Review of Analog Signal Processing Basics - Digital Signal Processing: Fundamentals and Applications - page 775

Appendix B: Review of Analog Signal
Processing Basics

B.1 FOURIER SERIES AND FOURIER TRANSFORM
Electronics applications require familiarity with some periodic signals such the square wave, rect-
angular wave, triangular wave, sinusoid, sawtooth wave, and so on. These periodic signals can be
analyzed in the frequency domain with the help of the Fourier series expansion. According to Fourier
theory, a periodic signal can be represented by a Fourier series that contains the sum of a series of sine
and/or cosine functions (harmonics) plus a direct current (DC) term. There are three forms of Fourier
series: (1) sine-cosine, (2) amplitude-phase, and (3) complex exponential. We will review each of them
individually in the following text. Comprehensive treatments can be found in Ambardar (1999),
Soliman and Srinath (1998), and Stanley (2003).

B.1.1 Sine-Cosine Form
The Fourier series expansion of a periodic signal xðtÞ with a period of T via the sine-cosine form is
given by

N
n¼ 1 a n cos ðnu 0 tÞþ

N
n¼ 1 b n sin ðnu 0 tÞ

x t ¼ a 0 þ

(B.1)

where u 0 ¼ 2 p=T 0 is the fundamental angular frequency in radians per second, while the funda-
mental frequency in terms of Hz is f 0 ¼ 1 =T 0 . The Fourier coefficients of a 0 , a n , and b n may be found
according to the following integral equations:

Z

1
T 0

a 0 ¼ xðtÞdt (B.2)

T 0

Z

2
T 0

a n ¼ xðtÞ cos ðnu 0 tÞdt (B.3)

T 0

Z

2
T 0

b n ¼ xðtÞ sin ðnu 0 tÞdt (B.4)

T 0

Notice that the integral is performed over one period of the signal to be expanded. From Equation
(B.1) , the signal xðtÞ consists of a DC term and sums of sine and cosine functions with their corre-
sponding harmonic frequencies. Again, note that nu 0 is the n th harmonic frequency.
http://dx.doi.org/10.1016/B978-0-12-415893-1.15002-4