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EEM.asp/0405/06pg/ Req:None UNIVERSITY OF SURREY School of Electronics and Physical Sciences ELECTRONIC ENGINEERING PROGRAMMES For MRes and MSc Degrees Level M Modular Examination Module EEM.asp ADVANCED SIGNAL PROCESSING Duration: 2 hours Spring 2004/05 READ THESE INSTRUCTIONS Answer THREE questions, ONE question from EACH section. If you attempt more than THREE questions, only your best THREE solutions will be taken into account. SEE NEXT PAGE EEM.asp/0405/06pg/ Req:None SECTION A A1 (a) Define an auto-regressive process in terms of its z-plane transfer function. [10%] (b) Derive the Yule-Walker Equations for an auto-regressive process from first principles, and show that the all-pole filter derived via these equations is optimum in the mean squared error sense. [40%] (c) The first four auto-correlation coefficients for a linear discrete-time process are; r0 = 1.0, r1 = 0.8, r2 = 0.45, r3 = 0.25 Write out the set of three Yule-Walker equations and hence calculate the coefficients of the third order linear prediction filter. (N.B. do not use Durbin’s Algorithm.) [50%] A2 (a) Define the difference equation for a Moving Average Process and show that the corresponding z-plane transfer function represents an all-zero (FIR) filter. [10%] (b) Derive, from first principles, the Wiener-Hopf Equations for a Moving Average Process; ℜ.a = p where ℜ, a and p have their usual meaning. [40%] (c) The steepest decent algorithm is given by; a(n+1) = a(n) +µ[p - ℜa(n)] Use this algorithm to estimate the coefficients of a second order moving ⎡0.2⎤ ⎡1.0 0.1⎤ average system where p = ⎢ ⎥ , ℜ = ⎢ ⎥ , µ = 0.5, and an initial ⎣0.1⎦ ⎣0.1 1.0⎦ ⎡0.3⎤ estimate of a(n) = ⎢ ⎥ . ⎣0.1⎦ Calculate four iterations of this algorithm and comment on the rate of convergence. [50%] SEE NEXT PAGE EEM.asp/0405/06pg/ Req:None SECTION B B3 (a) Briefly discuss two main optimality criteria used in signal detection: Bayes criterion and Neyman-Pearson criterion. [25%] (b) Statistically independent observations xi, i = 1, 2, 3, 4, are distributed according to one of the two alternative probability density functions p(x|H0) = a0 -1 < x < 1 p(x|H0) = a1(1 - |x|)5 -1 < x < 1 The choice of one of the two hypotheses, H0 and H1, is based on the following test: choose H1, if z1 + z2 + z3 + z4 > 2 choose H0, otherwise. where a binary variable zi, i = 1, 2, 3, 4, is defined as zi = 1, if |xi| < 0.2 zi = 0, otherwise. Determine: (i) distributions of zi, under H0 and H1, respectively. [35%] (ii) probability of detection PD, defined as PD = P(select H1|H1 true). [20%] (iii) probability of false alarm PF, defined as PF = P(select H1|H0 true). [20%] SEE NEXT PAGE EEM.asp/0405/06pg/ Req:None B4 (a) Write an expression for the Bayes risk in signal detection and briefly explain the parameters involved. [30%] (b) A linear filter is matched to a rectangular pulse s(t) with duration T = 10 µs. When s(t) alone is applied to the filter, the response z(t) achieves a maximum value of A at t0 = 10 µs. When zero-mean white Gaussian noise is applied to the filter, the rms value of the output noise is equal to 10 mV. (i) Sketch both the impulse response of the filter and the output of the filter excited by s(t). [10%] (ii) Sketch the autocorrelation function of the noise observed at the output of the filter. [15%] (iii) A signal x(t) to be detected in noise by the filter comprises two mutually delayed pulses; x(t) is of the form x( t ) = s( t ) + s( t - d ) where d = 50 µs. Assume that the detection of x(t) is based on the sum v v = z( t 1 ) + 2 z( t 2 ) where the samples are taken at t1 = 10 µs and t2 = 65 µs. Determine the value of A required to achieve the probability of detection PD = 0.975 for the probability of false alarm PF = 0.025. [45%] Hint (1 / 2π )∫∞ 1.96 exp ( - w2 / 2 ) dw = 0.025 SEE NEXT PAGE EEM.asp/0405/06pg/ Req:None SECTION C C5 (a) Briefly describe the Bayes approach to random parameter estimation and explain the consequence of using the three standard cost functions. [30%] (b) The time interval w between consecutive events is modelled by a two-parameter exponential distribution of the form 1 ⎛ w - m ⎞ p ( w ; m ,σ ) = exp ⎜ - ⎟ , w ≥ m, m ≥ 0,σ > 0 σ ⎝ σ ⎠ Assume that N statistically independent observations w1, w2, ... , wN have been obtained. (i) Write down the expression describing the log- likelihood function. [15%] (ii) Derive maximum likelihood (ML) estimators, mML and σML of the parameters m and σ, respectively. [35%] (iii) Next, assume that m = 0 and determine the number N of independent observations needed to achieve var (σ ML ) < 0.001 [20%] σ 2 C6 (a) Briefly describe the concept of a Generalized Likelihood Ratio (GLR) test. [25%] (b) A random variable X has the probability density function of the form 1 p( x ; α , h ) = , α -h < x < α +h , 0 < h < α 2h Assume that N statistically independent observations x1, x2, ... , xN have been obtained. (i) Apply the method of moments to derive estimators, αMOM and hMOM, of parameters α and h, respectively. [30%] (ii) Determine the bias and variance of αMOM. [25%] (iii) Derive maximum likelihood estimators of α and h. [20%] SEE NEXT PAGE EEM.asp/0405/06pg/ Req:None Examiners: E. Chilton W. J. Szajnowski External Examiners: A. Clark SEE NEXT PAGE