma$ only. (0.2 pts. each term)

Probability Term	Function
$P(r=0)$
$P(r=1)$
$P(r=*)$

2. Determine the mutual information on a per outcome basis. Fill up the table below but show your solutions. Answers should be in terms of $p$ and $\gamma$ only. Answers should be in 1 $\log _{2}(\cdot )$ term only . Not following instructions results in -0.05 pts.(0.2 pts. each term)

Information Term	Function
$I(r=0,s=0)$
$I(r=1,s=1)$
$I(r=*|s=0)$
$I(r=*|s=1)$

3. What is the average mutual information $I(R,S)$? (0.2 pts.)

4. What is the maximum channel capacity $C$? When does it happen? (0.2 pts.)

5. What are the pros and cons of BEC compared to BSC? Let's assume $\gamma$ is channel noise. (0.2 pts.)

Problem 3.b Binary Error and Erasure Channel (2 pts.)

Below is a Binary Error and Erasure Channel (BEEC) . It's a direct combination of BSC and BEC. Have fun 🐔

Binary Error Erasure Channel (Yikes...)

This channel is pretty much the same. The only difference is that bit flips are now accounted for. Just in case the erasure decoder is not perfect.

1. Fill up the table below but show your solutions. Solve for each probability first then summarize your results in the table. Answers should be in terms of $p$, $\gamma$, and $\epsilon$ only. (0.2 pts. each term)

Probability Term	Function
$P(r=0)$
$P(r=1)$
$P(r=*)$

2. Determine the following measures of information. Fill up the table below but show your solutions. Answers should be in terms of $p$, $\gamma$, and $\epsilon$ only. Answers should be programming friendly . Not following instructions results in -0.05 pts.(0.3 pts. each term). You can write "let $q$ be ... something" if necessary.

Information Term	Function
$H(R)$
$H(R|S)$
$I(R,S)$
$H(R,S)$

3. Based on your answers in 3b.2, analyze and interpret, what does an erasure channel do with noise? (0.2 pts.)

Problem 3.c Binary Cascade Channel (6 pts.)

The figure below shows a Binary Cascade Channel (BCC).

Binary Cascade Channel (BCC)

A BCC works like an intervening transmitter in between. Think like repeaters for routers. The source sends messages first to the first receiver. Then, the first receiver sends the same message to the second receiver. Assume that the source and the second receiver and independent of each other. Determine the following:

1. Fill up the table below but show your solutions. Solve for each probability first then summarize your results in the table. Answers should be in terms of $p$, $\epsilon _{1}$, and $\epsilon _{2}$ only. You may use $q=p+\epsilon _{1}-2\epsilon _{1}p$ to keep equations shorter. (0.5 pts. each term)

Probability Term	Function
$P(r=0)$
$P(r=1)$

2. Determine the following measures of information. Fill up the table below but show your solutions. Answers should be in terms of $p$, $\epsilon _{1}$, and $\epsilon _{2}$ only. Answers should be programming friendly . You may use $q=p+\epsilon _{1}-2\epsilon _{1}p$ to keep equations shorter. Not following instructions results in -0.05 pts. You can write "let $q$ be ... something" if necessary. (0.5 pts. each term)

Information Term	Function
$I(R_{1},S)$
$I(R_{1}|R_{2})$

3. Let's do some analysis, what would $I(R_{1},S)$ be if $p=0.5$ and $\epsilon _{1}=0$? Explain thoroughly how you came up with your answer. (1 pt.)

4. Follow up from 3, what is the the appropriate equality for $I(R_{1},S){\overset {?}{=}}I(R_{2},R_{1})$ if $0<\epsilon _{2}<1$ and $\epsilon _{2}\neq 0.5$? Explain thoroughly how you came up with your answer. (1 pt.)

5. Follow up from 4, what is the the appropriate equality for $I(R_{1},S){\overset {?}{=}}I(R_{2},R_{1})$ if $0<\epsilon _{1}<1$ and $\epsilon _{1}\neq 0.5$? Explain thoroughly how you came up with your answer. (1 pt.)

6. In general, what is the appropriate equality for $I(R_{1},S){\overset {?}{=}}I(R_{2},R_{1})$? Explain thoroughly how you came up with your answer. (1 pt.)

(Big hint: "data processing")

Huffman Coding

The table below shows the source symbols to be sent over a noiseless BSC.

Source Symbol	Probability
b	0.15
d	0.25
g	0.02
j	0.10
o	0.28
space	0.20

Determine the following. Show your complete solution and box your final answers.

1. Draw the final Huffman coding tree. Make sure that you follow the same algorithm from the discussions. These include:

• All left arrows are tagged as 0s while all right are tagged as 1s.
• When nodes with same probabilities occur, make sure the deepest tree takes the left path.

2. Provide the codebook for the source.

3. Calculate $L(X)$.

4. Calculate $\eta$.

5. Calculate $R_{\textrm {max}}$.

6. Calculate $R_{X}$.

7. Decode the message 100011110100100111101.