Review of Analog Signal Processing Basics - Digital Signal Processing: Fundamentals and Applications - page 778

FIGURE B.1

Square waveform in Example B.1.

T 0 =2

Z

b n ¼ T 0 xðtÞ sin ðnu 0 tÞdt

T 0 =2

0:25

¼ 2
1

10 sin ðn2ptÞdt

0:25

0:25
0:25 ¼ 0

10 cos ð n 2p t Þ
n2p

¼ 2
1

Thus, the Fourier series expansion in terms of the sine-cosine form is written as
x t ¼ 5 þ

N

cos n2pt

10 sin ð0:5p n Þ
0:5pn

n ¼1

7p cos 14pt þ /
b. Making use of the relations between the sine-cosine form and the amplitude-phase form, we obtain
A 0 ¼ a 0 ¼ 5

cos 2pt 20

3p cos 6pt þ 4 p cos 10pt 20

¼ 5 þ 20
p

q

a n þ b n

sin ð0:5pnÞ
0:5pn

A n ¼ ¼ ja n j ¼ 10

Again, noting that cos ðxÞ¼cos ðx þ 180

Þ, the Fourier series expansion in terms of the amplitude-phase form

is

x t ¼ 5 þ 20
p

cos 2pt þ 20

3p cos 6pt þ 180

þ 4

p cos 10pt þ 20 7p cos 14pt þ 180

þ /

c. First let us find the complex Fourier coefficients using the formula, that is,

T 0 =2

Z

x t e jnu 0 t dt c n ¼ T 0

T 0 =2

0:25

¼ 1
1

Ae jn2pt dt

0:25

e j0:5pn

e j0:5pn
jn2p

0:25
0:25 ¼ 10

e jn2pt
jn2p

¼ 10