FOURIER SERIES - Signals, Systems, and Transforms - page 167

Note that this is also the trigonometric form.
A comparison of the frequency spectrum of the square wave (Figure 4.6)
with that of the train of impulse functions (Figure 4.11) illustrates an important
property of impulse functions. For the square wave, the amplitudes of the har-
monics decrease by the factor 1/ k , where k is the harmonic number. Hence, we ex-
pect that the higher harmonics can be ignored in most situations and that a finite
sum of the harmonics is usually adequate to represent a square wave. This state-
ment cannot be applied to the impulse train. The amplitudes of the harmonics re-
main constant for all harmonic frequencies. Hence, usually all harmonics must be
considered for a train of impulse functions. This point is considered further in the
next section.
Given in Table 4.3 are the Fourier coefficients of seven periodic signals that
are important in engineering applications. Since the coefficient is the average, or
dc, value of the signal, this value is not unique for a particular form of a periodic sig-
nal. For example, if we add a constant value to a sawtooth signal, the result is still a
sawtooth signal, with only the average value

C 0

C 0 changed. This point is covered in

greater detail in Section 4.6.
A MATLAB program that verifies the first three coefficients of the triangular
wave in Table 4.3 is
syms Ck ker t
for k=1 : 3
w0=2*pi;
ker=exp (-j*k*w0*t);
Ck=int (2*t*ker, 0, 0.5) + int (2* (1 - t) * ker, 0.5, 1);
simplify (Ck)

end

This program can also be written in the general variable k , but the results must be
simplified. The coefficients of the remaining signals in Table 4.3 can be derived by
altering this program in an appropriate manner.
As a final example in this section, we consider the important case of a train of
rectangular pulses.

Frequency spectrum of a rectangular pulse train

EXAMPLE 4.6

For this example, the frequency spectrum of the rectangular pulse train of Figure 4.12 will be
plotted. This waveform is common in engineering. The clock signal in a digital computer is a
rectangular pulse train of this form. Also, in communications, one method of modulation is to
vary the amplitudes of the rectangular pulses in a pulse train according to the information to
be transmitted. This method of modulation, called pulse-amplitude modulation, is described
in Sections 1.3 and 6.6.
From Table 4.3, the Fourier series for this signal is given by

q
k=- q

TX 0
T 0

sinc Tkv 0
2

e jkv 0 t , x(t) = (4.28)