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Thus we have for all with nonzero RMS value
w
rms
kH
w
k
rms
kH
k
1
kw
k
which shows that
rms gn
. By concentrating the power spectral density
kH
k
kH
k
1
of near a frequency max at which ( max)
, we have
w
!
jH
j
!
j
kH
k
1
rms
kH
w
k
rms
kH
k
:
1
kw
k
(Making this argument precise establishes (5.10).)
We can contrast the
and 2 norms by considering the associated inequality
H
H
1
speci cations. Equation (5.11) implies that the
norm-bound speci cation
H
1
(5.12)
kH
k
M
1
is equivalent to the speci cation:
rms
for all with
rms 1
(5.13)
kH
w
k
M
w
kw
k
:
In contrast, the 2 norm-bound speci cation
H
2
(5.14)
kH
k
M
is equivalent to the speci cation
rms
for a white noise.
(5.15)
kH
w
k
M
w
The
norm is often combined with a frequency domain weight:
H
1
=
kH
k
kW
H
k
:
W
1
1
If the weighting transfer function
and its inverse
1 are both stable, the
;
W
W
speci cation
1 can be expressed in the more classical form: is stable
kH
k
H
W
1
and
( )
( ) 1
;
for all
jH
j
!
j
jW
j
!
j
!
depicted in gure 5.4(b).
5.2.7
Shifted
Norm
H
1
A useful generalization of the
norm of a transfer function is its -exponentially
H
a
1
weighted 2 norm gain, de ned by
L
= sup ~ 2
kz
k
kH
k
1
a
~ 2=0 ~ 2
kw
k
kw
k
6
5.2 NORMS OF SISO LTI SYSTEMS
101
30
30
20
20
(dB) 10
(dB) 10
)
)j
j
0
0
(
(j!
j!
10
10
W
;
;
allowed region
j
W
j
1=
20
20
;
for ( )
jH
j
!
j
;
30
30
;
1
10
100 ; 1
10
100
!
!
(a)
(b)
An example of a frequency domain weight
that enhances
Figure
5.4
W
frequencies above = 10 is shown in (a). The specication
1
!
kW
H
k
1
requires that the magnitude of ( ) lie below the curve 1
( ) , as
H
j
!
=jW
j
!
j
shown in (b). In particular, ( ) must be below 20dB for
12.
jH
j
!
j
;
!
where ~( ) =
( ), ~( ) =
( ), and =
. It can be shown that
at
at
w
t
e
w
t
z
t
e
z
t
z
H
w
= sup
( ) =
(5.16)
kH
k
jH
s
j
kH
k
1
a
a
1
<s>;a
where ( ) = (
).
is called the -shifted transfer function formed from
H
s
H
s
;
a
H
a
a
a
, and the norm
is called the -shifted
norm of a transfer function.
H
k
k
a
H
1
a
1
This is shown in gure 5.5.
From (5.16), we see that the -shifted
norm of a transfer function is nite
a
H
1
if and only if the real parts of the poles of are less than . For
0, then,
H
;a
a
<
the shifted
norm can be used to measure the size of some unstable transfer
H
1
functions, for example,
1 ( 1)
2 = 1
k
=
s
;
k
1
;
(whereas 1 ( 1) = , meaning that its RMS gain is in nite). On the other
k
=
s
;
k
1
1
hand if
0, the -shifted
norm-bound speci cation
a
>
a
H
1
kH
k
M
1
a
(or even just
) guarantees that the poles of lie to the left of the line
kH
k
<
1
H
1
a
=
in the complex plane.
<s
;a
We can interpret the shifted transfer function (
) as follows. Given a block
H
s
;
a
diagram for that consists of integrators (transfer function 1 ), summing blocks,
H
=s
and scaling ampli ers, we replace each integrator with a transfer function 1 (
)
=
s
;
a
(called a \leaky integrator" when
0). The result is a block diagram of (
),
a
<
H
s
;
a
as shown in gure 5.6. In circuit theory, where is some network function, this is
H
called a uniform loading of .
H
102