ECE 567 STATISTICAL SIGNAL PROCESSING SPRING 2008
Homework Assignment #6
Solutions
1. We have
p(x;θ) = (2πσ2)−N /2exp
−1
2σ2
N−1
X
n=0 "x(n)−
p−1
X
k=0
Aknk#2
ln(p(x;θ)) = −N
2ln(2πσ2)−1
2σ2
N−1
X
n=0 "x(n)−
p−1
X
k=0
Aknk#2
∂ln(p(x;θ))
∂A`
=1
σ2
N−1
X
n=0 Ãx(n)−
p−1
X
k=0
Aknk!n`
∂2ln(p(x;θ))
∂AmA`
=−1
σ2
N−1
X
n=0
nmn`
−E·∂2ln(p(x;θ))
∂AmA`¸=1
σ2
N−1
X
n=0
nm+`
so that
I(θ) = 1
σ2
NPN−1
n=0 nPN−1
n=0 n2· · · PN−1
n=0 np−1
PN−1
n=0 n· · · .
.
.
.
.
.
PN−1
n=0 np−1PN−1
n=0 np· · · · · · PN−1
n=0 n2p−2
.
2. We have
x=
x(0)
.
.
.
x(N−1)
∼ N
x(0)
.
.
.
x(N−1)
| {z }
µ
,
C0··· 0
0C.
.
.
.
.
....0
0· · · 0C
| {z }
C
Using (3.32) from the text we have
I(ρ) = 1
2tr "µC−1(ρ)∂C(ρ)
∂ρ ¶2#
