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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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