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

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Unformatted text preview: )e jOk dn y[k] a s a s um o f responses t o all i nput c omponents 640 lO Fourier Analysis o f Discrete- Time Signals which is precisely t he r elationship in Eq. (lO.57). Thus, F (n) is t he i nput s pectrum a nd Y (n), t he o utput s pectrum (of t he e xponential components), is F (n)H(n) . E xample 1 0.7 For a system with unit impulse response h[k] = (0.5)ku[k], determine the (zero-state) response Y[k] for the input J[k] = (0.8)ku[k]. According t o Eq. (10.57) 10.6 Signal processing U sing D FT a nd F FT 641 similar to what we did in the continuous-time case by generalizing the frequency variable j w to s = ( 7 + j w (from Fourier to Laplace transform) . • • Y (n) = F (n)H(n) From the results in Eq. (10.37), we obtain F (n) _ 1 (10.60) - 1- 0.8e-in Also, H (n) is the D TFT of (0.5)ku[k], which is obtained from Eq. (10.37) by substituting = 0.5: -y in 1 e - 1 - 0.5e-in = e in _ 0.5 H (n) _ (10.61) Therefore e i2n Y (n) = ( ein _ 0.5)(e in _ 0.8) 1 0.6 Signal processing by OFT and FFT I n t his s ection, we use D FT (developed i n C hapter 5) a s a t ool, which allows us t o utilize a digital c omputer for digital signal processing. T his s ignal processing includes s pectral a nalysis o f d igital signals a nd LTID system analysis. B y s pectral analysis, we m ean d etermining t he d iscrete t ime F ourier series ( DTFS) o f periodic signals a nd d etermining F (n) from I[k) ( and vice versa) for aperiodic signals. As a tool for LTID system analysis, D FT c an b e u sed as a software oriented solution t o d igital filtering. D FT c an b e i mplemented on a digital c omputer b y a n efficient algorithm, t he f ast Fourier transform ( FFT) also discussed i n C hapter 5. T he D FT ( using F FT) is t ruly t he workhorse o f m odern digital signal processing. 10.6-1 Computation o f Discrete-Time Fourier Series (DTFS) T he d iscrete-time Fourier series (DTFS) equations (10.8) a nd (lO.9) are identical t o t he D FT e quations (5.18b) a nd (5.18a) within a scaling c onstant No. I fwe l et I[k) = Nolk a nd V r = Fr i n Eqs. (lO.9) a nd (lO.8), we o btain We can express the right-hand side as a sum of two first-order terms (modified partial fraction expansion as discussed in Sec. B.5-5) as follows:t Y (n) ein N o-l Fr - dn = " I k e - jrflok ~ k =O (ein - 0.5)(e in - 0.8) 5 =~+ ejn - 0.5 no 8 _ _3 _ _ ein - 0.8 Consequently, 211" No =;- T his is precisely t he D FT p air i n Eqs. (5.18). For instance, t o c ompute t he D TFS for t he p eriodic signal in Fig. lO.2a, we use t he values of l k = I[k)/No as 1 lk = { 032 0 ::; k ::; 4 5::; a nd 2 8::; k ::; 31 k ::; 27 We use these values in t he F FT a lgorithm discussed i n Sec. 5.2-2 t o o btain F r , which is t he s ame as V r • According to Eq. (10.37), the inverse D TFT of this equation is This example demonstrates the procedure of determining an LTID system response using DTFT. I t is similar to the method of Fourier transform in analysis of LTIC systems. As in the case of Fourier transform, this method can be used only...
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