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

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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
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 =
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 9 23 Copyright © Orchard Publications
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 8

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online