Digital Signal Processing Reference
In-Depth Information
The equation for the Fourier coefficients is simple, because, in (4.19),
T
0
e
jkv
0
t
e
-jnv
0
t
dt = 0,
C
k
L
k Z n.
0
These two complex-exponential functions are said to be
orthogonal
over the inter-
val
(0, T
0
).
In general, two functions
f
(
t
) and
g
(
t
) are
orthogonal
over the interval
(
a
,
b
) if
b
f(t)g(t)
dt = 0.
L
a
Orthogonal functions other than complex exponentials exist, and these functions
may be used in a manner similar to that in (4.11) and (4.22) to express a general
function as a series [2]. We do not consider this general topic further.
Equation (4.22) is the desired result and gives the coefficients of the exponen-
tial form of the Fourier series as a function of the periodic signal The coeffi-
cients of the two trigonometric forms of the Fourier series are given in Table 4.2.
Because the integrand in (4.22) is periodic with the fundamental period the lim-
its on the integral can be generalized to and where is arbitrary. We ex-
press this by writing at the lower limit position of the integral and leaving the
upper limit position blank:
x(t).
T
0
,
t
1
t
1
+ T
0
,
t
1
T
0
1
T
0
L
T
0
x(t)e
-jkv
0
t
dt.
C
k
=
(4.23)
We now consider the coefficient
C
0
.
From (4.23),
1
T
0
L
T
0
C
0
=
x(t) dt.
Hence, is the
average value
of the signal This average value is also called the
dc value,
a term that originated in circuit analysis. For some waveforms, the dc value
can be found by inspection.
The exponential form and the combined trigonometric form of the Fourier se-
ries are probably the most useful forms. The coefficients of the exponential form
are the most convenient to calculate, while the amplitudes of the harmonics are di-
rectly available in the combined trigonometric form. We will usually calculate
from (4.23); if the amplitudes of the harmonics are required, these amplitudes are
equal to except that the dc amplitude is
In this section, we define the Fourier series for a periodic function and derive
the equation for calculating the Fourier coefficients. Table 4.2 gives the three forms
for the Fourier series. Examples of calculating the Fourier coefficients are given in
the next section.
C
0
x(t).
C
k
2 ƒC
k
ƒ ,
C
0
.
