Test 2 Electrical Engineering Bachelor Module 8 Signal Processing and Communications

Transcription

1 Test 2 Electrical Engineering Bachelor Module 8 Signal Processing and Communications ( ) Tuesday May 26, 2015, 14:00-17:00h This test consists of three parts, corresponding to the three courses in Theme 2: Part 1: Probability Theory; Part II: Random Signals and Noise; Part III: Digital Signal Processing. Problems in different parts can be solved independently. Grading: Part I and Part II each have a maximum score of 25 points, whereas Part III has a maximum score of 40 points; You earn 10 additional points by obeying all test rules outlined below and on the next page. The grade for the test is the total number of points divided by 10. Material to use during the test: All printed and handwritten material can be consulted during the test, including books, handouts, printed lecture slides, written notes, summaries and solutions. A simple pocket calculator can be used. Graphical and/ or programmable calculators, laptop computers, mobile phones, tablets, PDAs and any other electronic devices with wireless communication capability are not allowed on your table. See next page 1

2 Answering: Write down answers corresponding to different test parts (I, II and III) on separate answer papers. Fill out your full name and student number on every answer paper before writing down your answers. In case you need more than one answer paper per test part, complete all four pages of your first paper before proceeding to a new one. Possibly ask the exam supervisor for more papers. Although the problems are formulated in English, you can write your answers in Dutch. Please write in black or blue, using a pen. Do not use a pencil. Motivate all your answers. End of the test: No one leaves during the last 15 minutes of the test (i.e. after 16:45h). When it is 17:00h, do not leave your seat until your answer papers have been collected. Students with disabilities can use 45 minutes (25%) more time, provided that they show their pass and sit in the front row. Hand in at least one answer paper. Any remaining papers can be taken home, including this question paper. Once your answer papers have been collected, leave the room quickly and quietly. Good luck! See next page for Part I 2

3 Part I - Probability Theory 1. A shop in second-hand electronic equipment sells three types of microphones. Type i microphones are working properly with probability Pi, where p 1 = 0.8, p 2 = 0.9 and p 3 = 1. Arjan buys a microphone at random, meaning that with probability 1/3 he buys a type i microphone, i = 1, 2, 3. Define and use notation for relevant event(s) and/or random variable(s) to find the following. a. The probability that the purchased microphone is working properly. b. The probability that the purchased microphone is of type 3 when it turns out to be working properly. 2. Let Xn be the number of typos (errors) on page n of a certain set of lecture notes. The simultaneous probability distribution of the numbers of typos on pages 1 and 2 is given by e-2 P(X1 = i, X2 = j) = , i, j = 0, 1, 2,... L..J. a. Show that the distribution of X 1 is Poisson with expectation 1. b. Are X 1 and X 2 mutually independent? c. Determine the probability distribution of X 1 + X Suppose a researcher takes some time measurements X andy (in seconds), where X and Y have a joint density function: f(x, y) = 2x + 2y for 0 :::; x :::; 1 and 0 :::; y :::; x, while f(x, y) = 0 elsewhere. a. Find the marginal probability density of X. b. Knowing that the expected value of Y is 5/12 (so you do not need to compute this), find the covariance of X andy. What can be concluded from this value? c. What is the probability that the sum of X andy is less than 1 second? Scores per problem: Totaal alb alblc alb _I c End of Part I - See next page for Part II 3

4 Part II- Random Signals and Noise This part consists of 5 related problems. Consider a random signals X(t) that is given by X(t) = sin(27rfot + 8), where the frequency fo is a positive deterministic constant, and the phase 8 is a random variable that is uniformly distributed between 0 and 1r. 1. [3 points] Plot two possible realizations of X(t). Clearly mark the axes. 2. [3 points] Is X(t) first-order stationary? Motivate your answer. Suppose that the signal X(t) is passed through a squaring device, resulting in a random signal Y(t) = X 2 (t). 3. [8 points] Is Y(t) wide-sense stationary? Motivate your answer. 4. [4 points] Show that the power spectral density of Y(t) is given by Sy(J) = (1 + 2fo) + ~15(1) (1-2fo). Now suppose that Y(t) is passed through a linear time-invariant filter with impulse response f h(t) = 0 < t < 1/ f JO' - - JO' { 0, elsewhere, resulting in an output signal Z(t). 5. [7 points] Derive the autocorrelation function of Z(t). End of Part II - See next page for Part III 4

5 Part III - Digital Signal Processing Exercise 1 DTFT [5 points] Prove that if y[n] = x[-n] and :r[n] +--+ X(ei 0 ), then y[n] +--+ X(e-i 0 ) Exercise 2 DTFT [5 points] Prove that, for lal < 1, lnl 1- a2 a a cos(s1) + a 2 Exercise 3 DFT [5 points] Assume that x[n], n = 0,..., N- 1 is real-valued and x[n] +--+ X*[k] denote the complex conjugate of X[k]. Prove that X[k]. Let X*[k] = X[N- k], k = 0,..., N- 1. Hint: Use X[-k] = X[N- k]. Exercise 4 Filter design [5 points] Consider the impulse response h[n], n = 0,..., AI of an FIR lowpass filter with a cutoff frequency Slc. Explain how this impulse response can be modified to that of a high-pass filter with a cutoff frequency Jr- Slc. Exercise 5 Spectral DSP [10 points total] Consider an overlap-add implementation of a digital filter with impulse response h[n], n = 0,..., K -1. The signal is divided into segments xi of length N, that are processed in the DFT (FFT) domain separately. After processing the resulting segments Yi are added to reconstruct the filtered output y[n]. (a) [5 points] Explain why the segments xi have to be extended (padded) with zeros. What is the minimum number of zeros? (b) [5 points] \Vhat is the additional delay of this overlap add-implementation? Exercise 6 Sampling-rate conversion [5 points] Explain why the lowpass filter that is used after zero insertion, i.e. the block t K, in a K-times up has a gain K. Exercise 7 Sampling-rate conversion [5 points] Why is it important that the transition band of the lowpass filter used in a K-times down sampler is fully in the frequency range [0, f<: )? End of Test 5