outputs of (a)
(b)
13 and 13 with a white noise excitation. Thus, in the sense of RMS
H
H
response to white noise, we can say that (a)
(b)
13 is about half the size of 13 .
H
H
6
6
(b)
69
4
13 2 = 2
kH
k
:
(a)
4 ;
13 2 = 1 17
kH
k
:
2
;
2
;
;
() 0
() 0
t
t
z
z
2
2
;
;
4
4
;
;
6
6
;
0
2
4
6
8
10 ; 0
2
4
6
8
10
t
t
(a)
(b)
(a) shows a sample of the steady-state response of (a)
13 to a
Figure
5.10
H
white noise excitation, together with the value (a)
13 2 = 1 17. (b) shows
kH
k
:
a sample of the steady-state response of (b)
13 to a white noise excitation,
H
together with the value (b)
13 2 = 2 69.
kH
k
:
From the third row, we see that the worst case response of (a)
13 to inputs
H
bounded and slew-rate limited by 1 is similar to that of (b)
13 . Amplitude and slew-
H
rate limited input waveforms that produce outputs with peak values close to these
worst case values are shown in gure 5.11. Thus, in the sense of maximum peak
output in response to inputs bounded and slew limited by 1, (a)
13 is about the same
H
size as (b)
13 .
H
From the fourth row, we see that the peak output of (b)
13 with a worst case input
H
bounded by 1 is almost three times larger than (a)
13 . This is expected from the
H
step response total variation expression for the peak gain (see gures 5.3 and 5.8).
Input waveforms that produce outputs close to these worst case values are shown
in gure 5.12. Thus, in the sense of maximum peak output in response to inputs
bounded by 1, (a)
(b)
13 is less than one third the size of 13 .
H
H
From the fth row, we see that the RMS gain of (b)
13 is more than twice as
H
large as the RMS gain of (a)
13 . This can be seen from gure 5.9 input signals that
H
result in the largest possible ratio of RMS response to RMS input are sinusoids at
frequencies = 0 9 (for (a)
(b)
13 ) and = 6 3 (for 13 ).
!
:
H
!
:
H
Finally, the worst case square root energy in the output of (a)
13 after its unit-
H
energy input signal is turned o is about 48% lower than the worst case for (b)
13
H
(sixth row of table 5.1). Thus, in the sense of square root energy gain from past
inputs to future outputs we can say that (a)
(b)
13 is about half the size of 13 . Fig-
H
H
5.2 NORMS OF SISO LTI SYSTEMS
107
2
2
1 5
1 5
:
:
( )
z
t
1
1
@
@
R
0 5
0 5
:
:
0
( )
0
( )
w
t
z
t
0 5
;
0 5
;
;
;
;
:
( )
;
:
w
t
1
1
;
;
;
;
1 5
1 5
;
:
0
2
4
6
8
10 ; : 0
2
4
6
8
10
t
t
(a)
(b)
(a) shows an input signal with
= 1 and _
= 1
Figure
5.11
w
kw
k
kw
k
1
1
that drives the output of (a)
(a)
13 close to
13 wc = 1 60. (b) shows an input
H
kH
k
:
signal with
= 1 and _ = 1 that drives the output of (b)
13 close
w
kw
k
kw
k
H
1
1
to (b)
13 wc = 1 68.
kH
k
:
4
4
( )
w
t
2
2
( )
z
t
0
0
;
;
( )
w
t
;
;
2
2
;
;
4
4
;
;
;
;
( )
z
t
0
2
4
6
8
10
0
2
4
6
8
10
t
t
(a)
(b)
(a) shows an input signal with
= 1, together with
Figure
5.12
w
kw
k
1
the output produced when (a)
13 is driven by
(10) = 1 74 is very close
z
H
w
z
:
to (a)
13 pk gn. (b) shows an input signal
with
= 1, together with
kH
k
w
kw
k
1
the output produced when (b)
13 is driven by
(10) = 4 86 is close to
z
H
w
z
:
(b)
13 pk gn = 4 93.
kH
k
:
108