The z-Transform

Quote of the Day Such is the advantage of a well-constructed language that its simplified notation often becomes the source of profound theories.

Laplace

1 Some Special Functions

First consider the delta function (or unit sample function): 0, n  0 0, t  0 (n)   or (t)   1, n  0 1, t  0 This allows an arbitrary sequence x(n) or continuous- time function f(t) to be expressed as:

 x(n)   x(k) (n  k) k  f (t)  f ( ) (t  )d 

2 Convolution, Unit Step x(n)  x(n)(n)

These are referred to as discrete-time convolution, and is denoted by:  x(n)   x(k) (n  k) k Also consider the unit step function: 1, n  0 u(n)   0, n  0

Note also:  u(n)   (k) k 

3 z-Transform

. The z-transform is the most general concept for the transformation of discrete-time series.

. The Laplace transform is the more general concept for the transformation of continuous time processes.

. For example, the Laplace transform allows you to transform a differential equation, and its corresponding initial and boundary value problems, into a space in which the equation can be solved by ordinary algebra.

. The switching of spaces to transform calculus problems into algebraic operations on transforms is called operational calculus. The Laplace and z transforms are the most important methods for this purpose.

4 The Z Transforms The one-sided z-transform of a function x(n): Also known as unilateral z-transform.

 X (z)   x(n)z n n0 The two-sided z-transform of a function x(n): Bilateral z- transform  X (z)   x(n)z n n The Laplace transform of a function f(t):

 F(s)  f (t)estdt 0 5 Relationship to Fourier Transform

Note that expressing the complex variable z in polar form reveals the relationship to the Fourier transform:

 i.e. Z  rei  X (rei )   x(n)(rei )n , or n   X (rei )   x(n)r nein , and if r 1, n

  X (ei )  X ()   x(n)ein n

which is the Fourier transform of x(n).

6 The z-transform and the DTFT . The z-transform is a function of the complex z variable . It is convenient to describe on the complex z-plane . If we plot z=ej for =0 to 2 we get the unit circle

j Im Xe 

Unit Circle

r=1  0 Re  2 0 2

7 Copyright (C) 2005 Güner Arslan Region of Convergence(ROC) The power series for the z-transform is called a Laurent series:  X (z)   x(n)z n  X(z) ...... x(2)z2  x(1)z  x(0)  x(1)z1  x(2)z2..... n •The set of all values of z for which this series converges(attains the finite value) is known as its region of convergence. •Each value of r represents a circle of radius r.

 The region of convergence is made of Im circles.  Example: z-transform converges for values of 0.5

8 Poles and Zeros When X(z) is a rational function, i.e., a ratio of polynomials in z, then: 1. The roots of the numerator polynomial are referred to as the zeros of X(z), and 2. The roots of the denominator polynomial are referred to as the poles of X(z).

Note that no poles of X(z) can occur within the region of convergence since the z-transform does not converge at a pole. Furthermore, the region of convergence is bounded by poles.

9 Right-Sided Exponential Sequence Example

  n xn anun  X z  anunz n  az1  n n0 . For Convergence we require  n Region of convergence  az1   n0  . Hence the ROC is defined as a az1 1 z  a

. Inside the ROC series converges to . Geometric series formula

0  N2 N1 N2 1  1 1 n a  a 1 n az   az  1 X z  az   a    1 1 nN 1 a n0 1 az 1 az 1 • Region outside the circle of  n 1 z radius a is the ROC  X z  az1     1 • Right-sided sequence ROCs n0 1 az z  a extend outside a circle

Clearly, X(z) has a zero at z = 0 and a pole at z = a. 10 Left-Sided Exponential Sequence Example

 1 n xn bnu n 1  X z   bnu n 1z n   bz1  n n . We can alternatively write as  n  n Region of convergence X (z)   bz1   b1z   n1 n1  . Hence the ROC is defined as b b1z 1 z  b

. Inside the ROC series converges to . Geometric series formula

 1 0 1  N2 N1 N2 1 1 n b z b z n a  a X z 1 b z 1 a   1  1 a n0 1b z nN1 1 b  z  X z 1 1  • Region Inside the circle of radius 1b1z b  z b  z b is the ROC z • Left-sided sequence ROCs  X z  extend Inside a circle z b 11 Clearly, X(z) has a zero at z = 0 and a pole at z = b. Two-Sided Exponential Sequence Example n n  1  1  xn   un-  u- n -1  3  2  0  1 1  1 1   1 1  ROC :  z  1  n  z    z  3  1 1   3   3  1  z     1 1 1 n0  3  1 z 1 1 z 1  z 3 3 3

 0 1 1  1 1  1 1  ROC : z  1 n  z    z  1 2  1 1   2  2   1   z    n 2  1 1 1 1 1 1  z 1  z  z Im 2 2 2  1  z2z   z z 6   1 1 X z     1 1  1  1  3 2 z  z   z   z   x oo x Re 3 2  3 2  1 1 1 12 ROC :  z  3 2 Copyright (C) 2005 Güner Arslan Clearly, X(z) has a zero at z = 0,1/12 and a pole at z = ½,-1/3. Finite Length Sequence 2n 0  n  3 If xn   0 otherwise 3  X z  2n z n 1 2z 1  4z 2  8z 3 n0 X(z) is the series of –ve powers of z. Therefore the ROC for finite-length right sided sequence is the entire z-plane except at z= 2n  3  n  0 If xn   0 otherwise 0  X z  2n z n  23 z3  22 z 2  21 z 1 n3

X(z) is the series of +ve powers of z. Therefore the ROC for finite-length left sided sequence is the entire z-plane except at z=0 13 Finite Length Sequence contd.. an 3  n  0 xn  n b 0  n  3 0 3  X z  an z n  bn z n n3 n1 X(z)  a3z3  a2z2  a1z 1bz1 b2z2 b3z3 X(z) is the series of –ve powers & +ve powers of z. Therefore the ROC for finite-length left sided sequence is the entire z-plane except at z=0 & z=

an 0  n  N 1 xn   0 otherwise

N 1 N 1 1 N N N n n 1 n 1 az  1 z  a X z  a z  az   1  N 1 n0 n0 1 az z z  a • Region outside the circle of radius a is the ROC 14 Properties of The ROC of Z-Transform

. The ROC is a ring or disk centered at the origin. . The ROC cannot contain any poles. . The ROC for a right-handed sequence extends outward from the outermost pole possibly including z= . . The ROC for a left-handed sequence extends inward from the innermost pole possibly including z=0. . The ROC of a two-sided sequence is a ring bounded by poles . . The ROC must be a connected region. . The ROC for finite-length sequence is the entire z-plane – except possibly z=0 and z=. . A z-transform does not uniquely determine a sequence without specifying the ROC. 15 What is the z-transform of delta function? 0, n  0 x(n)   (n)   1, n  0  0 X (z)   x(n)z n   X (z)  1 z n 1 n n0 ZT (n)1 ROC is the entire z-plane

What is the z-transform of unit step function? 1, n  0 x(n)  u(n)   0, n  0  1 z n   X (z)  1 z  1 n0 1 z z 1 z u(n)ZT  • Region outside the circle of z 1 radius a is the ROC |Z|>1 16 Review

(n)ZT1 Roc is entire z plane z u(n)ZT  ROC is z 1 z 1 z anu(n)ZT  ROC is z  a z  a z bnu(n 1)ZT  ROC is z  b z b z z anu(n) bnu(n 1)ZT   ROC is a  z  b z  a z b If a  b else ZT does not exit. . Geometric series formula N2 a N1  a N2 1 an   1 a nN1 17