ABSTRACT
The Fourier coefficients a0, an and bn used in Chapters 101 to 104 all require functions to be integrated, i.e.
a0 = 12π ∫ π −π
f (x)dx = 1 2π
f (x)dx
= mean value of f (x) in the range−π to π or0 to2π
an =
= 1 π
f (x)cosnx dx = twice the mean value of f (x)cosnx in the range
0 to 2π
bn = 1 π
f (x)sinnx dx
= 1 π
f (x)sinnx dx = twice the mean value of f (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 the trapezoidal rule, can be used to evaluate the Fourier coefficients. Most practical waveforms to be analysed are periodic. Let the period of a waveform be 2π and be divided into p equal parts, as shown in Figure 105.1. The width of each interval is thus
2π p
. Let the ordinates be labelled y0, y1, y2, . . . yp (note that y0 = yp). The trapezoidal rule states:
Area = (width of interval) [ 1 2 (first+ last ordinate)
+ sum of remaining ordinates ]
≈ 2π p
[ 1 2 (y0 + yp)+ y1 + y2 + y3 + ·· ·
]
Since y0 = yp, then 12 (y0 + yp) = y0 = yp
Hence area ≈ 2π p
Mean value
≈ 1 2π
( 2π p
yk ≈ 1p p∑
However,a0 = mean value of f (x) in the range 0 to 2π .
