This is simply the 1 norm of with the time domain weight ( ) = , which serves
L
u
w
t
t
to emphasize the signal at large times, and de-emphasize the signal at small times.
u
One commonly used family of time domain weights is the family of exponential
weightings, which have the form ( ) = exp( ). If is positive, such a weighting
w
t
at
a
exponentially emphasizes the signal at large times. This may be appropriate for
measuring the size of a decaying signal. Alternatively, we can think of a speci cation
such as ~
(where ~( ) = exp( ) ( ), and
0) as enforcing a rate of
ku
k
M
u
t
at
u
t
a
>
1
decay in the signal at least as fast as exp( ). An example of a signal and the
u
;at
u
exponentially scaled signal ( ) is shown in gure 4.16.
;t
u
t
e
1
1
0 6
0 6:
e;t
:
0 2
0 2
:
:
u(t) 0 2
0 2
;
:
;
:
H
Y
u(t)e;t
H
0 6
0 6
;
:
;
:
1
1
;
0
0 5
1
1 5
2
2 5 ; 0
0 5
1
1 5
2
2 5
:
t :
:
:
t :
:
(a)
(b)
A signal u is shown in (a). The exponentially weighted 2
Figure
4.16
L
norm of u is the 2 norm of the signal ~u(t) = u(t)e , which is shown in
;t
L
(b).
If is negative, then the signal is exponentially de-emphasized at large times.
a
This might be useful to measure the size of a diverging or growing signal, where the
value of an unweighted norm is in nite.
There is a simple frequency domain interpretation of the exponentially weighted
2 norms. If the -exponentially weighted 2 norm of a signal is nite then its
L
a
L
u
Laplace transform ( ) is analytic in the region
, and in fact
U
s
fs
j
<s
>
;ag
Z
Z
~ 2
1
1
2 =
(exp( ) ( ))2 = 1
( + ) 2
(4.15)
ku
k
0
at
u
t
dt
2
jU
;a
j
!
j
d!
;1
(recall ~( ) = exp( ) ( )). The only di erence between (4.15) and (4.11) is that the
u
t
at
u
t
integral in the -exponentially weighted norm is shifted by the weight exponent .
a
a
The frequency domain calculations of the 2 norm and the -exponentially weighted
L
a
2 norm for the signal in gure 4.16(a) are shown in gure 4.17.
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4.2 COMMON NORMS OF SCALAR SIGNALS
85
( + )
jU
j
!
j
!
(a)
( + ) 2
jU
j
!
j
U(0 + j!) 2
j
j
U(1 + j!) 2
!
j
j
= 0
= 1
(b)
An exponentially weighted 2 norm can be calculated in the
Figure
4.17
L
frequency domain. Consider u 2 1 = ~u 2, where ~u(t) = e u(t) is shown
;t
k
k
k
k
;
in gure 4.16(b). U( +j!) , the magnitude of the Laplace transform of u,
j
j
is shown in (a) for
0. As shown in (b), u 22 is proportional to the area
k
k
under U(j!) 2, and u 22 1 is proportional to the area under U(1+j!) 2.
j
j
k
k
j
j
;
86