Digital Signal Processing Reference
In-Depth Information
l
g
r
,
y
i
d
.
,
©
,
L
s
Figure 1.6
The planimeter: a mechanical integrator.
Now consider the case in which all we have is a set of daily measure-
ments
c
1
,
c
2
,...,
c
D
as in Figure 1.1; the “average” temperature of our mea-
surements over
D
days is simply:
D
1
D
C
=
c
n
(1.3)
n
=
1
(as shown in the bottom panel of Figure 1.5) and this is an elementary sum
of
D
terms which anyone can carry out by hand and which does not depend
on how the measurements have been obtained: wickedly simple! So, obvi-
ously, the question is: “How different (if at all) is
C
from
C
?” In order to find
out we can remark that if we accept the existence of a temperature function
f
then the measured values
c
n
are
samples
of the function taken one day
apart:
(
t
)
c
n
=
f
(
nT
s
)
(where
T
s
is the duration of a day). In this light, the sum (1.3) is just the
Riemann approximation to the integral in (1.2) and the question becomes
an assessment on how good an approximation that is. Another way to look
at the problem is to ask ourselves how much information we are discarding
by only keeping samples of a continuous-time function.
R
C
Figure 1.7
The RC network: an electrical integrator.
