2
Z
1
rms =
(0) = 1
( )
(4.6)
kuk
R
S
!
d!
:
u
2
u
;1
We can interpret the last integral as follows: the average power in the signal is the
integral of the contribution at each frequency.
For stochastic signals, the analogs of the average-absolute or peak norm are less
often encountered than the RMS norm. For stationary, we de ne the average-
u
absolute norm as
aa =
( )
kuk
E
ju
t
j
which for ergodic signals agrees (with probability one) with our deterministic de -
nition of
aa. We interpret
aa as the expected or mean resource consumption.
kuk
kuk
The analog of the steady-state peak of is the essential sup norm,
u
ess sup = inf
( ( )
) = 0
kuk
fa
j
Prob
ju
t
j
a
g
or equivalently, the smallest number such that with probability one, ( )
.
a
ju
t
j
a
Under some mild technical assumptions about , this agrees with probability one
u
with the steady-state peak norm of de ned in section 4.2.1.
u
4.2.5
Amplitude Distributions
We can think of the steady-state peak norm, RMS norm, and average-absolute
norm as di ering in the relative weighting of large versus small signal values: the
steady-state peak norm is entirely dependent on the large values of a signal the
RMS norm is less dependent on the large values, and the average-absolute norm
less still.
This idea can be made precise by considering the notion of the amplitude distri-
bution ( ) of a signal , which is, roughly speaking, the fraction of the time the
F
a
u
u
signal exceeds the limit , or the probability that the signal exceeds the limit at
a
a
some particular time.
We rst consider stationary ergodic stochastic signals. The amplitude distribu-
tion is just the probability distribution of the absolute value of the signal:
( ) =
( ( )
)
F
a
Prob
ju
t
j
a
:
u
Since is stationary, this expression does not depend on .
u
t
We can also express ( ) in terms of the fraction of time the absolute value of
F
a
u
the signal exceeds the threshold . Consider the time interval 0 ]. Over this time
a
T
interval, the signal will spend some fraction of the total time with ( )
.
u
T
ju
t
j
a
( ) is the limit of this fraction as
:
F
a
T
!
1
u
( ) = lim
0
( )
ft
j
t
T
ju
t
j
ag
(4.7)
F
a
u
T
!1
T
4.2 COMMON NORMS OF SCALAR SIGNALS
77
where ( ) denotes the total length (Lebesgue measure) of a subset of the real line.
These two ideas are depicted in gure 4.9. The amplitude distribution of the signal
in gure 4.9 is shown in gure 4.10.
3
2
a = 1:5
1
()t
0
u
;1
H
Y
H
;2
a = 1:5
;
;3
0
1
2
3
4
5
6
7
8
9
10
t
Example of calculating F (1:5) for the signal u in gure 4.1.
Figure
4.9
u
For T = 10, t 0 t T u(t)
1:5 is the length of the shaded
f
j
j
j
g
intervals, and this length divided by T approximates F (1:5).
u
This last interpretation of the amplitude distribution in terms of the fraction of
time the signal exceeds any given threshold allows us to extend the notion of ampli-
tude distribution to some deterministic (non-stochastic) signals. For a deterministic
, we de ne ( ) to be the limit (4.7), provided this limit exists (it need not). All
u
F
a
u
of the results of this section hold for a suitably restricted set of deterministic signals,
if we use this de nition of amplitude distribution. There are many more technical
details in such a treatment of deterministic signals, however, so we continue under
the assumption that is a stationary ergodic stochastic process.
u
Clearly ( ) = 0 for
ss , and
( ) increases to one as decreases
F
a
a
>
kuk
F
a
a
u
1
u
to zero. Informally, we think of ( ) as spending a large fraction of time where
ju
t
j
the slope of ( ) is sharp if
decreases approximately linearly, we say ( ) is
F
a
F
ju
t
j
u
u
approximately uniformly distributed in amplitude. Figure 4.11 shows two signals
and their amplitude distribution functions.
We can compute the steady-state peak, RMS, and average-absolute norms of u
directly from its amplitude distribution. We have already seen that
ss = sup
( ) 0
(4.8)
kuk
fa
j
F
a
>
g:
1
u
The steady-state peak of a signal is therefore the value of at which the graph of
a
the amplitude distribution rst becomes zero.
78