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Unformatted text preview: entity 'P(t) and the
frequencydomain i dentity (Fourier spectra). T he two identities c omplement each
other; taken t ogether, t hey provide a b etter understanding o f a signal. 195 3.4 Trigonometric Fourier Series T his condition is known as t he w eak D irichlet c ondition. I f a function f (t) satisfies t he weak Dirichlet condition, t he existence of a Fourier series is g uaranteed,
b ut t he series may not converge a t every point. For example, if a function f (t) is
infinite a t some point, then obviously t he series representing t he function will be
nonconvergent a t t hat point. Similarly, if a function has a n infinite number of maxima a nd m inima in one period, t hen t he function contains a n appreciable a mount
of components of frequencies approaching infinity. Consequently, t he coefficients
in t he series a t higher frequencies do not decay rapidly, so t hat t he series will not
converge rapidly o r uniformly. Thus, for a convergent Fourier series, in addition to
condition (3.59), we require t hat
2. T he function f (t) have only a finite number of maxima and minima in one
period, a nd only a finite number of finite discontinuities in one period. These two
conditions are known as t he s trong D irichlet c onditions. We n ote here t hat any
periodic waveform t hat c an b e g enerated in a laboratory satisfies strong Dirichlet
conditions, a nd hence possesses a convergent Fourier series. Thus, a physical possibility of a periodic waveform is a valid a nd sufficient condition for the existence of
a convergent series.
• E xample 3 .4 Find the compact trigonometric Fourier series for the periodic square wave f (t) illus·
trated in Fig. 3.8a, and sketch its amplitude and phase spectra.
Here, the period To = 211' and wo = 211' ITo = 1. Therefore
f (t) = ao + L ancos n t+bnsin n t
n;::;;l where+ =..!.. ao To Series C onvergence a t J ump D iscontinuities
An interesting aspect of a Fourier series is t hat whenever there is a j ump
discontinuity i n f (t), t he series a t t he p oint of discontinuity converges t o a n average
of t he l efthand a nd r ighthand limits of f (t) a t t he i nstant of discontinuity·t In
Fig. 3.7b, for instance, 'P(t) is discontinuous a t t = 0 with 'P(O+) = 1 a nd 'P(O) =
e  rr / 2 = 0.20S. T he c orresponding Fourier series converges t o a value ( 1+0.20S)/2 =
0.604 a t t = O. T his conclusion is easily verified from Eq. 3.56b by setting t = O. r f(t) dt lTo In the above equation we may integrate f (t) over any interval of duration To = 211'. Figure
3.8a shows t hat the best choice for a region of integration is from 11' to 11'. Because
f (t) = 1 only over (~, ~) and f (t) = 0 over the remaining segment,
ao = ..!..17r/2
211' 7r/2 dt = ~ (3.60a) 2 We could have easily deduced that ao, the average value of f (t), is ~ merely by inspection
of f (t) in Fig. 3.8a. Also, E xistence o f t he Fourier Series: Dirichlet C onditions
an T here a re two basic conditions for the existence of t he Fourier series
1 . For t he series t o exist, t he coeffic...
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 Spring '13
 Bayliss
 Signal Processing, The Land

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