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

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Unformatted text preview: e, o r realizable with infinite time delay. Therefore, in practice, there is always a n e rror in reconstructing a signal from its samples. Moreover, practical signals are n ot bandlimited, which causes a n a dditional error (aliasing error) in signal reconstruction from its samples. Aliasing error can be eliminated by bandlimiting a signal t o its effective bandwidth. T he s ampling theorem is very i mportant i n signal analysis, processing, a nd t ransmission because it allows us t o replace a continuous-time signal with a discrete sequence o f numbers. Processing a continuous-time signal is therefore equivalent to processing a discrete sequence of numbers. This leads us directly into t he a rea of digital filtering (discrete-time systems). In t he field of communication, t he t ransmission o f a c ontinuous-time message reduces t o t he t ransmission of a sequence of numbers. This opens doors t o m any new techniques of communicating continuoustime signals by pulse trains. T he d ual of t he sampling theorem s tates t hat for a signal timelimited t o T seconds, its s pectrum F(w) c an be reconstructed from t he samples of F(w) t aken a t uniform intervals n ot g reater t han I /T Hz. I n o ther words, t he s pectrum should be sampled a t a r ate n ot less t han T s amples/Hz. To compute t he d irect or inverse Fourier transform numerically, we need a relationship between t he samples of j (t) a nd F(w). T he sampling theorem a nd i ts dual provide such a quantitative relationship in t he form o f a discrete Fourier transform (DFT). T he D FT c omputations are greatly facilitated by a fast Fourier transform ( FFT) a lgorithm, which reduces t he n umber of computations from something o f t he o rder o f N J t o No log No. W e show t hat 2::: N o-l e imflok = m = 0, ±No, ±2No, . .. { No 0 o therwise k=O Recall t hat n oNa = 2rr a nd N o-l 2::: k =O e imflok References (5.43) = 1 for m = 0, ± No, ± 2No , . .. , so t hat N o-l e imflok = 2::: 1 = No for m = 0, ±No, ±2No, . .. k=O To c ompute t he s um for o ther values of m, we note t hat t he s um o n t he l eft-hand side 3mflo of Eq. (5.43) is a geometric progression w ith common ratio a = e . T herefore, (see Sec. B.7-4) N o-l e imfloNo _ " " ei mflok = e3mflO - l . ~ 1 1. Linden, D.A., "A Discussion of S ampling Theorem," Proc. I RE, vol. 47, pp 1219-1226, J uly 1959. 2. Bracewell, R.N., The Fourier Transform and Its Applications, 2nd revised ed., McGraw-Hill, New York, 1986. 3. B ennett, W.R., Introduction to Signal T ransmission, McGraw-Hill, New York, 1970. 4. L athi, B.P., L inear S ystems and Signals, Berkeley-Cambridge Press, C armichael, CA, 1992. 5. Cooley, J .W., a nd J .W., Tukey, "An A lgorithm for t he Machine Calculation of Complex Fourier Series," M athematics o f C omputation, Vol. 19, pp. 297-301, April 1965. =0 Problems k =O t Actually, where W 5 .1-1 ~ is a conservative figure because some multiplications corresponding t o t he cases No = I ,j, e tc., a...
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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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