This preview shows page 1. Sign up to view the full content.
Unformatted text preview: ECE 745  Advanced Communication Theory, Spring 2009 Final Exam TakeHome (24 Hours) Procedure 1. Pick up the exam outside my door anytime between Tuesday, May 12th, 9am, and Sunday, May 17th, noon. Sign and date the envelope. 2. There are seven problems for 140 points. 3. For the exam, you may use the course textbook Elements of Information Theory, by Cover and Thomas, and your course notes. For anything from the text, you must establish any missing steps not done in the course notes. For example, if you use the result of a theorem or homework problem, you must provide all of the steps between the course notes and the result you want to use. No other references are allowed, including the WWW. You may not consult with anybody else for any reason  even for an issue of clariﬁcation. Instead, send me email. 4. Within 24 hours of picking up the exam, return it to one of the two following places: (1) If the Marcus 215 complex is open, slide it under the door of my ofﬁce (Marcus 215H). (2) If the Marcus 215 complex is not open, place it in an envelope with “Goeckel” written prominently on the outside. Slide the envelope under the door of the “ECE Mailroom” next to Marcus 210. Testmanship Full credit will be given only to fully justiﬁed answers. Giving the steps along the way to the answer will not only earn full credit but also maximize the partial credit should you stumble or get stuck. If you get stuck, attempt to neatly deﬁne your approach to the problem and why you are stuck. If part of a problem depends on a previous part that you are unable to solve, explain the method for doing the current part, and, if possible, give the answer in terms of the quantities of the previous part that you are unable to obtain. Start each problem on a new page. Not only will this facilitate grading but also make it easier for you to jump back and forth between problems. If you get to the end of the problem and realize that your answer must be wrong (e.g. the Fourier Transform of a real signal that is not conjugate symmetric), be sure to write “this must be wrong because . . . ” so that I will know you recognized such a fact. Academic dishonesty will be dealt with harshly  the minimum penalty will be an “F” for the course. for realvalued discrete random variables and . Show that (c) Let . Give an example where and another where 2. Find the capacity of the following channel 3. Recall the binary symmetric channel with crossover probability . Suppose that I hook two such channels together with no encoding or decoding between the channels; that is, the output from one channel is the input to the next channel: (a) The result of the above is another binary symmetric channel (BSC) with a modiﬁed crossover probability. Find its capacity as a function of . (b) Now, suppose I hook binary symmetric channels with crossover probability together with no encoding or decoding between the channels. Find the capacity as a function of and . What happens as ?. (c) Suppose that I again hook binary symmetric channels with crossover probability together, but I allow inﬁnitely complex encoding and decoding before and after each of the binary symmetric channels. Find the capacity from the input to the output. F 7 7 7 7 ¡ ¨§ )& 7 ¡ (b) Describe a pair of random variables and for which ¡ $ % ¨§ ¨ § % ¡ ©0& ( ¨§ & ¨©§ ¨ § ¡ ! ¡ $ ¨ § ¨ ©§ " #¨ ¥ ¦¢ £ ¤¢ ¢ ¥ ¡ ¨ ©©§ ¨ § ¡ 1. (a) Describe a pair of random variables ¨ ©§ 91 3 @8 8 ¢ £ ¡ 21 1 2 @A BA 4 ¥4 44 £ 44 5 3 164 4 2 and such that there exists and such that % 8 A 7 7 ¨ § 8A 8 7A A D E2 D E2 ¨ ©©§ F IG PHF 7 8 F A 8 £ ¥ A @8 8 91 1 8B [email protected] A 42 44 44 44 8 A 7 7 ¡ 8 A D E2 D E2 & ' 8A 7A 7 8 A 8 % A3 82 3 ¡ ¨ ©§ . . 4. Consider a parallel Gaussian channel: with zeromean independent and identically distributed Gaussian noise sequences and of reand , where . Consider the power constraint . For spective variances the optimal solution, at what power does the channel stop behaving like a single channel with noise and begin behaving like a pair of channels? 5. For the discretetime additive white Gaussian noise channel multipleaccess channel, prove, using typical sequences, that any rate pair can be achieved as long as:
£ ¥ ¥ & ¥ £F Give the codebook constructions, encoding and decoding rules, and show that the probability of error goes to zero. 6. Let be an independent and identically distributed (IID) sequence of random variables, with and . Let be an independent and identically distributed (IID) sequence, with and . Let , where denotes modulo2 addition. (a) At what rate pairs can and be separately encoded with arbitrarily small error probabiliy; in other words, what is the SlepianWolf region. (b) Sketch the rate region in (a) for the special cases:
2 ¥F % E2 FD 2 ¡ B % D2 3 2¨ % 6 A@ 4 23 ¨ 2 0 ¥ 8 9 & F 0 1 0 £ £ ¥ F F ) ) &2 &2 &2 ' ' (%# &$ ' (%# &$ £¡ ¥¡ 1
&$ %# 2 ¡ 1 2 2 2
BC % 7 2 2 ¨ D2 3 ¨ ¥ ¥ $ $ $ § ¥ "! ¨ © 2 3 £ ¥ ¥ £ ¥ £ ¨ § £F ¥F
& ¥¦ £¤ ¢ £¡ ¥¦ £¤ ¢ £¡ £ 1 1
2 ¥ ¨ ¥ ¥ "! § £ ¨ ¥ £ 7 2 72 5¨ 6 4 23 2 3¨ § ¥ £ 3 8 9 B G B G ¥ £ 07 ¥ £ 07 £ ¥ ¥ ¥ E § 7. Before starting this problem, recall the following couple of lines for bounding the error probability during the achievability part of the derivation of the Channel Coding Theorem for discrete memoryless channels (DMCs): where the ﬁrst term upper bounds the probability that the received sequence is not jointly typical with the transmitted codeword, and the second term upper bounds the probability that the received sequence is jointly typical with another codeword. At the time, we were not concerned what happens , we see that we when we got an error, but here we are interested in such. In particular, for have, on average, other codewords (other than the one transmitted) that are jointly typical with . Now, suppose that we have the following situation. Two sources and , with noninterfering channels with capacity and , respectively, to a common destination , desire to send a common message (i.e. a message that they both know) of rate to the destination. Obviously, if , the senders can split the message to accomplish reliable communication. But, alas, Sender 1 insists on sending at rate . Thus, it is your job as Sender 2 to come up with a scheme that transmits at the same time and uses no feedback from the destination (i.e. there is no way to know which “other codewords” are jointly typical with the output of channel 1 and are thus being confused with the correct one) to clear up the confusion. Give your scheme, prove it works, and indicate how big is required to be for your scheme. (Hint: Think about how many bits you would need to send to appropriately reduce the second term in the error probability above, and how you might set up a scheme to accomplish such.) 6 7¨ ¥ ¤B C 9 C @8 8 £B [email protected] A #( & §# 0)'%$" !§
A 8 #( & §# 1 0)'%$" 5443§ 2 6 £ E¨
6 £ F¨ £§¥£ ¤¦¦¨ ¤¥ £6 ¥6 £ © 2 & & 1 1
¢ ¢
$ $ £B ¥B #( & 9# § 1 0)A@$" 54'8 ¡
¥6 ¥6 £6 ¥6 & £6 $
¡ D ...
View
Full
Document
This note was uploaded on 03/30/2010 for the course EE 654 taught by Professor Jackrauster during the Spring '10 term at Universidad del Turabo.
 Spring '10
 jackrauster

Click to edit the document details