4.3
Common Norms of Vector Signals
The norms for scalar signals described in section 4.2 have natural extensions to
vector signals. We now suppose that : +
n
, i.e., ( )
n
for
0.
u
R
!
R
u
t
2
R
t
4.3.1
Peak
The peak, or
norm, of a vector signal is de ned to be the maximum peak of
L
1
any of its components:
= max
= sup max ( )
kuk
ku
k
ju
t
j
1
1
i
1
0 1
i
i
n
i
n
t
(note the di erent uses of the notation
). Thus, \
is small" means
k
k
kuk
1
1
that every component of is always small. A two-input peak detector is shown in
u
gure 4.18.
q
q
+
+
1
2
q
q
q
q
u
(t)
u
(t)
;
+
;
;
Vc
q
q
With ideal diodes and an ideal capacitor, the voltage on the
Figure
4.18
capacitor, V , tends to u for t large, where u = u1 u2]T is a vector of
k
k
c
1
two signals.
4.3.2
RMS
We de ne the RMS norm of a vector signal as
1
1 2
!
=
Z
T
rms =
lim
( )T ( )
(4.16)
kuk
0 u t u t dt
T
!1
T
provided this limit exists (see the Notes and References). For an ergodic wide-sense
stationary stochastic signal this can be expressed
2
1 Z 1
rms =
( ) 22 =
(0) =
( )
kuk
E
ku
t
k
T
r
R
T
r
S
!
d!
:
u
2
u
;1
4.3 COMMON NORMS OF VECTOR SIGNALS
87
(For vector signals, the autocorrelation is de ned by
( ) = ( ) ( + )T
R
E
u
t
u
t
u
(c.f. (4.4))). For such signals we have
1 2
!
=
rms =
n
X
2rms
(4.17)
kuk
ku
k
=1 i
i
i.e.,
rms is the square root of the sum of the mean-square values of the compo-
kuk
nents of .u
A conceptual model for the RMS value of a vector voltage signal is shown in
gure 4.19.
+
1
R
ambient
u
(t)
temperature Tamb
;
+
2
R
u
(t)
P
i
P
;
P
large thermal mass
temperature T
If u1 and u2 vary much faster than the thermal time constant
Figure
4.19
of the mass, then the long term temperature rise of the mass is proportional
to the average power in the vector signal u, i.e. T Tamb
u 2rms, where
;
/
k
k
Tamb is the ambient temperature, and T is the temperature of the mass.
4.3.3
Average-Absolute
The average-absolute norm of a vector signal is de ned by
1 Z n
T
X
aa = lim sup
( )
kuk
0
ju
t
j
dt:
i
T
T
!1
=1
i
This measures the average total resource consumption of all the components of .u
88