all of its poles have negative real part. The power spectral density of the output z
of is then
H
( ) = ( ) ( ) 2
S
!
S
!
jH
j
!
j
z
w
and therefore
1 2
Z
=
1
rms =
1
( ) 2 ( )
(5.2)
kz
k
2
jH
j
!
j
S
!
d!
:
w
;1
Thus we assign to the norm
H
1 2
Z
=
1
rms =
1
( ) 2 ( )
(5.3)
kH
k
jH
j
!
j
S
!
d!
:
w
2
w
;1
The right-hand side of (5.3) has the same form as (4.12), with substituted
H
for . The interpretations are di erent, however: in (5.3), is some xed signal,
W
w
and we are measuring the size of the LTI system , whereas in (4.12), is a xed
H
W
weighting transfer function, and we are measuring the size of the signal .
w
5.2.3
Norm: RMS Response to White Noise
H
2
Consider the RMS response norm above. If ( ) 1 at those frequencies where
S
!
w
( ) is signi cant, then we have
jH
j
!
j
1
1 2
Z
=
1
rms
( ) 2
kH
k
jH
j
!
j
d!
:
w
2 ;1
It is convenient to think of such a signal as an approximation of a white noise signal,
a ctitious input signal with ( ) = 1 for all (and thus, in nite power, which
S
!
!
w
we conveniently overlook).
This important norm of a stable system is denoted
1 2
Z
=
1
2 =
1
( ) 2
kH
k
2
jH
j
!
j
d!
;1
(we assign
2 =
for unstable ), and referred to as the 2 norm of .
kH
k
1
H
H
H
Thus we have the important fact: the 2 norm of a transfer function measures
H
the RMS response of its output when it is driven by a white noise excitation.
The 2 norm can be given another interpretation. By the Parseval theorem,
H
1 2
Z
=
1
2 =
( )2
=
2
kH
k
0 h t dt
khk
the 2 norm of the impulse response of the LTI system. Thus we can interpret
L
h
the 2 norm of a system as the 2 norm of its response to the particular input
H
L
signal , a unit impulse.
5.2 NORMS OF SISO LTI SYSTEMS
97
5.2.4
A Worst Case Response Norm
Let us give an example of measuring the size of a transfer function using the worst
case response paradigm. Suppose that not much is known about except that
w
ampl and _
slew, i.e.,
is bounded by ampl and slew-rate
kw
k
M
kw
k
M
w
M
1
1
limited by slew. If the peak of the output is critical, a reasonable measure of
M
z
the size of is
H
wc = sup
ampl _
slew
kH
k
fkH
w
k
j
kw
k
M
kw
k
M
g
:
1
1
1
In other words,
wc is the worst case (largest) peak of the output, over all inputs
kH
k
bounded by ampl and slew-rate limited by slew.
M
M
5.2.5
Peak Gain
The peak gain of an LTI system is
pk gn = sup kHwk1
(5.4)
kH
k
=0
:
kw
k
1
kw
k
6
1
It can be shown that the peak gain of a transfer function is equal to the 1 norm
L
of its impulse response:
Z
1
pk gn =
( ) =
1
(5.5)
kH
k
0 jh t j dt khk :
The peak gain of a transfer function is nite if and only if the transfer function is
stable.
To establish (5.5) we consider the input signal
( ) = sgn( (
)) for 0
h
T
;
t
t
T
w
t
0
otherwise
(5.6)
which has
= 1 (the sign function, sgn( ), has the value 1 for positive argu-
kw
k
1
ments, and 1 for negative arguments). The output at time is
;
T
Z
( ) = T (
) ( )
z
T
0 w T ; t h t dt
Z
= T sgn( ( )) ( )
0
h
t
h
t
dt
Z
= T ( )
0 jh t j dt
which converges to
1 as
. So for large (and
stable, so that
khk
T
!
1
T
H
pk gn
), the signal (5.6) yields
near
1 it is also possible to
kH
k
<
1
kz
k
=kw
k
khk
1
1
show that there is a signal such that
=
1.
w
kz
k
=kw
k
khk
1
1
98