Parallel Derivations of the Z and Laplace Transforms

MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.161 Signal Processing - Continuous and Discrete Parallel Derivations of the Z and Laplace Transforms 1 The following is a summary of the derivation of the Laplace and Z transforms from the continuousand discrete-time Fourier transforms: The Laplace Transform The Z transform (1) We begin with causal f (t) and find its Fourier transform (Note that because f (t) is causal, the integral has limits of 0 and ∞): (1) We sample f(t) at intervals ΔT to produce f ∗ (t). We take its Fourier transform (and use the sifting property of δ(t)) to produce F (jΩ) = � ∞ f (t)e−jΩt dt 0 F ∗ (jΩ) = ∞ � fn e−jnΩΔT n=0 (2) We note that for some functions f (t) (for ex­ ample the unit step function), the Fourier integral does not converge. (2) We note that for some sequences fn (for exam­ ple the unit step sequence), the summation does not converge. (3) We introduce a weighted function (3) We introduce a weighted sequence � w(t) = f (t)e−σt {wn } = fn r−n and note � and note lim {wn } = {fn } lim w(t) = f (t) σ →0 r→1 The effect of the exponential weighting by e−σt is to allow convergence of the integral for a much broader range of functions f (t). The effect of the exponential weighting by r−n is to allow convergence of the summation for a much broader range of sequences fn . (4) We take the Fourier transform of w(t) (4) We take the Fourier transform of wn W (jΩ) = F̃ (jΩ|σ) = = � ∞� �0∞ 0 � f (t)e−σt e−jΩt dt f (t)e−(σ+jΩ) dt W ∗ (jΩ) = F̃ ∗ (jΩ|r) = = ∞ � � n=0 ∞ � � fn r−n e−jnΩΔT � fn rejΩΔT �−n n=0 and define the complex variable s = σ + jΩ so that we can write F (s) = F̃ (jω|σ) = � ∞ 0 f (t)e−st dt F (z) = F̃ ∗ (jΩ|r) = ∞ � fn z −n n=0 F (s) is the one-sided Laplace Transform. Note that the Laplace variable s = σ + jΩ is expressed in Cartesian form. 1 and define the complex variable z = rejΩΔT so that we can write D. Rowell October 22, 2008 1 F (z) is the one-sided Z-transform. Note that z = rejΩΔT is expressed in polar form. The Laplace Transform (contd.) The Z transform (contd.) (5) For a causal function f (t), the region of con­ vergence (ROC) includes the s-plane to the right of all poles of F (jΩ). (5) For a right-sided (causal) sequence {fn } the region of convergence (ROC) includes the z-plane at a radius greater than all of the poles of F (z). jW Á { z } s - p la n e : z - p la n e R O C s : R O C : : -1 .0 1 .0  { z } u n it c ir c le We note that the mapping between the s plane and the z plane is given by z = esΔT and that the imaginary axis (s = jΩ) in the s-plane maps to the unit circle (z = ejΩΔT ) in the z-plane. Furthermore we note that the mapping of the unit circle in the z-plane to the imaginary axis in the s-plane is periodic with period 2π, and that the mapping of the jΩ axis to the unit circle produces aliasing for |Ω| > π/ΔT . If we define a normalized discrete-time frequency that is independent of ΔT , that is ω≤π ω = ΩΔT we can make the following comparisons: The Laplace Transform (contd.) The Z transform (contd.) (6) If the ROC includes the imaginary axis, the FT of f (t) is F (jΩ): (6) If the ROC includes the unit circle, the DFT of {fn }, n = 0, 1, . . . , N − 1. is {Fm } where F (jΩ) = F (s) |s=jΩ Fm = F (z) |z=ejωm = F (ejωm ), where ωm = 2πm/N for m = 0, 1, . . . , N − 1. (7) The convolution theorem states (7) The convolution theorem states f (t) ⊗ g(t) = � ∞ −∞ L f (τ )g(t − τ )dτ ⇐⇒ F (s)G(s) (8) For an LTI system with transfer function H(s), the frequency response is {fn } ⊗ {gn } = ∞ � m=−∞ (8) For a discrete LSI system with transfer func­ tion H(z), the frequency response is H(z) |z=ejω = H(ejω ) H(s) |s=jΩ = H(jΩ) if the ROC includes the imaginary axis. Z fm gn−m ⇐⇒ F (z)G(z) |ω| ≤ π if the ROC includes the unit circle. 2 The Laplace Transform (contd.) The Z transform (contd.) (9) The transfer function (9) The transfer function bm sm + bm−1 sm−1 + . . . + b1 s + b0 an sn + an−1 sn−1 + . . . + a1 s + a0 H(s) = is derived from the ordinary differential equation an dn y dn−1 y dy + an−1 n−1 . . . + a1 + a0 y n dt dt dt dm f df = bm m + . . . + b1 + b0 f dt dt H(z) = bm z −m + bm−1 z −(m−1) + . . . + b1 z −1 + b0 an z −n + an−1 z −(n−1) + . . . + a1 z −1 + a0 is derived from the difference equation a0 yk + a1 yk−1 + . . . + an−1 yk−(n−1) + a0 yk−n = b0 fk + b1 fk−1 + . . . + bm fk−m (10) Poles of H(s) in the rh-plane indicate insta­ bility in the continuous-time system. (10) Poles of H(z) outside the unit circle indicate instability in the discrete-time system. (11) The frequency response H(jω) may be in­ terpreted geometrically from the poles and zeros of H(s) according to the following diagram: (11) The frequency response H(jω), (ω = Ω/ΔT ) may be interpreted geometrically from the poles and zeros of H(z) according to the following dia­ gram: jW q w r1 x q 2 j r2 2 x 1 q j q Á { z } s - p la n e o 1 s q p 2 j x n � H(jΩ) = i=1 φi − q q j o 1 0 1  { z } 2 then H(ejω ) = H(z)|z=ejω and �m qi = K �ni=1 i=1 ri n � 1 r2 2 2 |H(jΩ)| W r1 x q 1 then H(jΩo ) = H(s)|s=jΩo and  z - p la n e u n it c ir c le � � � � �H(ejω )�  θi = H(ejω ) = i=1 �m qi K �ni=1 n � i=1 F i=1 ri n � φi − θi i=1 Z (12) If f (t) ⇐⇒ F (s) then (12) If {fn } ⇐⇒ F (z) then F Z f (t − τ ) ⇐⇒ e−sτ F (s), {fn−m } ⇐⇒ z −m F (z), which is the delay property of the Laplace trans­ form. which is the delay (shift) property of the Z trans­ form. 3