Digital Signal Processing Reference
In-Depth Information
q
k=-
q
C
k
e
jkv
0
t
,
[eq(4.11)]
x(t) =
where is a periodic function with the fundamental frequency We consider
the convergence of the right side of (4.11) in the next section. First, each side of
equation (4.11) is multiplied by
x(t)
v
0
.
e
-jnv
0
t
,
with
n
an integer, and then integrated from
t = 0
to
t = T
0
:
T
0
T
0
q
k=-
q
x(t)e
-jnv
0
t
dt =
L
B
C
k
e
jkv
0
t
R
e
-jnv
0
t
dt.
L
0
0
Interchanging the order of summation and integration on the right side yields the
equation
T
0
q
k=-
q
T
0
x(t)e
-jnv
0
t
dt =
e
j(k-n)v
0
t
dt
B
R
C
k
.
(4.19)
L
L
0
0
Using Euler's relation, we can express the general term in the summation on the
right side as
T
0
T
0
T
0
e
j(k-n)v
0
t
dt = C
k
L
C
k
L
cos(k - n)v
0
t
dt + jC
k
L
sin(k - n)v
0
t dt.
(4.20)
0
0
0
The second term on the right side of this equation is zero, since the sine function is
integrated over an integer number of periods. The same is true for the first term on
the right side, except for
k = n.
For this case,
T
0
T
0
`
C
k
L
cos(k - n)v
0
t dt
k=n
= C
n
L
dt = C
n
T
0
.
(4.21)
0
0
Hence, the right side of (4.19) is equal to
C
n
T
0
,
and (4.19) can be expressed as
T
0
x(t)e
-jnv
0
t
dt = C
n
T
0
.
L
0
This equation is solved for
C
n
:
T
0
1
T
0
L
x(t)e
-jnv
0
t
dt.
C
n
=
(4.22)
0
This is the desired relation between and the Fourier coefficients It can
be shown that this equation minimizes the mean-square error defined in Section
4.1 [2].
x(t)
C
n
.
