Review of Analog Signal Processing Basics - Digital Signal Processing: Fundamentals and Applications - page 785

EXAMPLE B.4
In the sawtooth waveform shown in Table B.1 and reprinted in Figure B.7 , ifT 0 ¼ 1 ms and A ¼ 10, use the
formula in the table to determine the Fourier series expansion in a magnitude-phase form, and determine the
frequency f 3 and amplitude value of A 3 for the third harmonic. Write the Fourier series expansion in a complex
exponential form also, and determine jc 3 j and jc 3 j for the third harmonic.
Solution:
a. Based on the information in Table B.1 , we have
x t ¼ 2 A
p

sin u 0 t 1

2 sin 2u 0 t þ 1 3 sin 3u 0 t 1 4 sin 4u 0 t þ /

Since T 0 ¼ 1 ms, the fundamental frequency is
f 0 ¼ 1=T 0 ¼ 1; 000 Hz;

and u 0 ¼ 2pf 0 ¼ 2; 000p rad=sec

Then, the expansion is determined as
x t ¼ 2 10
p

sin 2; 000pt 1

2 sin 4; 000pt þ 1 3 sin 6; 000pt 1 4 sin 8; 000pt þ /

Using the trigonometric identities
sin x ¼ cos x 90

and sin x ¼ cos x þ 90

and simple algebra, we finally obtain
x t ¼ 20
p

cos 2; 000pt 90

þ 10
p

cos 4; 000pt þ 90

3p cos 6; 000pt 90

þ 5

p cos 8; 000pt þ 90

þ /

þ 20

From the magnitude-phase form, we then determine f 3 and A 3 as follows:

A 3 ¼ 20 f 3 ¼ 3 f 0 ¼ 3; 000 Hz; and

3p ¼ 2:1221

b. From Table B.2 , the complex exponential form is
x t ¼ 10
jp

/ 1
3

þ 1
2

1
2

þ 1
3

e j6;000pt e j4;000pt e j2;000pt þ e j2;000pt e j4;000pt e j6;000pt þ /

From the expression, we have

FIGURE B.7
Sawtooth waveform for Example B.4.