A variation on the peak norm is the eventual peak or steady-state peak:
ss = lim sup ( ) = lim sup ( )
kuk
ju
t
j
ju
t
j:
1
T
!1
t!1
t
T
The steady-state peak norm measures only persistent large excursions of the signal
unlike the peak norm, it is una ected by transients, i.e., the addition of a signal
that decays to zero:
+ transient ss = ss
if lim transient( ) = 0
ku
u
k
kuk
u
t
:
1
1
t!1
4.2.2
Root-Mean-Square
A measure of a signal that re ects its eventual, average size is its root-mean-square
(RMS) value, de ned by
1
1 2
!
=
Z
T
rms =
lim
( )2
(4.1)
kuk
0 u t dt
T
!1
T
provided the limit exists (see the Notes and References). This is a classical notion
of the size of a signal, widely used in many areas of engineering. An example of a
signal and its RMS value
rms is shown in gure 4.3.
u
kuk
3
7
2
u rms
6
k
k
;
5
;
1
u 2rms
k
k
4
u(t) 0
2 (t)u 3
1
;
2
@
I
2 @
1
;
u rms
;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 RMS value u rms is shown in (a). u 2rms
Figure
4.3
k
k
k
k
is the average area under u2, as shown in (b).
In an early RMS ammeter, shown in gure 4.4, the torque on the rotor is propor-
tional to the square of the current its large rotational inertia, the torsional spring,
and some damping, make the rotor de ection approximately proportional to the
mean-square current,
2rms.
kuk
Another useful conceptual model for
rms is in terms of the average power dis-
kuk
sipated in a resistive load driven by a voltage , as in gure 4.5. The instantaneous
u
4.2 COMMON NORMS OF SCALAR SIGNALS
73
L1
L2
u(t)
An early RMS ammeter consists of a stator L1 and a rotor
Figure
4.4
L2, which is tted with a needle and restoring spring. The deection of the
needle will be approximately proportional to u 2rms, the square of the RMS
k
k
value of the input current u.
power dissipated in the load resistor is simply ( )2
the (long term) temperature
u
t
=R
rise of the large thermal mass above ambient temperature is proportional to the av-
erage power dissipation in the load,
2rms . This conceptual model shows that
kuk
=R
the RMS measure is useful in specifying signal limits that are due to steady-state
thermal considerations such as maximum power dissipation and temperature rise.
For example, the current through a voice coil actuator might be limited by a max-
imum allowable steady-state temperature rise for the voice coil this speci cation
could be expressed as an RMS limit on the voice coil current.
ambient
temperature Tamb
+
u(t)
R
large thermal mass
;
temperature T
)
If u varies much faster than the thermal time constant of the
Figure
4.5
mass, then the long term temperature rise of the mass is proportional to
the average power in u, i.e. T Tamb
u 2rms, where Tamb is the ambient
;
/
k
k
temperature, and T is the temperature of the mass.
Even if the RMS norm of a signal is small, the signal may occasionally have
large peaks, provided the peaks are not too frequent and do not contain too much
energy. In this sense,
rms is less a ected than
by large but infrequent
kuk
kuk
1
values of the signal. We also note that the RMS norm is a steady-state measure of
a signal the RMS value of a signal is not a ected by any transient. In particular,
a signal with small RMS value can be very large for some initial time period.
74