1 1 Chapter 3 The z-Transform

2 2  Consider a sequence x[n] = u[n]. Its Fourier transform does not converge.  Consider that, instead of e j , we use re j  in the transform, where r ≥ 0 is a real number. Then we have This transform will converge to, provided r>1. z-Transform (1)

3 3  In general, we have  Let z=re j . Then, the z-transform of a sequence x[n] is defined as with z being a complex variable. z-Transform (2)

4 4  Because z is a complex number, we often use the z-plane.  When |z|=1, that is, z takes value from the unit circle, the z-transform reduces to the Fourier transform. z-Transform (3) Unit circle Im Re 1 z-plane 

5 5  Properties of the system can be easily studied and characterized in the z-domain (stability, causality, …).  The solution process reduces to a simple algebraic procedure.  In the temporal domain, the output sequence is y[n]=x[n]*h[n] (convolution), whereas in the z-domain it becomes Y(z)=X(z)H(z) (multiplication). Why z-Transform

6 6  Absolute summability of z-transform  It is possible for the z-transform to converge even if the Fourier transform does not.  Convergence depends only on |z|. The region of convergence (ROC) consists of all values of z such that the last inequality holds.  If the ROC includes the unit circle, the Fourier transform of the sequence converges. Region of Convergence

7 7  Consider x[n]=a n u[n]. Because it is nonzero only for n ≥ 0, this is an example of a right-sided sequence. For convergence of X(z), we require Thus, the ROC is the range of values of z for which |az -1 | |a|. Inside the ROC, the infinite series converges to Example – Right-Sided Exponential Sequence (1)

8 8  The infinite sum becomes a simple rational function of z inside the ROC.  Such a z-transform is determined to within a constant multiplier by its zeros and its poles.  For this example, one zero: z=0 (plotted as o); one pole: z=a (plotted as x ).  When |a|<1, the ROC includes the unit circle. Unit circle Im Re 1 x z-plane ROC a Example – Right-Sided Exponential Sequence (2)

9 9  Consider x[n]= –a n u[–n –1]. Because it is nonzero only for n ≤ –1, this is an example of a left-sided sequence. ROC and Example – Left-Sided Exponential Sequence Unit circle Im Re 1 x z-plane ROC a

10  As can be seen from the two examples, the algebraic expression or pole-zero pattern does not completely specify the z-transform of a sequence; i.e., the ROC must also be specified. Notes on ROC

11  Consider the sequence ROC and Example – Two-Sided Exponential Sequence Im Re 1/2 x z-plane ROC x –1/3 1/12

12  Consider the sequence Then The ROC is determined by which requires Example – Finite-Length Sequence Unit circle Im Re 1 x z-plane 11 th order pole a ROC In this example N = 12 and 0< a <1. x

13 Some Common z-Transform Pairs (1)

14 Some Common z-Transform Pairs (2)

15  Property 1: The ROC is a ring or disk in the z-plane centered at the origin; i.e., 0 ≤r R <|z|<r L ≤ ∞.  Property 2: The Fourier transform of x[n] converges absolutely if and only if the ROC of the z-transform of x[n] includes the unit circle.  Property 3: The ROC cannot contain any pole.  Property 4: If x[n] is a finite-duration sequence, i.e., a sequence that is zero except in a finite interval –∞ < N 1 ≤ n ≤ N 2 < ∞, then the ROC is the entire z-plane, except possibly z = 0 or z = ∞. Properties of the ROC (1)

16  Property 5: If x[n] is a right-sided sequence, i.e., a sequence that is zero for n < N 1 < ∞, the ROC extends outward from the outmost (i.e., largest magnitude) finite pole in X(z) to (and possibly include) z = ∞.  Property 6: If x[n] is a left-sided sequence, i.e., a sequence that is zero for n > N 2 > –∞, the ROC extends inward from the innermost (smallest magnitude) nonzero pole in X(z) to (and possibly include) z = 0. Properties of the ROC (2)

17  Property 7: A two-sided sequence is an infinite-duration sequence that is neither right sided nor left sided. If x[n] is a two-sided sequence, the ROC will consist of a ring in the z-plane, bounded on the interior and exterior by a pole and, consistent with property 3, not containing any poles.  Property 8: The ROC must be a connected region. Properties of the ROC (3)

18 z-Transform with Different ROC (1) Im Re x z-plane Unit circle x x  For a system whose poles are shown in the figure, consider the stability and causality of the system.

19 z-Transform with Different ROC (2) Im Re x z-plane ROC x x Right-sided sequence Im Re x z-plane ROC x x Left-sided sequence

20 z-Transform with Different ROC (3) Im Re x z-plane ROC x x Two-sided sequence Im Re x z-plane ROC x x Two-sided sequence

21 Stability, Causality, and the ROC (1) Im Re x z-plane ROC x x Unit circle Im Re x z-plane x x If the system is stable h[n] is absolutely summable, and therefore has Fourier transform, the ROC must include the unit circle. h[n] is two-sided, and therefore, the system is not causal.

22 Stability, Causality, and the ROC (2) h[n] is right-sided. h[n] is not stable. Im Re x z-plane ROC x x Unit circle Im Re x z-plane x x If the system is causal

23 Im Re x z-plane Unit circle x x Stability, Causality, and the ROC (3) From the above discussion… There is no ROC that would imply that the system is both stable and causal.

24

25  Proof of the Initial-Value Theorem Since x[n]=0 for n<0, we have  If x[n]=0 for n>0, then we have

26  Inspection Method – use your familiar transform pairs The Inverse z-Transform - Inspection Method We know that then we would recognize Example: If we need to find the inverse z-transform of

27 When X(z) is expressed as as a ratio of polynomials in z -1 ; i.e., Such z-transforms arise frequently in the study of LTI systems.  Partial Fraction Expansion – obtain an alternative expression for X(z) as a sum of simple terms The Inverse z-Transform - Partial Fraction Expansion (1)

28 Zeros and poles for There are M zeros and N poles at nonzero positions. In addition, if M > N, there are M – N poles at z = 0, or if N > M, there are N – M zeros at z = 0. For the above expression, the z-transforms always have the same number of zeros and poles in the finite z-plane, and there are no poles or zeros at z = ∞. The Inverse z-Transform - Partial Fraction Expansion (2)

29 Note that X(z) can be written in the form where c k ’s are nonzero zeros of X(z), and d k ’s are nonzero poles of X(z). (case 1) If M < N and the poles are all first order, we have Multiplying both sides by (1 – d k z –1 ) and evaluating at z = d k, The Inverse z-Transform - Partial Fraction Expansion (3)

30 Example: Rewrite X(z) as where The Inverse z-Transform - Partial Fraction Expansion (4)

31 Therefore and Zeros and poles: Two zeros at z = 0, and first order poles at z = 1/4 and 1/2. The Inverse z-Transform - Partial Fraction Expansion (5)

32 (case 2) If M ≥ N and the poles are all first order, we have The B r ’s can be obtained by long division of the numerator by the denominator, with the division processing terminating when the remainder is of lower degree than the denominator. The Inverse z-Transform - Partial Fraction Expansion (6)

33 Example: Since M=N=2, X(z) can be represented as The constant B 0 can be found by long division The Inverse z-Transform - Partial Fraction Expansion (7)

34 (case 3) If X(z) has multiple-order poles. If X(z) has a pole of order s at z=d i and all other poles are first-order, then (there are no B r terms if M < N) with The Inverse z-Transform - Partial Fraction Expansion (8)

35  The defining expression for the z-transform is a Laurent series where the sequence values x[n] are the coefficients of z -n. Thus, if the z-transform is given as a power series in the form we can determine any particular value of the sequence by finding the coefficient of the appropriate power of z We have already used this approach in finding the inverse transform of the polynomial part of the partial fraction expansion when M ≥ N. - This method is also very useful for finite-length sequences where X(z) may have no simpler form than a polynomial in z -1. The Inverse z-Transform - Power Series Expansion (1)

36  Example 1 (Finite-length sequence) Express X(z) as we have The Inverse z-Transform - Power Series Expansion (2)

37  Example 2 (Inverse transform by power series expansion) Using the power series expansion for log(1+x), with |x|<1, we obtain Therefore The Inverse z-Transform - Power Series Expansion (3)

38  Obtain the inverse z-transform of by a different method. Since then We know therefore

39  Example 3 (Power series expansion by long division) Carrying out long division Therefore, x[n]= a n u[n]. The Inverse z-Transform - Power Series Expansion (4)

40  Example 3 (Power series expansion for a left-sided sequence) Because X(z) at z=0 is infinite, the sequence is zero for n>0. Thus, we divide, so as to obtain a series in powers of z as follows Therefore, x[n]= – a n u[ – n – 1]. The Inverse z-Transform - Power Series Expansion (5)

41 Homework (3)