E1C02 09/14/2010
13:35:18 Page 56
Equation 2.12 is integrated from Àp to p:
ð p
Àp
y t
ð Þdt ¼ A 0
ð p
Àp
dt þ
X 1
n¼1
A n
ð p
Àp
cos ntdt þ B n
ð p
Àp
sin ntdt
ð2:13Þ
Since
ð p
Àp
cos ntdt ¼ 0 and
ð p
Àp
sin ntdt ¼ 0
Equation 2.13 yields
A 0 ¼
1
2p
ð p
Àp
y t
ð Þdt
ð2:14Þ
The coefficient A m may be determined by multiplying Equation 2.12 by cos mt and integrating from
Àp to p. The resulting expression for A m is
A m ¼
1
p
ð p
Àp
y t
ð Þcos mtdt
ð2:15Þ
Similarly, multiplying Equation 2.12 by sin mt and integrating from Àp to p yields B m . Thus,
for a function y(t) with a period 2p, the coefficients of the trigonometric series representing y(t) are
given by the Euler formulas:
A 0 ¼
1
2p
ð p
Àp
y t
ð Þ
A n ¼
1
p
ð p
Àp
y t
ð Þcos ntdt
B n ¼
1
p
ð p
Àp
y t
ð Þsin ntdt
ð2:16Þ
The trigonometric series corresponding to y(t) is called the Fourier series for y(t), and the
coefficients A n and B n are called the Fourier coefficients of y(t). In the series for y(t) in Equation
2.12, when n ¼ 1 the corresponding terms in the Fourier series are called fundamental and have the
lowest frequency in the series. The fundamental frequency for this Fourier series is v ¼ 2p=2p ¼ 1.
Frequencies corresponding to n ¼ 2, 3, 4, . . . are known as harmonics, with, for example, n ¼ 2
representing the second harmonic.
Functions represented by a Fourier series normally do not have a period of 2p. However, the
transition to an arbitrary period can be affected by a change of scale, yielding a new set of Euler
formulas described below.
Fourier Coefficients for Functions Having Arbitrary Periods
The coefficients of a trigonometric series representing a function having an arbitrary period T are
given by the Euler formulas:
56 Chapter 2 Static and Dynamic Characteristics of Signals