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

Lob in practice we may use a realizable interpolation

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Unformatted text preview: t hat t his e quation is v alid for any value o f k. T herefore, replacing k b y k + 2, we o btain lOT (e) (d) ( c) y[k] T t- I~ ~ y[k] = y{kT) a nd (B.27) T he signals f [k] a nd y[k] a re t he i nput a nd t he o utput for t he d iscrete-time s ystem G. Now, we r equire t hat yet) = T herefore, a t t = kT t (see Fig. 8.20a) dJI y(kT) = -dt = l im . =kT ~ [J(kT) - J[(k - I)T]] T -oT (8.28) 566 8 D iscrete-time S ignals a nd S ystems By using the notation in Eq. (8.27), the above equation can be expressed as T { I[k] - f[k - I]} (8.29) T he approximation improves as T approaches O. A discrete-time processor G t o realize Eq. (8.29) is shown inside t he shaded box in Fig. 8.20b. T he system in Fig. 8.20b acts as a differentiator. This example shows how a continuous-time signal can be processed by a d iscrete-time system. To determine t he sampling interval T , we note t hat t he highest frequency t hat will a ppear a t t he input is 20 kHz; t hat is, Fh = 20,000. Hence, according t o Eq. (8.17c) 1 T:'O 40,000 = 25 J1.s To gain some insight into this method of signal processing, let us consider t he differentiator in Fig. 8.20b with a ramp input f (t) = t, depicted in Fig. 8.20c. I f t he system were to act as a differentiator, t hen t he o utput yet) o f t he s ystem should b e t he unit step function u(t). L et us investigate how the system performs this particular operation and how well it achieves t he objective. T he samples of t he i nput f (t) = t a t t he interval of T seconds a ct as the input to t he discrete-time system G. These samples, denoted by a compact notation J[kJ, are, therefore, f[k] = f(t)I'=kT = tl'=kT = kT t 5 67 1. T~O T his is t he input-output relationship for G required t o achieve our objective. In practice, t he s ampling interval T c annot be zero. Assuming T t o be sufficiently small, the above e quation can be expressed as y [k] ' " E xamples o f D iscrete-Time S ystems We s how i n C hapter 11 t hat t his is g enerally t rue o f d iscrete-time s ystems. T he d iscrete-time s ystems c an b e r ealized i n t wo ways: l y [k] = lim -T { I[k] - f[k - I]} 1 8.5 2.: 0 k 2.:0 F igure 8.20d shows the sampled signal f [k]. This signal acts as a n i nput t o t he discretetime system G. Figure 8.20b shows t hat t he operation of G consists of subtracting a sample from the previous (delayed) sample and t hen multiplying t he difference with l iT. F rom Fig. 8.20d, i t is clear t hat t he difference between t he successive samples is a constant k T - (k - I)T = T for all samples, except for t he sample a t k = 0 (because there is no previous sample a t k = 0). T he o utput o f G is l iT times t he difference T , which is unity for a ll values of k, except k = 0, where it is zero. Therefore, t he o utput y[k] of G consists of samples of unit values for k 2.: 1, as illustrated in Fig. 8.20e. T he Die (discrete-time t o continuous-time) converter converts these samples into a continuous-time signal yet), as shown in Fig. 8.20f. Ideally, t he o utput should have been yet) = u (t). T his d...
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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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