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734 Chapter 23 Fourier series
f(t)
f(t)
f(t)
(a)
t
(b)
Figure23.13
Fourier synthesisofasawtooth waveform: (a) f (t) = 2sint;
(b)f(t) =2(sint − 1 2 sin2t + 1 3 sin3t);
(c)f(t) =2(sint − 1 2 sin2t + 1 3 sin3t − 1 4 sin4t + 1 5 sin5t).
t
(c)
t
weseethattheseriesapproachesthedesiredsawtoothwaveform.Theprocessofadding
together sinusoids to form a new periodic function is called Fourier synthesis. We see
thatthesawtoothwaveformhasbeenexpressedasaninfiniteseriesofharmonicwaves,
sint beingthefundamentalorfirstharmonic,andtherestbeingwaveswithfrequencies
thatareintegermultiplesofthefundamentalfrequency.Thisinfiniteseriesiscalledthe
Fourier series representation of f (t) and what we have succeeded in doing is to break
down f (t) into its component harmonic waveforms. In this example, only sine waves
were required to construct the function. More generally we shall need both sine and
cosine waves.
Suppose the function f (t) is defined in the interval 0 < t < T and is periodic with
periodT. Then, under certainconditions, its Fourier series isgiven by
f(t)= a 0
2 + ∞
∑
orequivalently
n=1
(
a n
cos 2nπt
T
+b n
sin 2nπt )
T
f(t)= a ∞
0
2 + ∑
(a n
cosnωt +b n
sinnωt)
n=1
(23.2)
where a n
and b n
are constants called the Fourier coefficients. These are given by the
formulae
a 0
= 2 T
a n
= 2 T
b n
= 2 T
∫ T
0
∫ T
0
∫ T
0
f(t)dt (23.3)
f(t)cos 2nπt
T
f(t)sin 2nπt
T
dt fornapositive integer (23.4)
dt fornapositive integer (23.5)
The term a 0
represents the mean value or d.c. component of the waveform (see Section15.2).ThederivationoftheseformulaeappearsinExample23.16.Itisimportantto
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