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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 iscretetime 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 discretetime systems can be analyzed as if t hey were continuoustime systems.
So far we have considered t he discretetime 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
discretetime 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 discretetime 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 continuoustime 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 continuoustime case are directly
applicable t o discretetime systems, with integrators replaced by unit delays. T he
s econd canonical realization developed in Appendix 6.1 also applies t o discretetime
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 discretetime system with transfer function H [z], e xcept
t hat t he delays in H[z] a re replaced by elements t hat delay continuoustime signals
(such as voltages or currents). I f a continuoustime impulse o(t) is applied t o such
a delay of T seconds, t he o utput will be o(t  T). T he continuoustime 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 discretetime system H[z] o n j[k] (Fig. 11.8a) are also performed
by t he corresp...
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 Spring '13
 Bayliss
 Signal Processing, The Land

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