0195136047.pdf

14.1 SIGNALS AND SPECTRAL ANALYSIS 637

An

an

bn

φn

Figure 14.1.3Phasor diagram of Fourier coefficients.

EXAMPLE 14.1.5

The rectangular pulse train of Figure E14.1.4(a) consists of pulses of heightAand durationD.
Such pulse trains are employed for timing purposes and to represent digital information. For a
particular pulse train,A=3 and the duty cycleD/T= 1 /3.

(a) Find the Fourier-series expansion of the pulse train.

(b) Sketch the line spectrum of the rectangular pulse train whenT=1 ms.

Solution

(a) Using the solution of Example 14.1.4, we have for the particular pulse train:

a 0 =

DA

T

= 1 ; an=

2 A

πn

sin

2 Dn

T

=

6

πn

sin

πn

3

; bn= 0

Thus, the Fourier-series expansion results,

x(t)= 1 + 1 .65 cosωt+ 0 .83 cos 2ωt− 0 .41 cos 4ωt− 0 .33 cos 5ωt+...

Note that the terms corresponding to 3ω,6ω,... are missing becausean=0 forn=3,

6,....

(b)T =1ms= 10 −^3 s andf = 103 Hz=1 kHz. The line spectrum of the particular

rectangular pulse train is shown in Figure E14.1.5. As seen from the spectrum, most of

the time variation comes from the large-amplitude components below 6 kHz. The higher

frequency components have much smaller amplitudes, which account for the stepwise

jumps in the pulse train.

0

Phase

Amplitude

0

− 180 °

1

1

234

f, kHz

56789

Figure E14.1.5