Expert answer:EEL 6537 Detection Theory
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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
( | 1 ) =
2
exp �−
2
2 + 2
2 2
� 0 � 2 � ( )
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, 0 ( ) ≈
√2
, ≫ 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,
0 : = 0
1 : > 0
(b) Derive the expressions for PD = P (Y > λ β > 0 ) and PF = P (Y > λ β = 0 ) .
…
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