Digital Signal Processing Reference
In-Depth Information
Consider the rectangular functions of system 1 shown in Figure 16.1. They are
defined by
1
,
=
,
+
,
=
f
,
=
,
,
,...,
R
s
(
i
f
t
)
R
s
(
i
f
t
rT
i
)
T
i
r
0
1
2
i
R
s
i
T
i
4
,
=
,
+
,
=
,
,...,
R
c
(
i
f
t
)
f
t
i
1
2
(16.4)
R
s
(
i
f
,
t
)
=
R
s
(
i
f
,
−
t
)
,
R
s
(
i
f
,
t
)
∈
[1
,
−
1]
,
T
i
R
s
(
i
f
,
t
)d
t
=
0
0
The signal
x
(
t
) is first decomposed on the basis of these functions. Then
T
2
T
α
=
x
(
t
)
R
c
(
i
f
,
t
)d
t
,
i
0
(16.5)
T
2
T
β
=
x
(
t
)
R
s
(
i
f
,
t
)d
t
.
i
0
On the other hand, the Fourier series expansions of the functions
R
s
(
i
f
,
t
)
,
R
c
(
i
f
,
t
), given by
∞
b
s
r
sin(2
π
ri
R
s
(
i
f
,
t
)
=
ft
)
,
r
=
1
(16.6)
∞
R
c
(
i
f
,
t
)
=
a
c
r
cos(2
π
ri
ft
)
,
r
=
1
where
1)
r
+
1
π
(2
r
4(
−
4
a
s
r
=
1)
,
b
c
r
=
−
π
(2
r
−
1)
are the Fourier coefficients for the frequencies
ri
f
.
Substituting expressions (16.6) into Equations (16.5) yields
∞
T
2
T
α
=
x
(
t
)
a
c
r
cos(2
π
ri
ft
)d
t
i
0
r
=
1
∞
T
2
T
=
a
c
r
x
(
t
) cos(2
π
ri
ft
)d
t
.
(16.7)
0
r
=
1
