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

# I t c an be shown t hat 1 s en s 1 t he m aximum

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Unformatted text preview: ential s pectra r ange from - 00 t o 0 0. By definition, the frequency of a signal is always a positive quantity. Presence of a spectral component of a negative frequency - nwo merely indicates t hat t he Fourier series contains terms of t he form e - jnwot . T he s pectra o f t he t rigonometric a nd e xponential Fourier series are closely related, a nd one can be found by t he inspection of t he o ther. T he F ourier series coefficients C n o r D n may be computed numerically using t he discrete Fourier transform (DFT), which can be implemented by a n efficient F FT (fast F ourier t ransform) algorithm. This method uses No uniform samples of f (t) over one p eriod s tarting a t t = O. I n Sec. 3.7, w e discuss a method of finding t he response of a n LTIC s ystem t o a periodic input signal. T he periodic input is expressed as an exponential Fourier series, which consists of everlasting exponentials o f t he form e jnwot. We also know t hat response of a n L TIC system t o an everlasting exponential e jnwot is H (jnwo)ejnwot. T he s ystem response is t he s um of t he s ystem's responses t o all t he e xponential components in t he Fourier series for t he i nput. T he response is, therefore, also a n e xponential Fourier series. Thus, the response is also a periodic signal of t he same period as t hat o f t he i nput. P roblems 227 1 ~_ f<t) o X<I) <a) ~ (b) 1 o F ig. P 3.1-2 1 t_ 3 .1-2 ( a) F or t he s ignals f (t) a nd x (t) d epicted in Fig. P3.1-2, find t he c omponent of t he form x ( t) c ontained i n f ( t). I n o ther w ords find t he o ptimum value o f c in t he a pproximation f (t) ~ c x(t) so t hat t he e rror s ignal energy is m inimum. ( b) F ind t he e rror s ignal e(t) a nd i ts energy E .. Show t hat t he e rror signal is orthogonal t o x (t), a nd t hat E f = c2Ex + Ee. C an y ou explain t his r esult in t erms o f vectors. 3 .1-3 F or t he s ignals f (t) a nd x (t) s hown in Fig. P3.1-2, find t he c omponent o fthe form f (t) c ontained in x (t). I n o ther words, find t he o ptimum value o f c in t he a pproximation x (t) ~ c f(t) s o t hat t he e rror signal energy is m inimum. W hat is t he e rror signal energy? 3 .1-4 R epeat P rob. 3.1-2 if x (t) is a sinusoid pulse shown in Fig. P3.1-4. x U) References L athi, B .P., M odern Digital A nd A nalog C ommunication S ystems, 2nd Ed., Holt, R inehart a nd W inston, New York, 1989. 2. Bell, E. T ., M en o f M athematics, Simon a nd Schuster, New York, 1937. 3. D urant, W ill, a nd Ariel, T he Age o f Napoleon, History of Civilization, P art XI, Simon a nd Schuster, New York, 1975. 4. Calinger, R ., 4 th ed., Classics o f M athematics, Moore P ublishing Co., Oak Park, n., 1982. 5. Lanczos, C ., D iscourse o n F ourier S eries, Oliver Boyd Ltd., London, 1966. 6 . Walker, P .L., T he Theory o f F ourier S eries a nd Integrals, Wiley-Interscience, New York, 1986. 7. Churchill, R . V., a nd J. W. Brown, F ourier S eries and B oundary Value Problems, 3 rd E d., McGraw-Hill, New York, 1978. B. Guillemin, E . A., T heory o f L inear P hysical S ystems, Wiley, Ne...
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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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