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

Wtllw tw o 21r n oo to oo 46b t he s um o n t he

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Unformatted text preview: for any I (t) satisfying t he Dirichlet conditions mentioned on p. 194-195. T he first of these conditions i st 1.: I/(t)1 dt < 00 (4.13) 1.: Aperiodic Signal Representation by Fourier Integral DI D2 D3 D4 x2 x3 x4 X Linearity of t he Fourier Transform T he F ourier transform is linear; t hat is, if A nalogy f or F ourier t ransform. n W T= L Di i =l Consider now t he case of a continuously loaded beam, as depicted in Fig. 4.5b. In this case, although there appears t o be a load a t every point, t he load a t a nyone p oint is zero. This does n ot m ean t hat t here is no load on t he beam. A meaningful measure of load in this situation is n ot t he load a t a p oint, b ut r ather t he loading density per u nit l ength a t t hat point. Let F (x) b e t he loading density per u nit length of beam. I t t hen follows t hat t he load over a b eam l ength A x ( Ax --> 0), a t some point x, is F (x)Ax. To find t he t otal load on the beam, we divide t he b eam into segments o f interval A x ( Ax --> 0). T he load over the n th such segment of length A x is F (nAx)Ax. T he t otal load W T is given by Xn a nd WT t hen = Lllx_O~ F (nAx) A x im ' " Xl (4.14) T he p roof is t rivial a nd follows directly from Eq. (4.8a). This result can be extended t o a ny finite n umber of terms. 4 .1-1 Physical Appreciation o f t he Fourier Transform I n u nderstanding any aspect of t he Fourier transform, we should remember t hat Fourier representation is a way of expressing a signal in terms of everlasting sinusoids (or exponentials). T he Fourier spectrum of a signal indicates t he relative amplitudes a nd p hases of t he sinusoids t hat are required to synthesize t hat signal. A periodic signal Fourier s pectrum has finite amplitudes a nd exists a t discrete frequencies (wo a nd i ts multiples). Such a s pectrum is easy t o visualize, b ut t he s pectrum of a n a periodic signal is n ot easy t o visualize because i t has a continuous spectrum. T he c ontinuous s pectrum c oncept can be appreciated by considering a n analogous, more tangible phenomenon. O ne familiar example of a continuous distribution is t he loading o f a beam. Consider a b eam loaded with weights D l , D2, D 3,.·" D n u nits a t t he u niformly spaced points X l, X 2, . . . , xn, as shown in Fig. 4.5a. T he t otal load W T o n t he b eam is given by t he s um of these loads a t each of t he n points: t The r emaining Dirichlet conditions a re as follows: in any finite interval, I (t) m ay have only a finite n umber o f m axima a nd m inima a nd a finite n umber o f finite discontinuities. W hen t hese conditions a re s atisfied, t he Fourier integral o n t he r ight-hand side o f Eq. (4.8b) converges t o l (t) a t all points w here I (t) is continuous a nd converges t o t he average of t he r ight-hand a nd l eft-hand limits o f I (t) a t p oints w here I (t) is discontinuous. X II (b) F ig. 4 .5 This inequality shows t hat t he existence of t he Fourier transform is assured if condition (4....
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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