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API ABDUL KALAM TECHNOLOGICAL UNIVERSITY
SECOND SEMESTER M.TECH DEGREE EXAMINATION, APRIL/MAY 2018

Branch:Electronics & Communication Stream(s)..
1. Signal Processing
2. Microwave & Television Engineering 3. Telecommunication Engineering
Course Code & Nome:
00302, ESTIMATION AND DETECTION THEORY Answer any

two full questions from each pall Limit answers to the
required points.

Max. Marks: 60 Duration: 3 hours

PARTA

Consider the signal detection problem in which change in variance is used for 9
Hypothesis testing.

110 : x[n] 00 2), 1-0,1, ,N-l

HI: x[nJ N (0, 0 12 ), n Lica

Derive the detection rule for Neyman-Pearson (NP) detector
Consider the multiple hypothesis testing problem 9
110 : x[nl = —A+wf[n], nO, 1,
axel
H] : x[nl win], n=0,1,..., N-1H2x[nJ=A+w[n], ॥ = 0, 1,..-,where w[n) is
white Gaussian noise with variance G 2 . Find the detection rule for minimum
probability of error detector.
Consider the detection problem of DC level in noise with unknown variance. 9

118 : «(೧] = [೧], = 0,1, .,N-1
∙∙ ↾⋅⋅⋅⋅⇅−
∥∣⋅⋅≻≺⊏∩∃∶∧⊹∨∨⇂⊓∃∣⊓−∘∣↥∣⋅
⋀∣⋅⋝∣≺⊓∘⋁∨⊓∂⊓⊂⊴∧⊔⊖⊓⋅∨⊖⊂∃∈⊓∈↾∂∣∣⋅≵∈↺∐⇂≺⊖∣∣⋅↾↿∘∘⊄∣↸∂≖∣⋅∘⊺∈⋝↝∁≼∈∐↸⊺∏∘⊓↾↿⊖∱∘⊓↾↿⊖
detection problem

PART B
a Define unbiased estimator. Give an example. 3 0 Give the expression for Cramer- Rao
Lower bound foe vector parameter estimation. 6 In this context, explain the regularity
condition to be satisfied by the PDF p(x, 9) for the existence of Cramer-Rao bound. Also,
define Fisher information matrix.
Observed data samples { x(0), [1], -»X{N-1]} are IID under Laplacian PDF ബി 1/2
(0400-1). Find Best Linear Unbiased Estimator (BLUE) of the mean g.
We observe N IID data samples from Gaussian PDF with unit variance and unknown 9
mean. Obtain maximum likelihood estimator (MLE) for the unknown parameter.

PART C
Show that Wiener filter can be used for predicting the data sample x[n] from ‏اعلا‎ 12
previous observations {x(n-l}, x[n-21, ...., x[n-N+l]}. Derive the expression for Wiener filter
coefficients.
Consider the scalar state equation s[n) =a s[n-l) + u[nl, and scalar 12 observation equation
x[n] = s[n] + w[n] where u[n] is zero mean Gaussian noise with independent samples with
variance and w[n) is zero mean Gaussian noise with independent samples with variance
Derive the expression for Kalamn Gain for estimating [വ from x[n].

Discuss the applications of matched filter Estimators In Communication receivers, 12

1