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Unformatted text preview: de t his interval,
t he Fourier series repeats periodically w ith p eriod To.
Now, if t he f unction f (t) were itself t o b e periodic w ith p eriod To, t hen a
Fourier series representing f (t) over a n i nterval To will also represent f (t) for all t
(not j ust over t he i nterval To). A nother i nteresting fact, as seen in Fig. 1.7, is t hat a
periodic signal f (t) c an b e generated by a periodic r epetition of any of its segment
of d uration To. T herefore, t he t rigonometric Fourier series representing a segment
of f (t) o f d uration To s tarting a t a ny i nstant r epresents f (t) for all t. Therefore, it
follows t hat i n c omputing t he coefficients ao, an a nd bn , we may use any value for
t l i n Eqs. (3.51). I n o ther words, we may perform this integration over any interval
o f To. T hus t he F ourier coefficients o f a series representing a periodic signal f (t)
(for all t) c an b e expressed as T able 3.1 On + 2n7r) + On] n =l = 0.504 + 0.244 cos (2t - 75.96°) + 0.125 cos (4t - 82.87°) Cn cos [(nwot 00 n =l 1 n n =l T he v alues of C n a nd On for t he dc a nd t he first seven harmonics are computed from
t he above e quation a nd displayed in Table 3.1. Using these numerical values, we can
express f (t) i n t he c ompact trigonometric Fourier series as 0 LC 10 F ig. 3 .7 A periodic signal a nd i ts Fourier spectra. n for all t n =l 0.244 I 1 1 LJ
f (t) = 0.504 + 0.504 L Cn cos (nwot+ On) =;'°lTD f(t) dt
r (3.58a) t In r eality, t he s eries convergence a t t he p oints o f d iscontinuity s hows a bout 9% o vershoot ( Gibbs
p henomenon6 ) a s d iscussed l ater . 3 Signal Representation by Orthogonal Sets 194 an = ~ bn = ~ To a nd where ITo To r f(t) cos nwotdt n = 1 ,2,3, . .. (3.5Sb) r f(t) sin nwotdt n = 1 ,2,3, . .. (3.5Sc) lTo lTo m eans t hat t he integration is performed over any interval of To seconds. T he Fourier S pectrum
T he c ompact t rigonometric Fourier series in Eq. (3.54) indicates t hat a periodic
signal f (t) can be expressed as a sum of sinusoids of frequencies 0 (dc), wo, 2wo, . .. ,
nwo, . .. , whose amplitudes are Co, C l, C2, . .. , Cn, " ', a nd whose phases are 0, Ih,
I h, . .. , On, . .. , respectively. We can readily plot amplitude C n vs. w ( amplitude
s pectrum) a nd On vs. w ( phase s pectrum). T hese two plots together are the
f requency s pectra of f (t).
Figures 3.7c a nd 3.7d show t he a mplitude a nd phase s pectra for t he periodic
signal 'P(t) in Fig. 3.7b. These spectra tell us a t a glance the frequency composition
of 'P(t); t hat is, t he a mplitudes a nd phases of various sinusoidal components of
'P(t). K nowing the frequency spectra, we can reconstruct or synthesize 'P(t), as
shown on t he r ight-hand side of Eq. (3.56). Therefore, t he frequency spectra in
Figs. 3.7c a nd 3.7d provide a n a lternative d escription-the f requency-domain
d escription o f 'P(t). T he time-domain description of 'P(t) is d epicted in Fig. 3.7b.
A signal, therefore, h as a dual i dentity: the time-domain id...
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