Digital Signal Processing Reference
In-Depth Information
We can verify this result by testing the signal at several different values of time. Then, from
Table 4.8,
X
0
2
- 3 =-1
4
X
0
C
0y
= AC
0y1
+ B =
¢
≤
and
X
0
X
0
4
2
pk
e
-jp/2
, k Z 0.
2pk
e
-jp/2
=
C
ky
= AC
ky1
=
These results check those of Example 4.9, where these coefficients were calculated using only
an amplitude-reversal approach.
■
In this section, we consider amplitude and time transformations of periodic
signals, so as to extend the usefulness of Table 4.3. A second procedure for ex-
tending the usefulness of the table is stated in Section 4.4; the Fourier series of a
sum of periodic signals is equal to the sum of the Fourier series of the signals, pro-
vided that the sum is periodic. If the sum is not periodic, it does not have a Fourier
series.
In this chapter, we introduce the Fourier series, which is a representation of a peri-
odic signal by an infinite sum of harmonically related sinusoids. The Fourier series
is presented in three forms: the exponential form, the combined trigonometric form,
and the trigonometric form.
Several properties of the Fourier series are discussed. The Fourier series was
introduced using the property that the Fourier series minimizes the mean-square
error between a periodic function and its series.
The Fourier series of seven periodic functions that occur in engineering prac-
tice are given in Table 4.3. Procedures to expand the usefulness of this table are ap-
plied. These procedures include the independent variable transformations and the
amplitude transformations discussed in Chapter 2.
Frequency spectra give graphical representations of Fourier series. The spec-
tra are generated by plotting the coefficients of either the exponential form or the
combined trigonometric form of the Fourier series versus frequency. The plots for
the combined trigonometric form give the amplitudes of the sinusoidal harmonic
components directly.
As the final topic, the analysis of linear time-invariant systems with periodic
inputs is presented. The basis of this analysis is the sinusoidal steady-state response
of a system. The system response is the sum of the responses for each harmonic, by
superposition. This analysis procedure yields the frequency spectrum of the output
signal; however, it does not give a good indication of a plot of this signal as a func-
tion of time. See Table 4.9.
