4.2.3
Average-Absolute Value
A measure that puts even less emphasis on large values of a signal (indeed, the
minimum emphasis possible to still be a norm) is its average-absolute value, de ned
by
1 Z T
aa = lim
( )
(4.2)
kuk
0 ju t j dt
T
!1
T
provided the limit exists (see the Notes and References). An example of a signal u
and its average-absolute norm
aa is shown in gure 4.6.
aa can be measured
kuk
kuk
with the circuit shown in gure 4.7 (c.f. the peak detector circuit in gure 4.2).
3
3
2
u aa
2 5
k
k
:
u aa
;
1
k
k
2
;
u(t) 0
u(t)j 1 5:
1
j
1
;
@
I
@
2
0 5
;
u aa
:
;k
k
3
0
;
0
2
4 t 6
8
10
0
2
4 t 6
8
10
(a)
(b)
A signal u and its average-absolute value u aa is shown in (a).
Figure
4.6
k
k
u aa is found by nding the average area under u , as shown in (b).
k
k
j
j
R
q
+
u
+
(t)
q
q
q
C
Vc
;
;
ro
q
q
If u varies much faster than the time constant RC then V will
Figure
4.7
c
be nearly proportional to the average of the peak of the input voltage u, so
that V = u aa. The resistor r
R ensures that the output impedance
k
k
c
o
of the bridge is low at all times.
The average-absolute norm
aa is useful in measuring average fuel or resource
kuk
use, when the fuel or resource consumption is proportional to ( ) . In contrast, the
ju
t
j
4.2 COMMON NORMS OF SCALAR SIGNALS
75
RMS norm
2rms is useful in measuring average power, which is often proportional
kuk
to ( )2. Examples of resource usage that might be measured with the average-
u
t
absolute norm are rocket fuel use, compressed air use, or power supply demand in
a conventional class B ampli er, as shown in gure 4.8.
Vcc
power
supply
q
q
+
u(t)
R
V
;
;
cc
Idealized version of a class B power amplier, with no bias
Figure
4.8
circuitry shown. Provided the amplier does not clip, i.e., u
V , the
k
k
1
cc
average power supplied by the power supply is proportional to u aa, the
k
k
average-absolute norm of u the average power dissipated in the load R is
proportional to u 2rms, the square of the RMS norm of u.
k
k
4.2.4
Norms of Stochastic Signals
For a signal modeled as a stationary stochastic process, the measure of its size most
often used is
2
rms = ;
( )2 1=
(4.3)
kuk
E
u
t
:
Because the process is stationary, the expression in (4.3) does not depend on . For
t
stochastic signals that approach stationarity as time goes on, we de ne
2
rms = lim
( )2 1=
kuk
E
u
t
:
t!1
If the signal is ergodic, then its RMS norm can be computed either by (4.3)
u
or (4.1): with probability one the deterministic and stochastic RMS norms are
equal.
The RMS norm can be expressed in terms of the autocorrelation of ,u
( ) = ( ) ( + )
(4.4)
R
E
u
t
u
t
u
or its power spectral density,
Z
( ) = 1 ( ) ;j!
(4.5)
S
!
R
e
d
u
u
;1
76