N in example 1-1 – National Instruments NI MATRIXx Xmath User Manual
Page 21
Chapter 1
Introduction
1-14
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nonnegative hermitian for all
ω. If Φ is scalar, then Φ(jω)≥0 for all ω.
Normally one restricts attention to
Φ(·) with lim
ω→∞
Φ(jω)<∞. A key result
is that, given a rational, nonnegative hermitian
Φ(jω) with
lim
ω→∞
Φ(jω)<∞, there exists a rational W(s) where,
•
W(
∞)<∞.
•
W(s) is stable.
•
W(s) is minimum phase, that is, the rank of W(s) is constant in Re[s]>0.
In the scalar case, all zeros of W(s) lie in Re[s]
≤0, or in Re[s]<0 if Φ(jω)>0
for all
ω.
In the matrix case, and if
Φ(jω) is nonsingular for some ω, it means that
W(s) is square and W
–1
(s) has all its poles in Re[s]
≤ 0, or in Re[s]<0 if Φ(jω)
is nonsingular for all
ω.
Moreover, the particular W(s) previously defined is unique, to within right
multiplication by a constant orthogonal matrix. In the scalar case, this
means that W(s) is determined to within a ±1 multiplier.
Example 1-1
Example of Spectral Factorization
Suppose:
Then Equation 1-3 is satisfied by
, which is stable and
minimum phase.
Also, Equation 1-3 is satisfied by
and
,
and
, and
so forth, but none of these is minimum phase.
bst( )
and
mulhank( )
both require execution within the program of
a spectral factorization; the actual algorithm for achieving the spectral
factorization depends on a Riccati equation. The concepts of a spectrum
and spectral factor also underpin aspects of
wtbalance( )
.
Φ jω
( )
ω
2
1
+
ω
2
4
+
---------------
=
W s
( )
s 1
+
s 2
+
-----------
±
=
s 1
–
s 2
+
-----------
s 3
–
s 2
+
-----------
s 1
–
s 2
+
-----------
e
sT
–
s 1
+
s 2
+
-----------