Spectral factorization, Spectral factorization -13 – National Instruments NI MATRIXx Xmath User Manual
Page 20
Chapter 1
Introduction
© National Instruments Corporation
1-13
Similar considerations govern the discrete-time problem, where,
can be approximated by:
mreduce( )
can carry out singular perturbation. For further discussion,
refer to Chapter 2,
. If Equation 1-1 is balanced,
singular perturbation is provably attractive.
Spectral Factorization
Let W(s) be a stable transfer-function matrix, and suppose a system S with
transfer-function matrix W(s) is excited by zero mean unit intensity white
noise. Then the output of S is a stationary process with a spectrum
Φ(s)
related to W(s) by:
(1-3)
Evidently,
so that
Φ( jω) is nonnegative hermitian for all ω; when W( jω) is a scalar, so
is
Φ( jω) with Φ( jω) = |W( jω)|
2
.
In the matrix case,
Φ is singular for some ω only if W does not have full
rank there, and in the scalar case only if W has a zero there.
Spectral factorization, as shown in Example 1-1, seeks a W(j
ω), given
Φ(jω). In the rational case, a W(jω) exists if and only if Φ(jω) is
x
1
k 1
+
(
)
x
2
k 1
+
(
)
A
11
A
12
A
21
A
22
x
1
k
( )
x
2
k
( )
B
1
B
2
u k
( )
+
=
y k
( )
C
1
C
2
x
1
k
( )
x
2
k
( )
Du k
( )
+
=
x
1
k 1
+
(
)
A
11
A
12
I A
22
–
(
)
1
–
A
21
+
[
]x
1
k
( ) +
=
B
1
A
12
I A
22
–
(
)
1
–
B
2
+
[
]u k
( )
y
k
C
1
C
2
I A
22
–
(
)
1
–
A
21
+
[
]x
1
k
( ) +
=
D C
2
I A
22
–
(
)
1
–
B
2
+
[
]u k
( )
Φ s
( )
W s
( )W′ s
–
( )
=
Φ jω
( )
W j
ω
( )W
*
j
ω
( )
=