Topic 10: The Z Transform o Introduction to Z Transform o - PowerPoint PPT Presentation

ELEC361: Signals And Systems Topic 10: The Z Transform o Introduction to Z Transform o Relationship to the Fourier transform o Z Transform and Examples o Region of Convergence of the Z Transform o Inverse Z Transform and Examples o Properties of Z Transform and Examples o Analysis and characterization of LTI systems using z-transforms Dr. Aishy Amer o Geometric evaluation of the Fourier transform from the pole-zero plot Concordia University Electrical and Computer Engineering o Summary Figures and examples in these course slides are taken from the following sources: • A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 • M.J. Roberts, Signals and Systems, McGraw Hill, 2004 1 • J. McClellan, R. Schafer, M. Yoder, Signal Processing First, Prentice Hall, 2003

The Z Transform � The Laplace transform: an extension of the continuous-time Fourier transform � Advantage: to perform analysis of continuous-time signals & systems whose Fourier transform does not exist � The z − transform: an extension of the discrete-time Fourier transform � Let h[n] be the impulse response of a LTI system � The response of this system to a complex exponential input of the form z n is = n x [ n ] z ; � The expression H(z) is referred to as the z − transform of h[n] where z is a complex variable, z = re j ω Since z is a complex quantity, H(z) is a complex function � (Why do we deal with complex signals? They are often analytically simpler � to deal with than real signals.) 2

The Z Transform The Fourier transform of h[n] can be obtained by evaluating the z − transform at z = � e j ω with ω real ZT: � z = re j ω � � ω real � H(z) is defined for a region in z – called the ROC- for which the sum exists Since the Z-Transform is a power series, it converge when h[n] z -n is absolutely � ∞ summable, i.e., ∑ − < ∞ n | h [ n ] z | = −∞ n DT-FT � � z = e j ω (r=1) � ω real Recall: � h[n] is the impulse response of an LTI system � H(e i ω ) is the frequency response � H(z) is the transfer function � 3

The Z Transform: Rational function/ Poles and Zeros � The Z-transform will have the below structure, based on rational Functions: P ( z ) = H ( z ) Q ( z ) � For any two polynomials A and B, their ratio is called a rational function + − + + − − + − 1 M 2 L b b z b z z 4 ( z 2 )( z 2 ) = = = 0 1 M H ( z ) H ( z ) − − + + + + − + − 1 N 2 L a a z a z 2 z z 3 ( 2 z 3 )( z 1 ) 0 1 N � The numerator and denominator can be polynomials of any order � The rational function is undefined when the denominator equals zero, i.e., we have a discontinuity in the function � The z − transform is characterized by its zeros and poles � Zeros: The value(s) for z where P (z) = 0, i.e., the complex frequencies that make the transfer function zero � Poles: T he value(s) for z where Q(z) = 0, i.e., the complex frequencies that make the transfer function infinite 4

The Z Transform: The Z plane (complex plane) � The z-plane is a complex plane with an imaginary and real axis referring to the complex- valued variable z � Once the poles and zeros are found for the z transform, they can be plotted into the z plane � The position on the complex plane is given in a polar form by re j ω � ω : the angle from the positive real axis around the plane � H(z) is defined everywhere on this plane � H(e j ω ) on the other hand is defined only where |z| = 1 which is referred to as the unit circle This is useful because by representing the Fourier � transform as the z-transform on the unit circle, the periodicity of Fourier transform is easily seen 5

The Z Transform: The Z plane (complex clane) • Poles are denoted by “x” and zeros by “o” • We use shaded regions to indicates the Region of Convergence (ROC) for the z transform 6

The Z Transform: The Z plane (complex plane) − + ( z i )( z i ) = H ( z ) 1 1 1 1 + − − − + ( z 1 )( z ( i ))( z ( i )) 2 2 2 2 The zeros are : {i, - i} 1 1 1 1 − + The poles are : {-1, i , i } 2 2 2 2 In MATLAB you can easily create pole/zero � plots, e.g., % Set up vector for zeros z = [j ; -j]; % Set up vector for poles p = [-1 ; .5+.5j ; .5-.5j]; figure(1); zplane(z,p); title('Pole/Zero Plot'); 7

Outline � Introduction to Z Transform � Relationship to the Fourier transform � Z Transform and Examples � Region of Convergence of the Z Transform � Inverse Z Transform and Examples � Properties of Z Transform and Examples � Analysis and characterization of LTI systems using z-transforms � Geometric evaluation of the Fourier transform from the pole-zero plot � Summary 8

Relationships: DT-FT and the ZT � We first express the complex variable z in polar form as � z = re j ω � r is the magnitude of z and � ω is the phase of z � Representing z as such, can be expressed as or equivalently, 9

Relationships: DT-FT and the ZT � By comparing equations & we can see that H( re j ω ) is essentially the FT of the sequence x[n] multiplied by a real exponential r − n � The exponential r − n may be decaying or growing with increasing n depending on whether r is greater than or less than 1 j = ω j H ( z ) | H ( e ) � If we let r = 1, then ω = z e which suggests that the ZT reduces to the FT on the unit circle (i.e., the contour in the complex z − plane corresponding to a circle with a radius of unity) 10

Relationships: DT-FT and the ZT � For convergence of the z − transform, we require that the Fourier transform of h [n]r − n converge � For any specific sequence h[n], we would expect this convergence for some values of r and not for others (as in the Laplace transform) � The range of values for which the z − transform converges is referred to as the region of convergence (ROC) � If the ROC includes the unit circle, then the Fourier transform converges 11

Outline � Introduction to Z Transform � Relationship to the Fourier transform � Z Transform and Examples � Region of Convergence of the Z Transform � Inverse Z Transform and Examples � Properties of Z Transform and Examples � Analysis and characterization of LTI systems using z-transforms � Geometric evaluation of the Fourier transform from the pole-zero plot � Summary 12

The Z Transform: Examples right-sided 13

The Z Transform: Examples � x[n] is right-sided; it decays when a<1 (e.g., a=0.5) � It z − transform is a rational function with one zero at z = 0 and one pole at z = a 14

The Z Transform: ROC in the form |z| > |a| 0 < a < 1 a > 1 -1 < a < 0 a < -1 15

The Z Transform: Examples left-sided • = − − − n Let x [ n ] a u [ n 1 ] ∞ ∑ = − − − − − − = ∀ − − < n n Then X ( z ) a u [ n 1 ] z Note that u [ n 1 ] 0 n : n 1 0 , 1 4 2 4 3 = −∞ > − n n 1 ∞ − ∞ 1 ∑ ∑ ∑ = − − − − = − − − − − − − − n n n n n n X ( z ) a u [ n 1 ] z a u [ n 1 ] z a u [ n 1 ] z 1 4 2 4 3 1 4 2 4 3 = −∞ = −∞ = n n n 0 1 0 − − 1 1 ∑ ∑ = − − = − − − n n 1 n a z ( a z ) (by combining the power n) = −∞ = −∞ n n ∞ ∑ = − − 1 n az ( ) (by multiplyin g every n by - 1) = n 1 ∞ ∞ ∞ ∑ ∑ ∑ − − − ⇒ − = − 1 n 1 n - 1 0 1 n ( az ) 1 ( az ) (because 1 - (az ) - ( az ) ) = = = n 1 n 0 n 1 Finite Geometric Series Formula • < > − - 1 1 This sum converges if |a z| 1 or | z | | a | , consequent ly + − m m m 1 c(z z ) ∑ 1 2 2 1 1 z = n = − = = < cz X(z) 1 , |z| |a| − − − − − − 1 1 1 a z 1 az z a 1 z = n m 1 Infinite Geometric Series Formula Note: The algebraic expression of X(z) for ∞ m cz ∑ x[n]=a n u[n] and x[n]=-a n u[-n-1] = n cz − 16 are identical except for the ROCs 1 z = n m

The Z Transform: ROC in the form |z| < |a| 0 < a < 1 -1 < a < 0 a > 1 a < 1 17

The Z Transform: Examples Multiple Poles 18

The Z Transform: Examples � We first find the ROC for each term individually and then find the ROC of both terms combined (Similar to what we used to do in Laplace transform) � Provided that | ⅓ z -1 |<1 and |½z -1 |<1 or equivalently |z| > ⅓ and |z| > ½ � The ROC is |z| > ½ 19

Outline � Introduction to Z Transform � Z Transform and Examples � Region of Convergence of the Z Transform � Inverse Z Transform and Examples � Properties of Z Transform and Examples � Analysis and characterization of LTI systems using z-transforms � Geometric evaluation of the Fourier transform from the pole-zero plot � Summary 20