Chapter 70

A numerical method of

harmonic analysis

70.1 Introduction

Many practical waveforms can be represented by sim-
ple mathematical expressions, and, by using Fourier
series, the magnitude of their harmonic components
determined, as shown in Chapters 66 to 69. For wave-
forms not in this category, analysis may be achieved by
numerical methods.Harmonic analysisis the process
of resolving a periodic, non-sinusoidal quantity into a
series of sinusoidal components of ascending order of
frequency.

70.2 Harmonic analysis on data given

in tabular or graphical form

The Fourier coefficientsa 0 ,anandbnused in Chap-
ters 66 to 69 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 coef-

ficients cannot be determined by usingcalculus. Inthese

cases, approximate methods, such as thetrapezoidal

rule, can be used to evaluate the Fourier coefficients.

Most practical waveformstobeanalysedareperiodic.

Let the period of a waveform be 2πand be divided into

pequal parts as shown in Fig. 70.1. The width of each

f(x)

0

Period 52 

2 /p

2  x

yp



y 0 y 1 y 2 y 3 y 4

Figure 70.1