4.3.4
and
Norms
L
L
2
1
The 2 and 1 norms of a vector signal are de ned by
L
L
1 2
1 2
!
!
=
=
Z
n
1
X
2 =
( )2
= n
X
22
kuk
0
u
t
dt
ku
k
=1 i
=1 i
i
i
Z
n
1
X
1 =
( ) = n
X
1
kuk
0
ju
t
j
dt
ku
k
:
=1 i
=1 i
i
i
4.3.5
Scaling and Weighting
A very important concept associated with vector signal norms is scaling, which can
be thought of as assigning relative weights to the di erent components of the vector
signal. For example, suppose that 1 represents the output voltage and 2 the
u
u
output current of a power ampli er that voltage saturates (clips) at 100V and
current limits at 2A. One appropriate measure of the peak of this signal is
= supmax
1( )
2( )
ju
t
j
ju
t
j
kuk
D
1
0
100
2
= kDuk1
t
where
= 1 100 0
=
D
0
1 2 :
=
is referred to as a scaling matrix, and
the -scaled peak of . We can
D
kuk
D
u
D
1
interpret
as the size of , relative to ampli er voltage or current overload:
kuk
u
D
1
= 0 5 indicates 6dB of headroom before the ampli er overloads
=
kuk
:
kuk
D
1
D
1
1 0 indicates that the ampli er will just saturate or current limit.
:
When the di erent components of a vector signal represent di erent physical
u
quantities, as in the example above, the use of an appropriate scaling matrix is
crucial. It is useful to think of the scaling matrix as including the translation
factors among the various physical units of the components of : for our example
u
above,
= 1 100V
0
=
D
0
1 2A
=
which properly renders
unitless.
kuk
D
1
A simple rule-of-thumb is to use scale factors that are inversely proportional to
what we might consider typical, nominal, or maximum acceptable values for that
component of the signal. The scaling in our example above is based on this principle,
using the maximum acceptable (overload) values. We have already encountered this
idea in sections 3.6.1 and 3.6.3.
4.4 COMPARING NORMS
89
For average-absolute or 1 norms, the scale factors can be interpreted as rel-
L
ative costs or values of the resources or commodities represented by the di erent
components of the signal. For RMS or 2 norms, the scale factors might represent
L
di erent termination resistances in the conceptual model in gure 4.19.
Scaling is a very special form of weighting, since it consists of applying a linear
transformation to the signal before computing its norm the linear transformation
is just multiplication by a diagonal matrix. Of course, it is also possible to multiply
the signal vector by a nondiagonal matrix before computing the norm, as in
2 =
2
kuk
kAuk
A
where is some matrix. One familiar example of this is the weighted 2 norm,
A
L
1 2
Z
=
1
2 =
( )T ( )
kuk
t
R u
t
dt
A
0 u
where =
. is called a (constant) weight matrix. Constant weight matri-
T
R
A
A
A
ces can be used to emphasize some directions in n while de-emphasizing others,
R
whereas with (diagonal) scaling matrices we are restricted to directions aligned with
the axes. An example of this distinction is shown in gure 4.20 for the constraint
=
1
kuk
kAuk
:
A
1
1
When is diagonal the signal ( ) is constrained to lie in a rectangle at each time
A
u
t
. For a general matrix , the signal ( ) is constrained to lie in a trapezoid.
t
A
u
t
More generally, we can preprocess the signal by a weighting transfer matrix W
that has columns, as in
n
rms =
rms
kuk
kW
uk
:
W
Often, is square, i.e., it has rows as well.
might emphasize di erent \direc-
W
n
W
tions" at di erent frequencies.
4.4
Comparing Norms
We have seen many norms for signals. A natural question is: how di erent can they
be? Intuition suggests that since these di erent norms each measure the \size" of
a signal, they should generally agree about whether a signal is \small" or \large".
This intuition is generally false, however.
For scalar signals we have
rms
aa
(4.18)
kuk
kuk
kuk
1
for vector signals with components we have the generalization
n
1
1
rms
aa
(4.19)
p
kuk
kuk
kuk
:
1
n
n
90