2( )
u
t
1
A = 1:5 1:5
-
0:2 1:2
;
1
1( )
u
t
1
;
1
;
@
I
@
;
;
A = 1 0
0 1
A = 1:5 0
0 0:8
The constraint u
1 requires that the signal u(t) lie
Figure
4.20
k
k
A
1
inside a trapezoid at all times t. Three weighting matrices A are shown,
together with the corresponding limits on the signal u.
The rst inequality in (4.18) can be shown by replacing 2 by the upper bound
u
2 :
kuk
1
2
1 Z T
rms = lim
( )2
kuk
0 u t dt
T
!1
T
Z
lim 1 T
2
0 kuk dt
1
T
!1
T
=
2
kuk
:
1
The second inequality in (4.18) follows from the Cauchy-Schwarz inequality:
1 Z T
aa = lim
1 ( )
kuk
0 ju t j dt
T
!1
T
1 2
1 2
!
!
=
=
Z
Z
lim 1 T 12
1 T ( ) 2
0
dt
0 ju t j dt
T
!1
T
T
=
rms
kuk
:
It can also be shown that
ss
rms
aa
kuk
kuk
kuk
kuk
:
1
1
Another norm inequality, that gives a lower bound for
aa, is
kuk
2rms
aa
(4.20)
kuk
kuk
kuk
:
1
4.4 COMPARING NORMS
91
This inequality can be understood by considering the power ampli er shown in
gure 4.8, with a 1 load resistance. If we have
=
, then the ampli er
V
kuk
cc
1
does not saturate (clip), and the voltage on the load is the input voltage, ( ). The
u
t
average power delivered to the load is therefore
2rms. The average power supply
kuk
current is
aa, so the average power delivered by the power supply is
aa ,
kuk
kuk
V
cc
which is
aa
. Of course, the average power delivered to the load does not
kuk
kuk
1
exceed the average power drain on the power supply (the di erence is dissipated in
the transistors), so we have
2rms
aa
, which is (4.20).
kuk
kuk
kuk
1
If the signal is close to a switching signal, which means that it spends most of
u
its time near its peak value, then the values of the peak, steady-state peak, RMS
and average-absolute norms will be close. The crest factor of a signal gives an
indication of how much time a signal spends near its peak value. The crest factor
is de ned as the ratio of the steady-state peak value to the RMS value of a signal:
( ) =
ss
kuk
1
CF
u
rms :
kuk
Since
ss
rms, the crest factor of a signal is at least 1. The crest factor is
kuk
kuk
1
a measure of how rapidly the amplitude distribution of the signal increases below
=
ss
see gure 4.11. The two signals in gure 4.11 have crest factors of 1.15
a
kuk
1
and 5.66, respectively.
The crest factor can be combined with the upper bound in (4.20) to give a bound
on how much the RMS norm exceeds the average-absolute norm:
rms
kuk
( )
(4.21)
aa
CF
u
:
kuk
92