Multiple Input / Single Output Relationships
Consider the multiple input-multiple output system shown below. Here
are
assumed to be the stationary inputs with zero mean which are uncorrelated with the
uncorrelated extraneous output noise with zero mean,
.
Thus it holds
q
i
t
x
i
,
,
2
,
1
),
(
"
=
)
(
t
n
q
i
f
G
n
x
i
,
,
2
,
1
,
0
)
(
"
=
=
.
)
(
τ
q
h
#
)
(
2
t
x
)
(
1
τ
h
)
(
2
τ
h
)
(
1
t
v
)
(
2
t
v
)
(
t
v
q
)
(
1
t
x
)
(
t
n
)
(
t
x
q
)
(
t
y
From the relation
,
we obtain the auto and cross-spectral densities as
)
(
)
(
)
(
1
t
n
t
v
t
y
q
i
i
+
=
∑
=
nn
q
i
q
j
x
x
j
i
nn
q
i
q
j
v
v
yy
G
G
H
H
G
G
G
j
i
j
i
+
=
+
=
∑∑
==
11
*
∑
∑
=
=
=
=
=
q
i
x
x
i
q
i
v
x
y
x
q
j
G
H
G
G
i
j
i
j
j
1
1
,
,
2
,
1
,
"
Similarly to the previous single input-single output relationship, we can obtain the optimal
frequency response functions by minimizing the output noise power spectral density, which
can be expressed as
∑
∑
∑
∑
=
=
=
=
+
−
−
=
+
−
−
=
q
i
q
j
x
x
j
i
q
i
y
x
i
q
i
yx
i
yy
q
i
q
j
v
v
q
i
y
v
q
i
yv
yy
nn
j
i
i
i
j
i
i
i
G
H
H
G
H
G
H
G
G
G
G
G
G
*
1
*
1
1
1
Then, from
0
*
=
∂
∂
k
nn
H
G
, we derive the relation
∑
=
=
=
q
j
kj
j
ky
q
k
G
H
G
1
,
,
2
,
1
,
"
or, in matrix form,
=
−
qy
y
y
qq
q
q
q
q
q
G
G
G
G
G
G
G
G
G
G
G
G
H
H
H
#
"
#
#
#
"
"
#
2
1
1
2
1
2
22
21
1
12
11
2
1