This examination consists of 10 pages. Please check that you have a complete copy. Time: 2.5 hrs INSTRUCTIONS

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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 08 December 2009 This examination consists of 10 pages. Please check that you have a complete copy. Time: 2.5 hrs SURNAME Signature First Name Student ID Problem Points max Total: 105 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. Basic Definitions δ(t), u(t), δ[k], and u[k] denote the continuous time delta impulse (Dirac delta), the continuous time unit step function, the discrete time delta impulse (Kronecker delta), and the discrete time unit step function, respectively. E{ } denotes statistical expectation. 2. 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. 3. 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) 4. 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) 5. 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) 6. 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 7. Multi dimensional Gaussian Probability Density Function (pdf) The pdf of an n dimensional zero mean real Gaussian random vector x with covariance matrix M is given by ( 1 p(x) = exp 1 ) (2π) n/2 (det(m)) 1/2 2 xt M 1 x. (6)

3 3 8. Gaussian Q Function The Gaussian Q Function is defined as Some Fourier Transform Pairs Q(x) = 1 ( ) exp t2 dt, x 0. (7) 2π 2 In the following, X(f) denotes the Fourier transform of x(t) or x[k]. 1. x(t) = δ(t) 2. x(t) = { 1, t T1 0, otherwise x X(f) = 1 (8) 3. x(t) = sin(2πwt) πt 4. x(t) = e at u(t), R{a} > 0 X(f) = X(f) = sin(2πft 1) πf { 1, f W 0, otherwise X(f) = 1 a + j2πf (9) (10) (11) 5. x(t) = te at u(t), R{a} > 0 6. x(t) = e a t, a > 0 X(f) = X(f) = 1 (a + j2πf) 2 (12) 2a a 2 + (2πf) 2 (13) 7. x[k] = δ[k k 0 ] X(f) = e j2πftk 0 (14) 8. x[k] = 1 X(f) = 1 T δ(f k/t) (15) k=

4 4 Some Integrals 1. dx a 2 + x 2 = 1 a arctan x a (16) 2. dx a 2 x 2 = 1 a arctanhx a (17) 3. xe ax dx = (ax 1) eax a 2 (18) ( x x 2 e ax 2 dx = 0 a 2x a a 3 ) e ax (19) x n e ax dx = n! an+1, n 1 (20)

5 5 Problem 1 18 Points Consider two independent random variables x and y. Their probability density functions (pdfs) are given by p x (x) = c 1 e x u(x) and p y (y) = c 2 e 2y u(y). a) Find the values for c 1 and c 2 such that p x (x) and p y (y) are pdfs. b) Sketch p x (x) and p y (y). c) Find the joint pdf p xy (x, y) of x and y. d) Calculate and sketch the pdf of random variable z = x + 2y. e) Calculate and sketch the pdf of random variable w = x y.

6 6 Problem 2 18 Points The autocorrelation functions (ACFs) of two statistically independent and jointly stationary Gaussian random processes x(t) and y(t) are given by ϕ xx (τ) = e τ and ϕ yy (τ) = e 2 τ. a) Calculate and sketch the ACF of random process z(t) = x(t) + y(t). b) Is z(t) a Gaussian random process? Justify your answer. c) Calculate the power spectral density Φ zz (f) of z(t). d) Calculate and sketch the ACF of random process w(t) = x(t)y(t). e) Calculate and sketch the power spectral density Φ ww (f) of w(t). f) Is w(t) wide sense stationary? Justify your answer. g) Is w(t) a Gaussian random process? Justify your answer. h) Calculate the ACF of random process s(t) = e t x(t). i) Is w(t) wide sense stationary? Justify your answer.

7 7 Problem 3 The discrete time received signal of a digital communication system can be modeled as r[k] = h 0 x[k] + h 1 x[k 1] + n[k], 23 Points where x[k] {±1}, k, are binary phase shift keying (BPSK) symbols and n[k] is real valued additive white Gaussian noise (AWGN) with variance σ 2 n = N 0/2. h 0 > 0 and h 1 0 are real valued constants. x[k] = 1 and x[k] = 1 are transmitted with equal probability. a) What type of channel causes received signals that have the form of r[k]. We assume for subproblems b) and c) that h 1 = 0. b) Derive the optimal (maximum likelihood) decision rule for detection of x[k] and sketch the corresponding decision regions in the signal space. c) Derive the bit error rate (BER) for the optimal decision rule. Now, we turn to the general case where h 1 > 0. Assume for subproblems d) f) that the detector decides for ˆx[k] = 1 if r[k] 0 and for ˆx[k] = 1 otherwise. We denote the probability of making a decision error (when detecting x[k]) for a given x[k 1] by P e (x[k 1]). d) Calculate P e (x[k 1] = 1) and express the final result in terms of Q functions. Consider the cases h 0 h 1 and h 0 < h 1. e) Calculate the BER for h 1 > 0. f) Quantify the (approximate) performance loss caused by h 1 > 0 compared to h 1 = 0 for small noise variances σ 2 n. Consider again the cases h 0 h 1 and h 0 < h 1. Assume now that x[k 1] is known at the receiver. e) Sketch a simple receiver structure which achieves a better performance than the detector considered in d)-f). This new structure should make symbol by symbol decisions and exploit the knowledge of x[k 1] for detection of x[k]. g) Calculate the BER of your proposed receiver structure.

8 8 Problem 4 20 Points The receiver of a digital communication system observes the discrete time signals r 1 = h 1 x + n 1, and r 2 = h 2 x + n 2, where x {±1} denotes the transmitted symbol and h 1 and h 2 are complex channel gains. n 1 and n 2 are independent complex additive white Gaussian noise (AWGN) samples with variances σ 2 1 = E{ n 1 2 } and σ 2 2 = E{ n 2 2 }, respectively. a) Determine the joint conditional probability density function (pdf) of r 1 and r 2 given x, p(r 1, r 2 x). b) Determine the optimal (maximum likelihood) decision rule for detection of x. Simplify your decision rule to a simple threshold decision, i.e., find a decision variable d (which depends on r 1 and r 2 ) such that ˆx = 1 if d > 0 and ˆx = 1 otherwise. c) Calculate the bit error rate (BER) of the optimal decision rule. A suboptimal receiver makes a decision based on r = h 1 r 1 + h 2 r 2. d) Derive the conditional pdf of r given x, p(r x), and the corresponding maximum likelihood decision rule. e) Calculate the BER for the decision rule in d). f) Quantify the performance loss of the suboptimal decision rule in d) compared to the optimal decision rule in b). g) Under what conditions are the suboptimal decision rule in d) and the optimal decision rule in b) equivalent?

9 9 Problem 5 26 Points Consider the following equivalent low pass transmit signal s(t) = k= b[k] g T (t kt), where b[k], g T (t), and T denote the transmitted symbols, the transmit pulse, and the symbol duration, respectively. The Fourier transform of g T (t) is denoted by G T (f). The transmit symbols are given by b[k] = a[k] + ρ a[k 1] where a[k] {±1} and ρ is a real valued constant. The symbols a[k] are temporally independent and assume the values +1 and 1 with equal probability. a) Is the random process s(t) stationary in the strict sense, wide sense stationary, or cyclo stationary? b) Calculate the power spectral density (PSD) Φ bb (f) of b[k]. c) Provide the PSD Φ ss (f) of s(t). d) For what value of ρ has the PSD of s(t) a zero at f = 0? For all following subproblems, is valid. At the receiver, the received signal ρ = 0 r(t) = s(t) + z(t) is filtered by a filter with impulse response g R (t), where z(t) denotes complex additive white Gaussian noise (AWGN) with PSD N 0. The Fourier transform of g R (t) is denoted by G R (f). The resulting signal is sampled to yield r[k] = g R (t) r(t) t=kt. e) What criterion have G T (f) and G R (f) to fulfill to avoid intersymbol interference in r[k]? Provide the name of this criterion and the corresponding formula.

10 10 and For the remainder of this problem, we assume that x(t) = g T (t) g R (t) has the property that x(0) = 1 x(kt) = 0, k {..., 2, 1, 1, 2,...}. Furthermore, we assume that g T (t) has energy E g and the Fourier transform of x(t), X(f), is non zero only for B f B, where B denotes the bandwidth. f) Give the optimal G R (f) which maximizes the signal to noise ratio (SNR) of r[k] as a function of X(f). g) Calculate the SNR achieved with the optimal g R (t). h) Assume now that g R (t) is a low pass filter with bandwidth B. Give G T (f) as a function of X(f). i) Calculate the SNR of r[k] for the filters in h). j) Form the ratio of the SNRs calculated in g) and i) and express this ratio as a function of G T (f) and B.