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EEL 6537 – Detection Theory
1. Consider the signals
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
𝑝𝑝𝑅𝑅 (𝑟𝑟|𝐻𝐻1 ) =
𝑟𝑟 2 +𝑎𝑎2
� 𝐼𝐼0 �𝜎𝜎2 � 𝑢𝑢(𝑟𝑟)
− 2 u (r )
(a) Form a likelihood ratio test and derive expressions for PF and PD .
(r H0 )
(b) Use the approximation, 𝐼𝐼0 (𝑥𝑥) ≈
, 𝑥𝑥 ≫ 1, to simplify your result.
3. The generalized Marcum Q-function may be defined as
r2 + a2
QM ( a, b ) ∫ r exp −
I M −1 ( ar ) dr
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
distribution with specular component α . If Y = ∑ X i 2 , then it can be shown
that the pdf of Y is given by
pY ( y )
σ 2 β
2y + β
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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