Digital Signal Processing Reference
In-Depth Information
x
(
t
)
1
T
0
/2
0
T
0
/2
T
0
3
T
0
/2
t
1
(a)
x
(
t
)
4
sin t
x
(
t
)
1
0
T
0
/2
T
0
/2
T
0
3
T
0
/2
t
1
(b)
e
(
t
)
1
0
T
0
t
1
Figure 4.3
(c)
Functions for Example 4.1.
Hence, the best approximation of a square wave with unity amplitude by the function
is to choose we used the minimum mean-square error as the criterion for best.
Shown in Figure 4.3(b) is the square wave and the approximating sine wave, and Figure 4.3(c)
shows the error in the approximation. This error is given by
B
1
sin v
0
t
B
1
= 4/p;
4
p
sin v
0
t,
1 -
06 t 6 T
0
/2
4
p
sin v
0
t =
e(t) = x(t) -
c
.
4
p
sin v
0
t, T
0
/2 6 t 6 T
0
-1 -
■
Figure 4.3(c) illustrates very well why we minimize the average squared error
rather than the average error. In this figure, we see that the average error is zero,
while the approximation is not an especially good one. Furthermore,
any
value of
will give an
average error of zero,
with large values of negative error canceling large
values of positive error. However, the squared error is a nonnegative function, and
no cancellation can occur.
Other error functions can give reasonable results in the minimization proce-
dure; one example is the average value of the magnitude of
e
(
t
):
B
1
T
0
1
T
0
L
ƒ
e(t)
ƒ
dt.
J
m
[e(t)] =
0
