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

Z y z 5 z 2 z 2 z 3 c omment a nd e

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Unformatted text preview: s um of t he four inputs to t hat summer. Therefore so t hat (11.41) Moreover, Y[z], t he o utput of the second summer, is e qual t o t he s um o ffour signals t o t hat summer. Therefore C onnection between t he l aplace and t he Z- Transform We now show t hat d iscrete-time systems also c an b e analyzed using t he Laplace transform. In fact, we shall see t hat t he z -transform is the Laplace t ransform in disguise a nd t hat discrete-time systems can be analyzed as if t hey were continuoustime systems. So far we have considered t he discrete-time signal as a sequence of numbers a nd n ot as a n electrical signal (voltage o r c urrent). Similarly, we have considered a discrete-time system as a mechanism t hat processes a sequence o f n umbers (input) t o yield a nother sequence o f numbers (output). T he s ystem was built by using delays (along with adders a nd multipliers) t hat delay sequences o f numbers, n ot electrical signals (voltages o r c urrents). A digital computer is a perfect example: every signal is a sequence o f numbers, a nd t he processing involves delaying sequences of numbers (along with addition a nd m ultiplication). Consider a discrete-time system with transfer function H[z] a nd a n i nput j [k], as shown in Fig. 11.8a. We <:,an t hink o f (or generate, for t hat m atter) a corresponding continuous-time signal j (t) consisting of impulses spaced T seconds a part. L et t he k th impulse o f s trength b e j[k] as depicted in Fig. 11.8b. T hus 00 i (t) = (11.42) F rom Eqs. (11.41) a nd (11.42), i t follows t hat Y[z] F[z] b3Z3 + b2Z2 + bIZ + bo z3 + a2z2 + a iz + ao T his result shows t hat Fig. 11.7 is indeed a realization of H[z] in Eq. (11.40). Similarly, the cascade a nd parallel realizations of t he continuous-time case are directly applicable t o discrete-time systems, with integrators replaced by unit delays. T he s econd canonical realization developed in Appendix 6.1 also applies t o discrete-time c ase w ith l /s replaced by l /z. V' L j[k]o(t - kT) (11.43) k=O F igure 11.8 shows j[k] a nd corresponding J(t). L et us now consider a system identical in s tructure t o t he discrete-time system with transfer function H [z], e xcept t hat t he delays in H[z] a re replaced by elements t hat delay continuous-time signals (such as voltages or currents). I f a continuous-time impulse o(t) is applied t o such a delay of T seconds, t he o utput will be o(t - T). T he continuous-time transfer function of such a delay is e - sT [see Eq. (6.54)]. Hence, t he delay elements with transfer function 1 / z in t he realization of H [z] will be replaced by t he delay elements with transfer function e - sT in t he realization of t he corresponding i I (s). T his s tep is t he s ame as z being replaced by esT. Therefore, t he t ransfer function of this system is H[z] w ith z replaced by esT. T hus i I(s) = H[e sT ]. Now whatever operations are performed by t he discrete-time system H[z] o n j[k] (Fig. 11.8a) are also performed by t he corresp...
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