Signal Processing and Linear Systems-B.P.Lathi copy

Pt need b e e qual only over t hat i nterval o f to

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Unformatted text preview: entity 'P(t) and the frequency-domain 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 eft-hand a nd r ight-hand 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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