Digital Signal Processing Reference
In-Depth Information
When the Fourier coefficients characterizing the respective rectangular functions
are given, expression (16.12) allows all the elements
ν
ij
to be found. Specifically,
it follows from this expression that
ν
11
=
ν
22
=
ν
33
= ··· =
ν
nn
=
a
c1
.
If the functions
R
c
(
i
f
,
t
) are even, the equation describing the relationship
between the coefficients
{
α
i
}
and the Fourier coefficients
{
a
i
}
is given as
∞
α
i
=
a
c(2
r
+
i
)
a
(2
r
+
1)
i
.
(16.13)
r
=
0
In the case where the signal spectrum is restricted and the Fourier coefficients at
frequencies exceeding
n
f
can be considered to be equal to zero,
a
(2
r
+
1)
i
=
0
for (2
r
+
1)
i
>
n
(16.14)
and Equation (16.13) is then given by
[
n
/
2
i
−
1
/
2]
α
=
.
a
c(2
r
+
i
)
a
(2
r
+
1)
i
(16.15)
i
r
=
0
It follows that
α
=
a
c1
a
i
for all
i
satisfying the inequality
n
2
i
i
1
2
<
−
1
.
Therefore
α
=
a
c1
a
i
for
i
>
[
n
/
3]
.
(16.16)
i
In the case where, for example,
n
=
16
,
the relationships can be described by
the following system of equations:
α
1
=
a
c1
a
1
+
a
c3
a
3
+
a
c5
a
5
+
a
c7
a
7
+
a
c9
a
9
+
a
c11
a
11
+
a
c13
a
13
+
a
c15
a
15
α
2
=
a
c1
a
2
+
a
c3
a
6
+
a
c5
a
10
+
a
c7
a
10
α
3
=
a
c1
a
3
+
a
c3
a
9
+
a
c5
a
15
α
=
a
c1
a
4
+
a
c3
a
12
(16.17)
4
α
=
a
c1
a
5
+
a
c3
a
15
5
α
=
a
c1
a
6
..........
6
α
=
a
c1
a
16
.
16
From this example, for
n
=
16 all the Fourier coefficients
a
i
,
i
>
5
,
are ob-
tained simply by normalizing the respective coefficient
α
i
. On the basis of
