Communications and Signal Processing Spring 2017 MSE Exam Please obtain your Test ID from the following table. You must write your Test ID and name on each of the pages of this exam. A page with missing Test ID or missing name may be assigned a zero score.
Spring 2017 Comm/SP MSE Exam Part I Test ID: Name: 1 Communications and Signal Processing Spring 2017 MSE Exam Name: Test ID: Please put your Test ID and name on each page of this test. Only write in space provided and do not write on the back of the pages. This test is composed of two parts. You must work four questions from Part 1, and three questions from Part 2. If you answer more than seven questions, we will choose randomly which seven to score. So, please mark below those problems that you wish to have graded. Part I (work 4 problems) 1 2 3 4 5 6 7 Part II (work 3 problems) 8 9 10 11 12 13 14 15 16 17 You may use a calculator, a book of math tables (e.g., CRC tables) and tables of Fourier transform pairs, but no other reference material. The test is three hours long.
Spring 2017 Comm/SP MSE Exam Part I Test ID: Name: 2 1. EEE 203 Signals and Systems Consider a continuous-time linear time-invariant system with impulse response h(t) = δ(t n) where δ(t) is the unit impulse function. n=0 (a) Find and sketch the step response of this system; i.e. the output when the input is u(t), the unit step function. (b) Is this system causal? Justify your answer. (c) Is this system stable? Justify your answer.
Spring 2017 Comm/SP MSE Exam Part I Test ID: Name: 3
Spring 2017 Comm/SP MSE Exam Part I Test ID: Name: 4 2. EEE 350 Random Signal Analysis A 2 1 random vector is given by Answer the following questions: X = [ X1 X 2 ]. (1) (a) Let X 1 = U and X 2 = 3U where var(u) = 1. Find the covariance matrix of the resulting X in in equation (1). You can denote the matrix by any capital letter you chose. (b) For the same X 1 = U and X 2 = 3U as in problem (2a), find the correlation coefficient ρ X 1,X 2. (c) Is the covariance matrix of X computed above in problem (2b) positive definite? Explain why or why not. (d) Now consider arbitrary X 1 and X 2 as in equation (1), i.e., the entries of X are not limited to those given in problem (2a). Under what conditions on X 1 and X 2 is the var(x 1 + X 2 ) =var(x 1 ) without X 2 being a constant? (e) If X 1 and X 2 are jointly Gaussian with the joint PDF [ ] ([ ] [ ]) X1 1 2 1 N, (2) X 2 2 1 2 find the joint PDF of the transformed [ ] [ random ] [ vector ] W1 1 1 X1 =. (3) 2 3 W 2 X 2
Spring 2017 Comm/SP MSE Exam Part I Test ID: Name: 5
Spring 2017 Comm/SP MSE Exam Part I Test ID: Name: 6 3. EEE 404 Real-Time DSP Consider the following input signal: x(n) = 1, n = 0, 2 1, n = 1, 3 0, otherwise Consider the following impulse response of an LTI system: h(n) = 1, n = 0 1, n = 1 0, otherwise (a) Indicate how the output of the system can be computed using the circular convolution. (b) Compute the output of the system using the circular convolution.
Spring 2017 Comm/SP MSE Exam Part I Test ID: Name: 7
Spring 2017 Comm/SP MSE Exam Part I Test ID: Name: 8 4. EEE 407 Digital Signal Processing Use analog filter approximations to convert a 1st order R-C lowpass filter circuit (RC=1) to a digital filter with: (a) the impulse invariance method. (b) the bilinear transformation. In both cases, give the transfer function of the digital filter.
Spring 2017 Comm/SP MSE Exam Part I Test ID: Name: 9
Spring 2017 Comm/SP MSE Exam Part I Test ID: Name: 10 5. EEE 407 Digital Signal Processing Show that the Inverse Discrete Fourier Transform (DFT) can be derived by minimizing e(n) in the least-squares sense, where N 1 e(t) = x(n) c(k)e j2πkn/n n = 0, 1, 2,..., N 1. Define c(k). k=0
Spring 2017 Comm/SP MSE Exam Part I Test ID: Name: 11
Spring 2017 Comm/SP MSE Exam Part I Test ID: Name: 12 6. EEE 455 Communication Systems Consider the following signal set: s 1 (t) = A cos(2πf c t) where t [0, T ). s 2 (t) = A sin(2πf c t) (a) What are the basis functions for this signal set? (b) Draw the constellation diagram for this signal set. (c) Find the average energy and minimum distance of this signal set in terms of A.
Spring 2017 Comm/SP MSE Exam Part I Test ID: Name: 13
Spring 2017 Comm/SP MSE Exam Part I Test ID: Name: 14 7. EEE 455 Communication Systems Consider the design of an OFDM modulated signal. Assume that the transmitter operates at a 10 MHz complex sample rate, and that the maximum expected delay spread is less than 1 µs. (a) What is the complex sample period, T s (which is not the duration of the OFDM symbol)? (b) To assure orthogonality, what is the minimum number of samples to use for the cyclic prefix? (c) If the OFDM symbol has 100 subcarriers, what is the subcarrier spacing in frequency?
Spring 2017 Comm/SP MSE Exam Part I Test ID: Name: 15
Spring 2017 Comm/SP MSE Exam Part II Test ID: Name: 16 8. EEE 459/591 Communication Networks Consider TCP slow start. Suppose a new TCP connection is just starting up, i.e., sends one maximum size segment (MSS) in the first transmission round. Suppose the Slow Start Threshold is initially 16 MSS. Suppose the connection has an infinite number of MSSs to send. Further, suppose that a triple duplicate ACK occurs at the end of transmission round 8. No other triple-duplicate ACKs or time-outs occur. Tabulate or plot the congestion window (in units of MSS) as a function of time (in units of transmission rounds) for TCP Tahoe and for TCP Reno from transmission round 1 up to and including transmission round 16. Include the Slow Start Threshold in your table or plot.
Spring 2017 Comm/SP MSE Exam Part II Test ID: Name: 17
Spring 2017 Comm/SP MSE Exam Part II Test ID: Name: 18 9. EEE 505 Time-Frequency Signal Processing The spectrogram time-frequency representation of a signal x(t), with analysis window h(t), is defined as: 2 S x (t, f) = x(τ) h (τ t) e j2πτf dτ. τ (a) Compute and sketch the spectrogram of the sinusoid x(t)=e j600πt using the rectangular window h(t)=u(t) u(t T ), where u(t) is the unit step function and T is the duration of the window. (b) Discuss the time-frequency resolution of the spectrogram in (a) as the window duration T increases.
Spring 2017 Comm/SP MSE Exam Part II Test ID: Name: 19
Spring 2017 Comm/SP MSE Exam Part II Test ID: Name: 20 10. EEE 506 Digital Spectral Analysis Consider the transmitted communications signal represented by the sequence s m, m Z such that it looks like each element of the sequence was drawn independently from a unit-variance circularly symmetric white complex Gaussian signal. The signal propagates through a dispersive channel, so that the observed signal (where we have ignored any noise) at the receiver z m is given by x m = s m + i s m 1 + 1 2 s m 2. (a) Write down the transfer function H(z) for the channel. (b) Is H(z) stable? Why? (c) Evaluate the power spectral density P (f) in terms of the normalized frequency. You can leave it in terms of exponentials.
Spring 2017 Comm/SP MSE Exam Part II Test ID: Name: 21
Spring 2017 Comm/SP MSE Exam Part II Test ID: Name: 22 11. EEE 507 Multidimensional Signal Processing Consider the following 2D signal x(n 1, n 2 ) given by: x(n 1, n 2 ) = u(n 1 + 2, n 2 3) δ(n 1 4) Is x(n 1, n 2 ) separable? Justify your answer to receive proper credit.
Spring 2017 Comm/SP MSE Exam Part II Test ID: Name: 23
Spring 2017 Comm/SP MSE Exam Part II Test ID: Name: 24 12. EEE 508 Digital Image/Video Processing and Compression (a) Rods facilitate sharp vision. (Circle One) Ans: True False (b) Brightness is linearly proportional to luminance. (Circle One) Ans: True False (c) The color of an object depends on the luminance of the surround. (Circle One) Ans: True False (d) There are more cones than rods in the retina. (Circle One) Ans: True False (e) Simple cells have a circular receptive field. (Circle One) Ans: True False
Spring 2017 Comm/SP MSE Exam Part II Test ID: Name: 25
Spring 2017 Comm/SP MSE Exam Part II Test ID: Name: 26 13. EEE 510 Multimedia Signal Processing A simple two-band Quadrature Mirror Filter (QMF) bank is used for sub-band signal processing. The low band analysis filter is H(z) = 1 + z 1. Assuming no quantization/finite precision noise determine the rest of the filters for alias-free reconstruction.
Spring 2017 Comm/SP MSE Exam Part II Test ID: Name: 27
Spring 2017 Comm/SP MSE Exam Part II Test ID: Name: 28 14. EEE 552 Digital Communications Find the parameters (b, T ) of the filter h(t) = δ(t) bδ(t T ) to suppress a carrier signal at 900MHz.
Spring 2017 Comm/SP MSE Exam Part II Test ID: Name: 29
Spring 2017 Comm/SP MSE Exam Part II Test ID: Name: 30 15. EEE 554 Random Signal Theory Consider the random process X(t)=A u(t) + B cos (2πf 0 t), where f 0 is a known frequency and u(t) is the unit step function (which is defined as u(t)=1 for t > 0 and u(t)=0 for t < 0). The two continuous random variables A and B are assumed independent; A is uniformly distributed between -5 and 5, and B has a Gaussian distribution with zero mean and variance 25. (a) Find the mean of the random process X(t). (b) Find the autocorrelation function of the random process X(t). (c) Is the random process X(t) stationary in the wide sense? Justify your answer.
Spring 2017 Comm/SP MSE Exam Part II Test ID: Name: 31
Spring 2017 Comm/SP MSE Exam Part II Test ID: Name: 32 16. EEE 557 Broadband Networks Let G denote the offered traffic to a slotted Aloha channel and assume that the merged packet arrivals form a Poisson process. That is, P [k transmissions in one slot] = (G) k k! e G. (a) What is the proportion of slots wasted due to collisions? What is the optimal value G to minimize this proportion (i.e., minimize the collisions)? (b) What is the proportion of slots wasted due to either idle slots or collisions? What is the optimal value for G to minimize the wasted slots? (c) What can you conclude from the results you obtain in (1) and (2)? Can you conclude that as long as we minimize the collisions, we can always achieve the maximum throughput?
Spring 2017 Comm/SP MSE Exam Part II Test ID: Name: 33
Spring 2017 Comm/SP MSE Exam Part II Test ID: Name: 34 17. EEE 558 Wireless Communications Answer the questions below and be as brief as possible while justifying your answer. For each of the following, please justify your answer by explaining why. (a) Describe a process to estimate the power delay profile. What would the transmitter send, and what would the receiver do to estimate it? (b) Why is the lognormal distribution appropriate to model shadow fading? (c) Name one advantage, and one disadvantage to standardization in wireless. Each should be at most a couple of sentences. (d) Does the Rayleigh fading model fit better for an urban channel model, or a rural channel model? Justify.
Spring 2017 Comm/SP MSE Exam Part II Test ID: Name: 35