The peak gain of a system can also be expressed in terms of its step response :s
pk gn =
( )
kH
k
Tv
s
where ( ), the total variation of a function , is de ned by
Tv
f
f
1
( ) =
sup N;
X
( )
( +1)
Tv
f
0
jf
t
;
f
t
j
:
1 ...
i
i
t
t
=1
N
i
Roughly speaking, ( ) is the sum of all consecutive peak-to-valley di erences in
Tv
f
this is shown in gure 5.3.
f
1:4
1:2
1
0:8
()ts
0:6
0:4
0:2
0
0
1
2
3
4
5
6
7
8
9
10
t
The peak gain of a transfer function is equal to the total
Figure
5.3
variation of its step response , , the sum of all the consecutive peak-to-
s
i.e.
valley dierences (shown as arrows) of .s
It turns out that the peak gain of a SISO transfer function is also the average-
absolute gain:
pk gn =
aa gn = sup
aa
kH
w
k
(5.7)
kH
k
kH
k
aa=0
aa :
kw
k
kw
k
6
5.2.6
Norm: RMS Gain
H
1
An important norm of a transfer function is its RMS gain:
rms gn = sup
rms
kH
w
k
(5.8)
kH
k
rms=0
rms :
kw
k
kw
k
6
5.2 NORMS OF SISO LTI SYSTEMS
99
The RMS gain of a transfer function turns out to coincide with its 2 gain,
L
rms gn = sup
2
kH
w
k
(5.9)
kH
k
2=0
2
kw
k
kw
k
6
and is equal to the maximum magnitude of the transfer function,
rms gn = sup ( )
(5.10)
kH
k
jH
j
!
j
!
when is stable for unstable we have
rms gn = . For this reason, the RMS
H
H
kH
k
1
gain is sometimes called the maximum magnitude norm or the Chebychev norm of
a transfer function. We note that the right-hand side of (5.10) can be interpreted as
a worst case response norm of : it is the largest steady-state peak of the response
H
of to any unit amplitude sinusoid ( ( ) = cos ).
H
w
t
!
t
Equations (5.8{5.10) show that four reasonable interpretations of \the transfer
function is small" coincide:
H
the RMS value of its output is always small compared to the RMS value of
its input
the total energy of its output is always small compared to the total energy of
its input
the transfer function ( ) has a small magnitude at all frequencies
H
j
!
the steady-state peak of the response to a unit amplitude sinusoid of any
frequency is small.
The RMS gain of a transfer function can be expressed as its maximum mag-
H
nitude in the right half of the complex plane:
rms gn =
= sup ( )
(5.11)
kH
k
kH
k
s
j
1
0 jH
<s>
which is called the
norm of . (Note the very di erent meaning from
,
H
H
kw
k
1
1
the
norm of a signal.)
L
1
Let us establish (5.10) in the case where is stochastic and is stable it is
w
H
not hard to establish it in general. Let ( ) denote the power spectral density of
S
!
w
. The power spectral density of the output is then ( ) = ( ) 2 ( ), and
w
z
S
!
jH
j
!
j
S
!
z
w
therefore
2
Z
1
rms = 1
( )
kz
k
2
S
!
d!
z
;1
Z
= 1 1
2
2
( )
( )
jH
j
!
j
S
!
d!
w
;1
Z
sup ( ) 2 1 1 ( )
jH
j
!
j
2
S
!
d!
w
!
;1
=
2
2rms
kH
k
kw
k
:
1
100