Review of Analog Signal Processing Basics - Digital Signal Processing: Fundamentals and Applications - page 777

where c n represents the complex Fourier coefficients, which may be found from

Z

x t e jnu 0 t dt

1
T 0

c n ¼ (B.10)

T 0

The relationship between the complex Fourier coefficients c n and the coefficients of the sine-cosine
form are

c 0 ¼ a 0 (B.11)

c n ¼ a n jb n
2

; for n > 0 (B.12)

Considering a real signal x ( t )( x ( t ) is not a complex function) in Equation (B.10), c n is equal to the
complex conjugate of c n , that is, c n . It follows that
c n ¼ c n ¼ a n þ jb n
2

; for n > 0 (B.13)

Since c n is a complex value that can be written in the magnitude-phase form, we obtain
c n ¼ jc n j : f n (B.14)

where jc n j is the magnitude and f n is the phase of the complex Fourier coefficient. Similar to the
magnitude-phase form, we can create the spectral plots for jc n j and f n . Since the frequency index n
goes from N to N , the plots of the resultant spectra are two-sided.

EXAMPLE B.1
Consider the square waveform x(t) shown in Figure B.1 ,whereT 0 represents a period. Find the Fourier series
expansions in terms of (a) the sine-cosine form, (b) the amplitude-phase form, and (c) the complex exponential form.
Solution:
From Figure B.1 , we notice that T 0 ¼ 1 second and A ¼ 10. The fundamental frequency is
f 0 ¼ 1=T 0 ¼ 1Hz or u 0 ¼ 2p f 0 ¼ 2p rad=sec
a. Using Equations (B.1) to (B.3) yields

T 0 =2

Z 0:25

x t dt ¼ 1
1

a 0 ¼ T 0 10dt ¼ 5

T 0 =2 0:25

T 0 =2

Z

a n ¼ T 0 xðtÞ cos ðnu 0 tÞdt

T 0 =2

0:25

¼ 2
1

10 cos ðn2ptÞdt

0:25

0:25
0:25 ¼ 10 sin ð0:5pnÞ

10 sin ð n 2p t Þ
n2p

¼ 2
1

0:5pn