# Exam Paper 2005 - UNIVERSITY OF SURREY ELECTRONIC ENGINEERING by dfsdf224s

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UNIVERSITY OF SURREY
School of Electronics and Physical Sciences

ELECTRONIC ENGINEERING PROGRAMMES
For MRes and MSc Degrees

Level M Modular Examination

Module EEM.asp

Duration: 2 hours                                                          Spring 2004/05

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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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%]

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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.
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%]

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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%]

Hint
(1 /   2π    )∫∞

1.96
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.
[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%]

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

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