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fs02 - 166 Fourier Series the weight of each impulse...

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the weight of each impulse function equal to the area under a pulse. As a sec ample of calculating Fourier coefficients, we consider a train of impulse tunr ni ,a"i,' Fourier series for an impulse train The Fourier series for the impulse train shown in Figure 4.10 will be calculated. Fron co=+ [x1r1e-ik.,ra1 l o J T " t r T d 2 , l l + | E1t1p-i*-,,' 47 = !r-i*.",I : + I oJ-rlz To I f,o t o This result is based on the property of the impulse function l-* rur^, - todt = f(to), 166 Fourier Series Ieq(2.a1)] provided that f(t) is continuous at t : t0. The exponential form of the Fourier series is s x(t) = 3 !,'o'"'. *--1*Tu- A line spectrum for this functionis given in Figure 4.11. Because the Fourier coeffici real, no phase plot is given. From (4.13), the c;bined trigonometric form for the trair pulse functions is given by Figure 4.10 Impulse train Figure 4.11 Frequency spectrum t, impulse train. +.fi+coska;6r.

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Sec. 4.3 Fourier Series and Frequency Spectra Note that this is also the trigonometric form. A comparison of the frequency spectrum of the square wave (Figure 4.6) with that of the train of impulse functions (Figure 4.11) illustrates an important property of impulse functions.For the squarewave, the amplitudes of the har- monics decrease by the factor l"lk,where ft is the harmonic number. Hence, we ex-
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fs02 - 166 Fourier Series the weight of each impulse...

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