Higher Engineering Mathematics

L

Fourier series

73

A numerical method of harmonic

analysis

73.1 Introduction

Many practical waveforms can be represented by
simple mathematical expressions, and, by using
Fourier series, the magnitude of their harmonic com-
ponents determined, as shown in Chapters 69 to 72.
For waveforms not in this category, analysis may be
achieved by numerical methods.Harmonic analysis
is the process of resolving a periodic, non-sinusoidal
quantity into a series of sinusoidal components of
ascending order of frequency.

73.2 Harmonic analysis on data given

in tabular or graphical form

The Fourier coefficients a 0 , an and bn used in
Chapters 69 to 72 all require functions to be
integrated, i.e.

a 0 =

1

2 π

∫π

−π

f(x)dx=

1

2 π

∫ 2 π

0

f(x)dx

= mean value off(x)

in the range−πtoπor 0 to 2π

an=

1

π

∫π

−π

f(x) cosnxdx

=

1

π

∫ 2 π

0

f(x) cosnxdx

=twice the mean value off(x) cosnx

in the range 0 to 2π

bn=

1

π

∫π

−π

f(x) sinnxdx

=

1

π

∫ 2 π

0

f(x) sinnxdx

=twice the mean value off(x) sinnx

in the range 0 to 2π

However, irregular waveforms are not usually

defined by mathematical expressions and thus the

Fourier coefficients cannot be determined by using

calculus. In these cases, approximate methods, such

as thetrapezoidal rule, can be used to evaluate the

Fourier coefficients.

Most practical waveforms to be analysed are

periodic. Let the period of a waveform be 2πand

be divided intopequal parts as shown in Fig. 73.1.

The width of each interval is thus

2 π

p

. Let the ordi-

nates be labelledy 0 ,y 1 ,y 2 ,...yp(note thaty 0 =yp).

The trapezoidal rule states:

Area=(width of interval)

[

1

2

(first+last ordinate)

+sum of remaining ordinates

]

≈

2 π

p

[

1

2

(y 0 +yp)+y 1 +y 2 +y 3 +···

]

Sincey 0 =yp, then

1

2

(y 0 +yp)=y 0 =yp

Hence area≈

2 π

p

∑p

k= 1

yk

f(x)

y 0 y 1 y 2 y 3 y 4

yp

0 π 2 π x

2 π/p

Period = 2 π

Figure 73.1