To approximate an unknown system (or the behavior of that
system) as closely as possible
The optimal solution is
R1P=W
R
P
W
Suppose the unknown system is a causal, linear timeinvariant filter:
d
k
=
x
k
*
h
k
=∑
i
=0∞
x
k

i
h
i
d
k
x
k
h
k
i
0
x
k

i
h
i
Now
P=(
E
d
k
x
k

j
)=(
E∑
i
=0∞
x
k

i
h
i
x
k

j
)=(
∑
i
=0∞
h
i
E
x
k

i
x
k

j
)=(
∑
i
=0∞
r
xx
j−i
)=(
r
xx
0r1……rM−1rMrM+1…
r1r0⋱⋱⋮⋮⋮…
r2r1⋱⋱⋮⋮⋮…
⋮⋮…r0r1r2r3…
rM−1rM−2…r1r0r1r2…
)h0h1h2⋮
P
d
k
x
k

j
i
0
x
k

i
h
i
x
k

j
i
0
h
i
x
k

i
x
k

j
i
0
h
i
r
xx
j
i
r
xx
0
r
1
…
…
r
M
1

r
M
r
M
1
…
r
1
r
0
⋱
⋱
⋮

⋮
⋮
…
r
2
r
1
⋱
⋱
⋮

⋮
⋮
…
⋮
⋮
…
r
0
r
1

r
2
r
3
…
r
M
1
r
M
2
…
r
1
r
0

r
1
r
2
…
h
0
h
1
h
2
⋮
(1)
If the adaptive filter
HH is a
length
MM FIR filter
(
hm=hm+1=…=0
h
m
h
m
1
…
0
), this reduces to
P=Rh1
P
R
h
and
W
opt
=R1P=R1(Rh)=h
W
opt
R
P
R
R
h
h
FIR adaptive system identification thus converges in the mean
to the corresponding
MM samples of the
impulse response of the unknown system.
"A good introduction in adaptive filters, a major DSP application."