linearly weighted emphasis on signal amplitudes and the average-absolute norm
puts uniform weight on all signal amplitudes.
4.2.6
: Square Root Total Energy
L
2
The previous sections dealt with the sizes of signals that persist. In this section
and the next we examine some norms appropriate for transient signals, which decay
to zero as time progresses such signals have zero as their steady-state peak, RMS,
and average-absolute norms.
The total energy or 2 norm of a signal is de ned by
L
1 2
Z
=
1
2 =
( )2
kuk
0 u t dt
:
This norm is the appropriate analog of the RMS norm for decaying signals, i.e.,
signals with nite total energy as opposed to nite steady-state power.
A useful conceptual model for the 2 norm is shown in gure 4.12. Here we
L
think of as a driving voltage to a resistive load immersed in a thermal mass.
u
This thermal mass is isolated (adiabatic), unlike the mass of gure 4.5, which is
connected to the ambient temperature (a very large thermal mass) through a nite
thermal resistance. The eventual temperature rise of the isolated thermal mass is
proportional to
22.
kuk
+
u(t)
R
P
i
P
good insulation
P
;
P
i
P
P
large thermal mass
temperature T
The long term temperature rise of the mass is proportional
Figure
4.12
to the total energy in u, i.e. T Tinit
u 22, where Tinit is the initial
;
/
k
k
temperature of the mass, and T is the nal temperature of the mass.
As a practical example, suppose that represents the current through a voice
u
coil drive during a step input in commanded position, and the thermal time constant
of the voice coil is longer than the time over which is large. Then
22 is a measure
u
kuk
of the temperature rise in the voice coil during a step command input.
By Parseval's theorem, the 2 norm can be computed as an 2 norm in the
L
L
4.2 COMMON NORMS OF SCALAR SIGNALS
81
frequency domain:
1 2
Z
=
1
2 =
1
( ) 2
(4.11)
kuk
2
jU
j
!
j
d!
:
;1
4.2.7
: Total Fuel or Resource Consumption
L
1
The 1 norm of a signal is de ned as
L
Z
1
1 =
( )
kuk
0 ju t j dt:
Just as the 2 norm measures the total energy in a signal, while the RMS norm
L
measures its average power, the 1 norm of a signal can be thought of as measuring
L
a total resource consumption, while the average-absolute norm measures a steady-
state average resource consumption. For example, if represents the compressed
u
gas ow through a nozzle during a particular spacecraft maneuver, then
1 is
kuk
proportional to the total gas consumed during the maneuver.
4.2.8
Frequency Domain Weights
The norms described above can be combined with an initial linear transformation
that serves to emphasize or de-emphasize certain aspects of a signal. Typically,
this initial transformation consists of passing the signal through an LTI lter with
transfer function , which we refer to as a frequency domain weight, as shown in
W
gure 4.13.
u
W
Wu
k
k
k
k
A frequency domain weighted norm is computed by passing
Figure
4.13
the signal u through an LTI lter W, and then determining the (unweighted)
norm of this ltered signal.
The idea is that the weighting lter makes the norm more \sensitive" (i.e., assign
larger values) to signals that have a large power spectral density at those frequencies
where ( ) is large. This idea can be made precise for the -weighted RMS
jW
j
!
j
W
norm, which we will denote
rms. The power spectral density of the ltered
k
k
W
signal
is
W
u
( ) = ( ) ( ) 2
S
!
S
!
jW
j
!
j
W
u
u
so the RMS norm of
is
W
u
1 2
Z
=
1
rms =
rms =
1
( ) ( ) 2
(4.12)
kuk
kW
uk
S
!
jW
j
!
j
d!
:
W
2
u
;1
82