ECE 564/645 - Digital Communications, Spring 2018 Midterm Exam #1 March 22nd, 7:00-9:00pm Marston 220

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1 ECE 564/645 - Digital Communications, Spring 08 Midterm Exam # March nd, 7:00-9:00pm Marston 0 Overview The exam consists of four problems for 0 points (ECE 564) or 5 points (ECE 645). The points for each part of each problem are given in brackets - you should spend your two hours accordingly. The exam is closed book, but you are allowed one page-side of notes. Calculators are not allowed. I will provide all necessary blank paper. 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, 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.

2 . Huffman Coding: [5] (a) Let a source have alphabet X = {A, B, C, D, E, F }, with respective probabilities of the letters 0.4, 0.35, 0., 0.05, 0.05, and Find a Huffman code for this source for N = letters/block. Find the rate of your Huffman code. What can you say about the entropy H(X) of the source? [0] (b) I have a source with X = {A, B, C, D, E, F, G}, but I do not know the source probabilities. A friend comes to me and tells me that he/she has a Huffman code for this source with rate 3. bits/symbol. Is this possible?. Consider the -dimensional vector channel r = s + n where r is the received vector, s = s i if message m i is to be sent on (0, T s ), and n = (n, n ) T. Let n and n be independent Gaussian random variables with mean 0 and variance N 0 (i.e. this is an AWGN vector channel). Suppose there are eight possible equally likely messages (M = 8) and s = (, 0) T s = (, 0) T s 3 = (, 0) T s 4 = (, 0) T s 5 = (0, ) T s 6 = (0, ) T s 7 = (0, ) T s 8 = (0, ) T [5] (a) Draw the signal space and decision regions for the MAP receiver. [0] (b) In terms of the units of the vector space (you do not have to convert to E s in this part!) and N 0 : Find the Union Bound to the conditional symbol error probability P (E s ). Remove as many summands as possible from the Union Bound for P (E s ), while still guaranteeing an upper bound is preserved. [5] (c) Convert your answers from the second half of (b) to be in terms of E s and N 0, where E s is the average energy per signal. [5] (d) We want to compare to QPSK: Convert your answer from part (c) to be in terms of E b and N 0. How much better (or worse) it this system s performance at high signal-to-noise ratio (SNR) than QPSK? (I have been intentionally vague about the word performance here; I want you to tell me how you would describe the relative performance of the two systems at high SNR.)

3 [5] (e) We desire to use our 8-ary signal set to transmit 3 bits per symbol (of course). The receiver is still the MAP receiver from (a). Assign the sets of three bits (i.e. 000, 00, 00, 0, 00, 0, 0, ) to the signals so as to minimize the bit error probability at large signal-to-noise ratios. Only minimal justification is required. [5] (f) Without changing the energy of any of the signals, change the signal set to one whose MAP receiver has a smaller symbol error probability P (E) at large Es N 0. An approximate sketch is sufficient, but be sure to justify your answer. 3. Consider the waveform channel: r(t) where is additive white Gaussian noise with power spectral density N 0, r(t) is the received waveform, and = s i (t) when message m i is to be sent during time t (0, T s ). Your boss wants to employ quadrature phase-shift keying (QPSK) over the interval (0, T s ), which would have signal set: s (t) = cos(πf c t) + sin(πf c t) s (t) = cos(πf c t) sin(πf c t), s 3 (t) = cos(πf c t) + sin(πf c t), s 4 (t) = cos(πf c t) sin(πf c t), where f c T s is the carrier frequency. Unfortunately, the in-phase and quadrature branches are not balanced; hence, for some constant a > 0, your actual signal set (over (0, T s )) is: s (t) = cos(πf c t) + a sin(πf c t) s (t) = cos(πf c t) a sin(πf c t), s 3 (t) = cos(πf c t) + a sin(πf c t), s 4 (t) = cos(πf c t) a sin(πf c t), This problem deals with a system employing s (t), s (t), s 3 (t), and s 4 (t). You can assume the four signals are equally likely (i.e. P (s i (t) sent) = /4, for all i) throughout this problem. [0] (a) Find an orthonormal basis φ (t), φ (t),..., φ N (t) for s (t), s (t), s 3 (t), and s 4 (t) with minimum N. (You need not use Gram-Schmidt if you see an easier solution), and (do not forget this part!), give the vector representation of the signals in that basis. [5] (b) Give an optimal receiver that takes as input r(t) and decides the signal most likely to be sent given r(t). You can assume that the receiver knows the value of a. [0] (c) Suppose a = /. Find the exact probability of symbol error (P E ) for your receiver

4 in (b) in terms of E b, the energy per bit, and N 0. [5] (d) At high signal-to-noise ratios (SNRs), how much better or worse (in db) is this system (with a = /) than QPSK? 4. [564 only] In this problem, we consider what happens when the scaling of the correlators at the receiver for an AWGN channel is not as given in class. Consider a binary signal set {s 0 (t), s (t)}, where the information bit selects which of these two signals will be sent over the waveform channel: r(t) where is additive white Gaussian noise with power spectral density N 0, and r(t) is the received waveform. Suppose the orthonormal basis for {s 0 (t), s (t)} requires two basis functions φ (t) and φ (t), and the front-end of the receiver calculates (different than in class because of the out front!): r = r = 0 0 r(t)φ (t)dt r(t)φ (t)dt yielding the vector r = (r, r ) T for us to use to decide which signal was sent. Define the vectors x 0 and x, as follows: the j th component of x 0 and x is given by, respectively: x 0,j = 0 s 0 (t)φ j (t)dt x,j = 0 s (t)φ j (t)dt [5] (a) Suppose x 0 = (, 0) T and x = (0, ) T. Draw the optimal decision regions in r-space. [0] (b) Now, suppose I give you some x 0 and x (not necessarily those in part (a)). Find the probability of error of an optimal receiver (that has as input r, as given above) as a function of x 0 x and N 0. Be sure to justify your answer fully. [5] (c) As in class, let s i be the representation of s i (t) on the orthonormal basis. For any s 0 and s, what is the probability of error of the optimal receiver as a function of s 0 s and N 0?

5 5. [645 only] In this problem, we consider how to deal with a (very) poor front-end to our receiver. Consider the waveform channel: r(t) where is additive white Gaussian noise with power spectral density N 0, r(t) is the received waveform, and = s i (t) when message m i is to be sent during time t (0, T s ). Suppose there are M = 4 possible equally likely messages and the corresponding signals are: s (t) s (t) t t s 3 (t) s 4 (t) t t You, of course, recognize immediately that this is a two-dimensional signal space, and thus you should build a correlation receiver with two basis functions for the front-end of the optimal receiver. But, alas, the person designing the front-end of your receiver does not, and simply multiplies by the function: { t, 0 t x(t) = 0, else and integrates, providing you with the single number r = 0 r(t)x(t)dt. [0] (a) Find the optimal processing of r to decide which signal was sent. (Note: The fractions get a little messy in this part and the next - sorry!) [5] (b) Find the probability of symbol error (choosing the wrong symbol) as a function of the average signal energy E s and N 0. (Remember that the signal energy of s i (t) is 0 s i (t)dt.). [0] (c) For high SNRs, roughly how much performance (in db) is lost by using this (very) poor front-end versus an optimal front-end? (You can use quick d min arguments here.)

ECE 564/645 - Digital Communications, Spring 2018 Homework #2 Due: March 19 (In Lecture)

ECE 564/645 - Digital Communications, Spring 018 Homework # Due: March 19 (In Lecture) 1. Consider a binary communication system over a 1-dimensional vector channel where message m 1 is sent by signaling