We warn the reader that this de nition di ers slightly from the standard de nition
of a norm see the Notes and References at the end of this chapter.
We will generally use the notation
x = ( )
kv
k
v
where x is some distinguishing mark or mnemonic for . This notation emphasizes
that a norm is a generalization of the absolute value for real or complex numbers,
and the Euclidean length of a vector. Note that we allow norms to take on the
value + , just as we allow our functionals on transfer matrices to do. We interpret
1
= as \ is in nitely large" as measured by the norm
.
kv
k
1
v
k
k
In the next few sections we survey some common norms used to measure the
size of signals. The veri cation that these norms do in fact satisfy the required
properties is left as an exercise for the reader (alternatively, the reader can consult
the references).
4.2
Common Norms of Scalar Signals
4.2.1
Peak
One simple but strict interpretation of \the signal is small" is that it is small at
u
all times, or equivalently, its maximum or peak absolute value is small. The peak
or
norm of is de ned as
L
u
1
= sup ( )
kuk
t
j:
1
0 ju
t
An example of a signal and its peak
is shown in gure 4.1.
u
kuk
1
The peak norm of a signal is useful in specifying a strict limit on the absolute
value of a signal, e.g., the output current of a power ampli er, or the tracking error
in a disk drive head positioning system.
The peak norm of a signal depends entirely on the extreme or large values the
signal takes on. If the signal occasionally has large values,
will be large a
kuk
1
statistician would say
depends on outliers or \rare events" in the signal .
kuk
u
1
We shall soon see other norms that depend to a lesser extent on occasional large
signal values.
It is useful to imagine how various signal norms might be measured. A full wave
recti er circuit that measures the peak of a voltage signal is shown in gure 4.2.
The peak norm can be used to describe a signal about which very little is known,
or willing to be assumed, other than some bound on its peak or worst case value.
Such a description is called an unknown-but-bounded model of a signal: we assume
only
. An example is quantization error, the di erence between a signal
kuk
M
1
and its uniformly quantized value. This error can be modeled as unknown but
bounded by one-half of the quantization interval.
4.2 COMMON NORMS OF SCALAR SIGNALS
71
3
H
Y
H
u
k
k
1
2
1
()t
0
u
;1
;2
u
;k
k
1
;3
0
1
2
3
4
5
6
7
8
9
10
t
A signal u and its peak norm u .
Figure
4.1
k
k
1
q
+
u(t)
+
q
q
Vc
;
;
q
With ideal diodes and an ideal capacitor the voltage on the
Figure
4.2
capacitor, V , tends to u for t large.
k
k
c
1
72