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

For instance if we wish t o r epresent f t t over a n

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Unformatted text preview: 3.4-3, find t he e xponential F ourier series a nd s ketch t he c orresponding s pectra. 3 .5-2 I T he t rigonometric F ourier series of a c ertain p eriodic signal is given by f (t) = 3 + V 3cos 2 t + s in 2 t + s in 3 t - ~ cos (5t + %) ( a) S ketch t he t rigonometric F ourier s pectra. ( b) B y i nspection o f t he s pectra i n p art a , s ketch t he e xponential F ourier series s pectra. ( c) B y i nspection of t he s pectra i n p art b , w rite t he e xponential F ourier series for f (t). H int: To express t he F ourier series in c ompact form, combine t he s ine a nd cosine t erms o f t he s ame frequency. Moreover, all t erms m ust a ppear i n t he cosine form w ith p ositive a mplitudes. T his c an a lways b e d one by s uitably a djusting t he p hase. F ig. P 3.4-11. F igure P 3.4-11 s hows t he first eight functions in this set. R epresent f (t) in Fig. P3.411 o ver t he i nterval [0, 1J u sing a W alsh F ourier series using t hese 8 basis functions. C ompute t he e nergy of e(t), t he e rror i n t he a pproximation u sing t he first N n on-zero t erms i n t he s eries for N = 1, 2, 3 a nd 4. How does t he W alsh series c ompare w ith t he t rigonometric s eries in P rob. 3.4-10 from t he v iewpoint of t he e rror e nergy for a given N ? 3 .4-12 A s et o f L egendre p olynomials Pn(t), (n = 0 ,1,2,3,···) forms a complete s et o f orthogonal f unctions over t he i nterval - 1 < t < 1. These polynomials a re defined by 3 .5-3 T he e xponential F ourier series o f a c ertain p eriodic function is given as f (t) = (2 + j 2)e- j3t + j 2e- jt + 3 - j2e j ' + (2 - j2)e j3 ' ( a) S ketch t he e xponential F ourier s pectra. ( b) B y i nspection o f t he s pectra i n p art a , s ketch t he t rigonometric F ourier s pectra for f (t) . ( c) F ind t he c ompact t rigonometric F ourier series from t hese s pectra. ( d) F ind t he s ignal b andwidth. 3 .5-4 I f a p eriodic signal f (t) is expressed a s a n e xponential F ourier series n 1 d (2 )n Pn ( t ) = n! 2n dtn t - 1 n = 0, 1 ,2,··· ( a) Show t hat t he e xponential F ourier series for j (t) = f (t - T) is given b y T hus Po(t) = 1, 1 =t 1 3 P3(t) = 2 (5t - j (t) P1(t) 2 P2(t) = 2 (3t - 1), 3t) e tc. in which = I: Dnejnwo' 2 34 S ignal R epresentation b y O rthogonal S ets 3 This result shows t hat time shifting of a periodic signal by T seconds merely changes the phase spectrum by nwoT. T he amplitude spectrum is unchanged. ( b) Show t hat the exponential Fourier series for j ( t ) = j (at) is given by L 00 j(t) = D nejn(awo)t This result shows t hat time compression of a periodic signal by a factor a expands its Fourier s pectra by the same factor a . Similarly, time expansion of a periodic signal by a factor a compresses its Fourier spectra by the factor a. 3 .5-5 ( a) T he Fourier series for t he periodic signal in Fig. 3.lOa is given in Exercise E3.6. Verify Parseval's theorem for this series, given t hat ( b) I f j (t) i s approximated by the first N terms in this series, find N so t hat t he power of t he error signal is less t ha...
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