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