Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part46

# Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part46

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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 9 21 Copyright © Orchard Publications Computation of the Z Transform with Contour Integration for , and this is the same as (9.47), Page 9 15. Example 9.2 Derive the Z transform of the discrete exponential function using the residue theo- rem. Solution: From Chapter 2, Then, by residue theorem of (9.60), for and this is the same as (9.48), Page 9 16. Example 9.3 Derive the Z transform of the discrete unit ramp function using the residue theorem. Solution: From Chapter 2, Since has a second order pole at , we need to apply the residue theorem applicable to a pole of order n . This theorem states that (9.61) Fz () sp k k lim Fs 1z 1 e sT ----------------------- s0 lim 1s 1 e == s lim 1 e 1 1 e ------------------------- lim 1 1 --------------- z z1 ----------- = = > e naT u 0 n [] L e at u 0 t 1 sa + ---------- = k k lim 1 e + lim 1sa + 1 e 1 1 e lim 1 1 e a T -------------------------- z ze a T ------------------ = > nu 0 n L tu 0 t 2 = = 1 n1 !   k k lim d ds -------------- 1 e =

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Chapter 9 Discrete Time Systems and the Z Transform 9 22 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications Thus, for this example, for , and this is the same as (9.56), Page 9 18. 9.5 Transformation Between s and z Domains It is shown in complex variables textbooks that every function of a complex variable maps (trans- forms) a plane to another plane . In this section, we will investigate the mapping of the plane of the complex variable , into the plane of the complex variable . Let us reconsider expressions (9.6) and (9.1), Pages 9 2 and 9 1 respectively, which are repeated here for convenience. (9.62) and (9.63) By comparison of (9.62) with (9.63), (9.64) Thus, the variables and are related as (9.65) and (9.66) Therefore, (9.67) Since , and are both complex variables, relation (9.67) allows the mapping (transformation) of regions of the - plane into the - plane. We find this transformation by recalling that and therefore, expressing in magnitude-phase form and using (9.65), we obtain Fz () d ds ----- s0 lim s 2 1s 2 1z 1 e sT ----------------------- = d lim 1 1 e z z1 2 ------------------ == > xy uv sz Gs fn [] n0 = e nsT = z n = = ze = = = s 1 T --- z ln = s 1 T -- z ln = = s σ j ω + = z
Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 9 23 Copyright © Orchard Publications

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## This note was uploaded on 11/20/2009 for the course EE EE 102 taught by Professor Bar during the Fall '09 term at UCLA.

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Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part46

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