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Chapter 3
Multiplicative Error Reduction
© National Instruments Corporation
3-21
For
mulhank( )
, this translates for a scalar system into
and
The bounds are double for
bst( )
.
The error as a function of frequency is always zero at
ω = ∞ for
bst( )
(or at
ω = 0 if a transformation s → s
–1
is used), whereas no such particular
property of the error holds for
mulhank( )
.
Imaginary Axis Zeros (Including Zeros at
∞
)
When G(j
ω) is singular (or zero) on the jω axis or at ∞, reduction can be
handled in the same manner as explained for
bst( )
.
The key is to use a bilinear transformation [Saf87]. Consider the bilinear
map defined by
map defined by
where 0 < a < b
–1
and mapping G(s) into
through
86.9
v
i
dB 20log
10
<
Gˆ
nsr
G
⁄
i
nsr 1
+
=
ns
∑
–
8.69
v
i
i
nsr 1
+
=
ns
∑
<
dB
phase error
v
i
radians
i
nsr 1
+
=
ns
∑
<
s
z a
–
bz
–
1
+
-------------------
=
z
s a
+
bs 1
+
---------------
=
G˜ s
( )
G˜ s
( )
G
s a
–
bs
–
1
+
-------------------
⎝
⎠
⎛
⎞
=
G s
( )
G˜
s a
+
bs 1
+
---------------
⎝
⎠
⎛
⎞
=