M. TECH DEGREE EXAMINATIONSecond Semester

Branch: Applied Electronics and Instrumentation EngineeringSpecialization: Signal Processing

MAESP 202 DETECTION & ESTIMATION THEORY(2011 Admission onwards)

MODEL QUESTION PAPER

Time : 3 Hours Max. Marks: 100

1. a) Answer briefly in a few sentences each: 10 Marksi) A dice is thrown 2 times. What is the probability that the sum of the two throws is a) 3? b) 9? c) 13? ii) Write down and explain Bayes Ruleiii) Define the term “probability of false alarm”.iv) Give the expression for probability of error in a binary detection problemv) What is an ROC curve? Explain its dependence on the SNR.

b) Derive the likelihood ratio test (LRT), under the Neyman Pearson (NP) criterion for a binary hypothesis problem. 10 Marks c) When does the LRT test under minimum probability of error criterion become identical to that under NP criterion? 5 Marks

OR 2.

a) Consider the following detection problem: Under hypothesis H0, the measured data is x[0]= w[0]; where w[0] is zero mean Gaussian noise with variance 1. Under hypothesis H1, x[0]= 2 + w[0] . A detector decides H1 if x[0] > 1 and H0 otherwise. What is the probability of false alarm in this case. Under what criteria is the detector optimal? Explain.

10 marks

b) Show that the M-ary hypothesis test can be reduced to a set of LRT tests with likelihood functions defined corresponding to all of the M hypotheses. 10 Marks

c) Discuss briefly what you understand by Composite Hypothesis Testing. 5

Marks 3.

a) Answer briefly in a few sentences each: 10 Marks

i. Distinguish between estimation and detectionii. What is an estimator? list important properties of estimators

iii. What is the Craemer-Rao lower bound (CRLB)? Explain.iv. What do you understand by “linear model” in estimation? v. State and explain the Neyman-Fisher factorization theorem.

b) Show that the sample mean x is a sufficient statistic for the estimation of the level of a DC signal corrupted by additive Gaussian noise. What is the CRLB for this problem? 10 Marks

c) Show that x in b) above is also an efficient estimator of the average DC level. 5 Marks

OR

4. a) Formulate the general curve fitting problem x[n] = itn

(i 1), n =0,1,2…..(N-1), where tn are the time instants when the samples x[n] are observed, as a linear model. and find the MVU estimator for the parameter vector 15 Marks

b) The N observations x[n], n = 0,1,…N-1, are i.i.d. samples from a Rayleigh distribution p(x[n];) = (x[n]/.exp(-x[n]2/2) for x[n] > 0, and 0 otherwise. Find a sufficient statistic for estimation of

10 Marks

5. a) Define the likelihood function and explain the method of Maximum

Likelihood (ML) estimation. 10 Marks

b) The radar echo from an aircraft at distance R can be modeled as

r(t) = A cos [(t 2R/c)]+n(t) 0 < t < T

where, is the frequency and T the duration of transmitted radar pulse, c being the speed of EM waves in air. The amplitude constant A is a measure

of the aircraft’s back-scattering strength. The echo is corrupted by AWGN n(t) having zero mean, and variance 2..Explain how R can be estimated from N measured samples of r(t), using the ML method. What is the variance of the estimate? 15 Marks

OR

6. a) Discuss the following estimation method brieflyi) Least Square estimationII) Recursive Least Squares estimationiii) Best Linear Unbiased Estimation (BLUE)

15 Marksb) Consider the problem of finding the linear fit to the data set { xi, yi}, i =1,2,….N, using the relation y = A + Bx.. Find the MMSE estimates for

A and B. 10 Marks

7.a) Discuss the Bayesian approach to estimation. What are the typical risk

functions used ? Show that use of the absolute error cost function leads to the median of the posterior density function p(|x) as the optimal estimate of 15 Marks

b) Based on N statistically independent samples of a Gaussian process of variance 2 and unknown mean , we wish to find a MAP estimator of the mean. If can be assumed to be greater than 0, find the probability density function, and the estimator. 10 Marks

OR8.

a) Briefly answer the following : 10 Marks

i) What is a dynamical system ?ii) Define the state of a dynamical systemiii) Distinguish between Weiner and Kalman filtersiv) What is meant by ‘innovation” w.r.t. the Kalman filter v) What is the Extended Kalman Filter (EKF) ?: .

b) Assume we observe the data x[k] = Ark +w[k] for k = 0,1,….N.;where A is the realization of a random variable with p.d.f. N(A,2

A), 0 <r <1, and the w[k[‘s are samples of WGN, with variance2 .Also assume that A is independent of the w[k]’s. Find the sequential MMSE estimator of A based on {x[0], x[1],…….x[N]}

15 marks