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EEL 6537 – Detection Theory

Homework #3

Fall 2017

1. Consider the signals

n

,

=

sn ( t ) t =

n 0,1, 2,3.

Taking as your interval ( −1,1) instead of ( 0,T ) , use the Gram-Schmidt procedure to

generate the four Legendre polynomials P0 ( t ) , P1 ( t ) , P2 (t ), P3 ( t ) .

2. The observation R of a hypothesis testing problem has the following conditional

distributions

𝑝𝑝𝑅𝑅 (𝑟𝑟|𝐻𝐻1 ) =

2

exp �−

𝜎𝜎2

𝑟𝑟 2 +𝑎𝑎2

2𝜎𝜎2

𝑎𝑎𝑎𝑎

� 𝐼𝐼0 �𝜎𝜎2 � 𝑢𝑢(𝑟𝑟)

r2

exp

− 2 u (r )

σ2

2σ

(a) Form a likelihood ratio test and derive expressions for PF and PD .

pR =

(r H0 )

r

(b) Use the approximation, 𝐼𝐼0 (𝑥𝑥) ≈

𝑒𝑒 𝑥𝑥

√2𝜋𝜋 𝑥𝑥

, 𝑥𝑥 ≫ 1, to simplify your result.

3. The generalized Marcum Q-function may be defined as

M −1

∞

r2 + a2

r

QM ( a, b ) ∫ r exp −

I M −1 ( ar ) dr

=

2

a

b

where I M −1 (.) is the modified Bessel function of the first kind of order M − 1 .

(a) Find QM ( a, 0 ) .

(b) Find QM ( 0, b ) .

4. Let X = ( X 1 , X 2 ,…, X N ) be a sequence of i.i.d. observations, each with a Rice

N

distribution with specular component α . If Y = ∑ X i 2 , then it can be shown

i =1

that the pdf of Y is given by

1 2y

pY ( y )

=

σ 2 β

N −1

2

2β y

2y + β

exp −

I

u y

1

N

−

2

2

( )

2σ

σ

where β = Nα 2 .

(a) Derive the Neyman-Pearson test for the hypothesis testing problem,

𝐻𝐻0 : 𝛽𝛽 = 0

𝐻𝐻1 : 𝛽𝛽 > 0

(b) Derive the expressions for PD = P (Y > λ β > 0 ) and PF = P (Y > λ β = 0 ) .

…

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