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1 THE UNIVERSITY OF BRITISH COLUMBIA Department of Electrical and Computer Engineering EECE 564 Detection and Estimation of Signals in Noise Final Examination 6 December 2006 This examination consists of 11 pages. Please check that you have a complete copy. Time: 2.5 hrs SURNAME Signature First Name Student ID Problem Points max Total: 135 INSTRUCTIONS 1. All writings must be on the paper provided. 2. Each candidate should be prepared to produce, upon request, his/her Library/AMS card. 3. Read and observe the following rules: No candidate shall be permitted to enter the examination room after the expiration of one half hour, or to leave during the first half hour of the examination. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination-questions. Caution Candidates guilty of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination and shall be liable to disciplinary action: Making use of any books, papers or memoranda, calculators, audio or visual cassette players or other memory aid devices, other than as authorized by the examiners. Speaking or communicating with other candidates. Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness shall not be received. 4. Show all your work. Justify your answers. Partial credit is possible for an answer, but only if you show the intermediate steps in obtaining the answer.
2 2 Definitions 1. Characteristic Function The characteristic function Ψ X (jv) of random variable X is defined as Ψ X (jv) = p X (x) e jvx dx, (1) where p X (x) denotes the probability density function (pdf) of X. 2. Continuous Time Fourier Transform The Fourier transform X(f) of x(t) is defined as X(f) = F{x(t)} = x(t) e j2πft dt. (2) 3. Inverse Continuous Time Fourier Transform The inverse Fourier transform x(t) of X(f) is defined as x(t) = F 1 {X(f)} = X(f) e j2πft df. (3) 4. Discrete Time Fourier Transform The Fourier transform X(f) of x[k] is defined as X(f) = F{x[k]} = k= x[k] e j2πftk. (4) 5. Gaussian Probability Density Function (pdf) The pdf of a zero mean real Gaussian random variable X with variance σ 2 is given by ( ) 1 p(x) = exp x2. (5) 2πσ 2 2σ 2 6. Gaussian Q Function The Gaussian Q Function is defined as Q(x) = 1 ( ) exp t2 dt. (6) 2π 2 x
3 3 Some Fourier Transform Pairs In the following, X(f) denotes the Fourier transform of x(t) or x[k]. 1. x(t) = δ(t) X(f) = 1 (7) 2. x(t) = { 1, t T1 0, otherwise X(f) = sin(2πft 1) πf (8) 3. x(t) = sin(2πwt) πt X(f) = { 1, f W 0, otherwise (9) 4. x(t) = e a t, a > 0 X(f) = 2a a 2 + (2πf) 2 (10) 5. x[k] = δ[k k 0 ] X(f) = e j2πftk 0 (11) 6. x[k] = 1 X(f) = 1 T k= δ(f k/t) (12) Some Integrals 1. dx a 2 + x 2 = 1 a arctan x a (13) 2. dx a 2 x 2 = 1 a arctanhx a (14) 3. xe ax dx = (ax 1) eax a 2 (15)
4 4 4. ( x x 2 e ax 2 dx = a 2x a a 3 ) e ax (16) 5. 0 x n e ax dx = n! an+1, n 1 (17)
5 5 Problem 1 17 Points Given are N statistically independent complex random variables (RVs) X i, 1 i N, with means m Xi = E{X i } and variances σ 2 X i = E{ X i m Xi 2 }. Based on the X i, N new RVs are generated. Z i = i X n, 1 i N, a) Calculate the mean m Zi and the variance σ 2 Z i of Z i, 1 i N. n=1 b) Calculate ρ ij = E{Z i Z j }, 1 i N, 1 j N. c) Calculate the covariance µ ij between Z i and Z j, 1 i N, 1 j N. d) Are Z i and Z j statistically independent for j i? Justify your answer. Denote the probability density function (pdf) of X i by p Xi (x), 1 i N. e) Develop a recursive formula that can be used to calculate the pdf p Zi (x) of Z i based on the pdf p Zi 1 (x) of Z i 1. In the following, we assume that all X i follow the same pdf. In addition, we assume m Xi = 0 and σ 2 X i = 1/N, 1 i N, and N. f) Give the probability density function of Z N. Justify your answer.
6 6 Problem 2 22 Points We consider transmission over an additive complex non Gaussian noise channel (equivalent complex baseband model). The probability density function (pdf) p z (z) of the noise z is given by p z (z) = p xy (x, y) = c e b z = c e b x 2 +y 2, where x and y denote the real and the imaginary part of z = x + jy, respectively. b and c are constants. The variance of z is given by σ 2 z = N 0. a) Are the real part and the imaginary part of z statistically independent? Justify your answer. b) Calculate b and c as functions of N 0. Assume in the following b = 6/N 0 and c = 3/(πN 0 ). Note that this is not necessarily the correct answer to question b). The received signal can be modeled as where s 1 = 0 and s 2 = a, a > 0. r = s m + z, m {1, 2}, c) What is the name of the adopted modulation scheme? d) Calculate the average transmitted energy per bit E b as a function of a. e) Derive the (coherent) maximum likelihood (ML) decision rule for the considered signaling scheme. f) Sketch the optimum decision regions in the signal space. It is difficult to calculate the exact error probability for the considered channel and signaling scheme. However, simple (but non trivial) upper and lower bounds on the error probability if s 1 and s 2 are transmitted, respectively, can be obtained. g) Calculate an upper bound on the probability that s 2 = a is detected if s 1 = 0 was transmitted (as function of E b /N 0 ). h) Calculate a lower bound on the probability that s 1 = 0 is detected if s 2 = a was transmitted (as function of E b /N 0 ).
7 7 Problem 3 27 Points The received signal of a digital communication system is given by r(t) = s m (t) + n(t), m {1, 2}, where n(t) is real valued additive white Gaussian noise (AWGN) with power spectral density Φ nn (f) = N 0 /2. The waveforms s 1 (t) and s 2 (t) are given by { A 0 t T/2 s 1 (t) = 0 otherwise and s 2 (t) = { A T/2 < t T 0 otherwise, where A is a constant and T is the symbol duration. a) Sketch s 1 (t) and s 2 (t). b) Give the basis functions for the considered waveforms. c) Calculate the transmitted energy E b per bit. We consider first a correlation demodulator for the signaling scheme under investigation. d) Sketch the correlation demodulator (consisting of two multiplicators and two integrators) for the considered signaling scheme. e) Assume that s 1 (t) was transmitted and calculate the signal and noise powers at the outputs of the two integrators of the correlation demodulator. f) Derive the (coherent) maximum likelihood (ML) decision rule for the considered signaling scheme and simplify it as much as possible. g) Calculate the probability of error for the considered signaling scheme as a function of E b /N 0. Now we consider a matched filter demodulator. h) Sketch the matched filter demodulator for the considered signaling scheme. i) Give the probability that s 2 (t) is detected if s 1 (t) was transmitted if the matched filter demodulator is used. Justify your answer (no mathematical derivation required).
8 8 Problem 4 20 Points We consider ternary transmission over a complex additive white Gaussian noise (AWGN) channel. The received signal is given by r = e jθ s m + z, m {1, 2, 3}, where s 1 = 0, s 2 = A, and s 3 = 2A, A > 0. Θ is an unknown phase which is uniformly distributed in the interval [ π, π). The complex AWGN z has variance σ 2 z = N 0. a) Calculate the average transmitted energy per symbol E s as a function of A. The probability density function (pdf) of r conditioned on s m is given by ) ( ) 1 p(r s m ) = ( (πn 0 ) exp r 2 + s m 2 2 I 2 0 r s m, N 0 where I 0 ( ) denotes the zeroth order modified Bessel function of the first kind. N 0 b) Derive the optimum noncoherent maximum likelihood (ML) decision rule for the considered signaling scheme. Simplify your result as far as possibly without using any approximations. For the following, the approximation ln I 0 (x) x may be useful. c) Further simplify the decision rule derived in b) using the above approximation. d) Sketch the decision regions for the simplified decision rule in the signal space. e) Calculate the probability P 1 that s 1 is detected if s 1 was transmitted as function of E s /N 0. f) Calculate the probability P 3 that s 3 is detected if s 1 was transmitted as function of E s /N 0. g) Calculate the probability P 2 that s 2 is detected if s 1 was transmitted as function of E s /N 0.
9 9 Problem 5 26 Points Consider the following equivalent low pass transmit signal v(t) = k= a[k] g(t kt), where a[k], g(t), and T denote the data symbols, the transmit pulse, and the symbol duration, respectively. The binary data symbols a[k] {±1} are statistically independent random variables, i.e., a[k] is independent from a[n] for n k. Furthermore, we assume that a[k] has zero mean. a) Is the random process v(t) stationary in the strict sense, wide sense stationary, cyclo stationary, or ergodic? b) Give the power spectral density Φ aa (f) of a[k]. c) Calculate the power density spectrum Φ vv (f) of transmit signal v(t). Consider in the following the pulse shape g(t) shown below (A > 0 is the amplitude of the pulse). A g(t) T t d) Calculate and sketch Φ vv (f) if the above pulse shape is used. e) Assume we want to have a null in the spectrum at f = 1/3T. This can be achieved by transmitting the precoded sequence b[k] = a[k]+α a[k 3] instead of a[k]. Find the α that provides the desired null. f) Can we bandlimit the signal v(t) if the above pulse shape g(t) is used? If your answer is no, explain why this is not possible. If your answer is yes, explain how it can be done. Assume for all remaining problems that the new pulse shape g(t), whose Fourier transform G(f) is shown on the top of the next page, is used (A and b are positive constants). g) How do we have to choose b if intersymbol interference is to be avoided at the receiver? Justify your answer. h) A matched filter demodulator is used at the receiver. Calculate the impulse response of the matched filter h m (t) (the filter does not have to be causal).
10 10 A G(f) b f i) Calculate the impulse response of the overall channel h(t) [cascade of transmit pulse shape g(t) and matched filter h m (t)]. j) Would you recommend the considered pulse shape g(t) for practical implementation? Justify your answer. h) Sometimes it is desirable to have overall impulse responses h(t) that decay fast with increasing t. How can this be achieved and what price has to be paid?
11 11 Problem 6 23 Points a) Which binary modulation scheme(s) would you recommend for transmission over the following channels. Justify all your answers. i) Channel phase is completely constant. ii) Channel phase is slowly time variant. Channel phase is practically constant over two symbol intervals. iii) Channel phase changes rapidly from symbol interval to symbol interval. b) Continuous phase modulation (CPM) has certain advantages but also disadvantages when compared with linear modulation schemes. i) Give two advantages of CPM. ii) Give two disadvantages of CPM. c) Linear equalization (LE), decision feedback equalization (DFE), and maximum likelihood sequence estimation (MLSE) are the three basic equalization schemes. i) Explain under which conditions equalization is necessary. ii) Give two different filter optimization criteria for LE and explain which one is preferable in practice. iii) Give one advantage and one disadvantage for LE, DFE, and MLSE, respectively. iv) Consider transmission with 8 ary phase shift keying (8PSK) over a channel with L = 10 coefficients. The channel transfer function has several zeros close to the unit circle. Which of the three considered equalization schemes would you recommend for implementation? Give a detailed explanation for your choice.
This examination consists of 10 pages. Please check that you have a complete copy. Time: 2.5 hrs INSTRUCTIONS
THE UNIVERSITY OF BRITISH COLUMBIA Department of Electrical and Computer Engineering EECE 564 Detection and Estimation of Signals in Noise Final Examination 08 December 2009 This examination consists of
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