ENGR 461: Digital Communications Midterm Exam MIDTERM EXAM ENGR 461: Digital Communications School of Engineering The University of British Columbia — Okanagan Campus Total Time: 75 min including canvas upload time Total Marks: 40 (Questions are not with equal marks) In the first page of your answer sheet, write Name: UBC ID: Academic honesty and integrity pledge Academic honesty and integrity are essential principles of the University of British Columbia and engineering as a profession. All UBC students are expected to behave as honest and responsible members of an academic community. Engineering students have an even greater responsibility to maintain the highest level of academic honesty and integrity as they prepare to enter a profession with those principles as a cornerstone. Cheating on exams or projects, labs, plagiarizing or any other form of academic dishonesty are clear violations of these principles. As a student of the School of Engineering at UBC Okanagan, I solemnly pledge to follow the policies, principles, rules, and guidelines of the University with respect to academic honesty. In particular, I commit to upholding the academic integrity and the professionalism as an engineering student. By agreeing this pledge, I promise to adhere to exam/quiz requirements and maintain the highest level of ethical principles during the exam period. I also promise that I am doing this quiz in closed book, closed note and lab manuals. I am doing also without web search and help from other. Read above academic honesty and integrity pledge. Then in the first page of your sheet your answer sheet, if you agree, write. ”I agree with the attached academic honesty and integrity pledge” and then sign. October 26, 2020 1 ENGR 461: Digital Communications Midterm Exam Problem 1 (Marks: 4+4+2) Signal X(t) has a bandwidth of 3000 Hz, and its amplitude at any time is a random variable whose PDF is given by fX(x) = { 1 10(1− |x|/10) |x| ≤ 10 0 otherwise. (1) We want to transmit this signal using a uniform PCM system. 1. Determine power in X(t). 2. What is the SQNR in decibels if a PCM system with 32 levels is employed? 3. What is the minimum data rate that we need to transmit this signal? Problem 2 (Marks: 2+4+4+2) Consider the three waveforms shown in the following. x1(t) = { +1 0 ≤ t ≤ 1 −1 1 < t ≤ 2 0 otherwise. (2) x2(t) = { −1 0 ≤ t ≤ 1 +1 1 < t ≤ 2 0 otherwise. (3) x3(t) = { +2 3 ≤ t ≤ 4 0 otherwise. (4) 1. Plot these signals, x1(t), x2(t) and x3(t). 2. Design and plot (with proper amplitude) the basis function(s) to represent these signals. 3. In order to validate your design, express x1(t), x2(t) and x3(t) (with proper scaling factors) in terms of basis signals you designed in previous question. 4. Find the distance between two signals x1(t), x2(t). Problem 3 (Marks: 2+2+2+4) Assume that if a signal s1(t) is passed over a noisy channel, the output of the correlation-type demodulator sampled at t = T , r is given by the following equation r = −1 + n (5) where the noise variable n is characterized by the Laplacian probability density function (PDF) with zero mean and variance 2. In general, a Laplacian PDF with mean mX and variance is σ2X is given by fX(x) = 1√ 2σ2X e −|x−mX |√ σ2 X /2 . (6) October 26, 2020 2 ENGR 461: Digital Communications Midterm Exam 1. Determine the mean of the random variable at the demodulator output, r assuming that the signal s1(t) is transmitted. (Hints: the mean of n is zero and the variance is 2σ) 2. Determine the variance of the random variable at the demodulator output, r assuming that the signal s1(t) is transmitted. (Hints: the mean of n is zero and the variance is 2σ) 3. Write down the PDF of the random variable at the demodulator output, r assuming that the signal s1(t) is transmitted. (Hints: it will be a shifted Laplacian PDF) 4. Let us assume that the detector compares the output of the demodulator, r with a thresh- old α = 0 and makes a decision that the signal s1(t) is transmitted if r < 0. On the other hand, if r ≥ 0, the detector decides that another signal has been transmitted. Find the probability of error assuming that the signal s1(t) is transmitted. Problem 4 (Marks: 5+3) Consider the channel with frequency (magnitude) response C(f) = 1√ 1+(f/W )2 |f | ≤W 0; otherwise (7) where W = 4800Hz. This channel is equalized by a zero-forcing equalizer at the receiver. Assuming that the receiving filter gR(t) is an ideal low pass filter with cutoff frequency of W Hz i.e., |GR(f)| = { 1 |f | ≤W 0; otherwise (8) 1. Assuming an additive white Gaussian noise channel with zero mean and power spectral density N0/2, determine the noise variance/power at the sampling instant at the output of the zero-forcing equalizer in terms of N0. [Hint: zero-forcing equalizer has frequency response GE(f) = 1/C(f)]. 2. If there was no equalizer at the receiver, what would be the noise power at the output of the receive filter? October 26, 2020 3
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