Digital Signal Processing Reference
In-Depth Information
TABLE 4.4
Peak Values of Sinc
x
x
ƒ
sinc x
ƒ
1/
x
0.0
1.0
—
4.493
0.2172
0.2226
7.725
0.1284
0.1294
10.90
0.0913
0.0917
14.07
0.0709
0.0711
sin
x
is unity, the maximum magnitude of sinc
x
is approximately 1/
x
. For example,
in the vicinity of the maximum magnitude of sinc
x
is approximately
Values of 1/
x
are also given in Table 4.4.
In this section, we introduce one of the basic concepts of signals and systems
engineering, that of frequency spectra. Many analysis and design procedures for sig-
nals and systems are based on this concept. We expand on this concept in the re-
mainder of this chapter, and in Chapter 5 extend the concept to aperiodic signals.
x = 25,
1/25 = 0.04.
4.4
PROPERTIES OF FOURIER SERIES
In this section, some properties of the Fourier series are stated. These properties are
then discussed, with examples for illustration.
Any single-valued periodic function that satisfies the
Dirichlet
conditions
can be expanded into a Fourier series. The Dirichlet conditions are [2]
x(t)
1.
x
(
t
) has at most a finite number of discontinuities in one period;
2.
x
(
t
) has at most a finite number of maxima and minima in one period;
3.
x
(
t
) is bounded.
The third condition has been expanded to include singularity functions and may be
stated as [3]
L
T
0
ƒ x(t) ƒ
dt 6
q
3a.
.
Any function of time that appears in physical systems will satisfy these conditions.
Several properties of the Fourier series will now be given. The readers inter-
ested in the proofs of these properties should see Refs. 2, 4, and 5. For
x(t)
satisfy-
ing the Dirichlet conditions 1, 2, and 3, the following are true:
1.
The Fourier series converges to the value of
x
(
t
) at every point of continu-
ity where
x
(
t
) has a right-hand and a left-hand derivative, whether these derivatives
are the same or different. The right-hand derivative of
x
(
t
) at
t = t
a
is defined as the
derivative as
t
approaches
t
a
from the right. The left-hand derivative is the deriva-
tive as
t
approaches
t
a
from the left.
