EEM.asp/0405/06pg/ Req:None

                        UNIVERSITY OF SURREY
                          School of Electronics and Physical Sciences

                              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.

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EEM.asp/0405/06pg/ Req:None

                                           SECTION A

A1     (a)     Define an auto-regressive process in terms of its z-plane transfer function.
       (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.)

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⎦
               estimate of a(n) = ⎢ ⎥ .
               Calculate four iterations of this algorithm and comment on the rate of
               convergence.                                                                 [50%]

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                                          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.

               (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%]

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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%]

                        (1 /   2π    )∫∞

                                             exp ( - w2 / 2 ) dw = 0.025

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                                           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.

       (b)     A random variable X has the probability density function of the form
                       p( x ; α , h ) =      ,     α -h < x < α +h , 0 < h < α
               Assume that N statistically independent observations x1, x2, ... , xN have been
               (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%]

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EEM.asp/0405/06pg/ Req:None

                              Examiners: E. Chilton
                                         W. J. Szajnowski
                              External Examiners: A. Clark

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