1. 1 Z-Transform. CHAPTER 5 School of Electrical System Engineering, UniMAP School of Electrical System Engineering, UniMAP NORSHAFINASH BT SAUDIN norshafinash@unimap.edu.my EKT 230

2. 2 5.1 Introduction. 5.2 The z-Transform. 5.2.1 Convergence. 5.2.1 Convergence. 5.2.2 z-Plane. 5.2.2 z-Plane. 5.2.3 Poles and Zeros. 5.2.3 Poles and Zeros. 5.3 Properties of Region of Converges (ROC). 5.4 Properties of z-Transform. 5.5 Inverse of z-Transform. 5.6 Transfer Function. 5.7 Causality and Stability. 5.8 Discrete and Continuous Time Transformation. 5.9 Unilateral z-Transformation. 5.0 Z-Transform.

3. 3 5.1 Introduction.  In Laplace Transform we evaluate the complex sinusoidal representation of a continuous signal.  In the z-Transform, it is on the complex sinusoidal representation of a discrete-time signal.

4. 4 5.2 The z-Transform.  Let z = re j  be a complex number with magnitude r and angle . The signal x[n]=z n is a complex exponential signal.  We may write, exponential damped cosine  The real part of x[n] is an exponential damped cosine. exponential damped sine  And the imaginary part is an exponential damped sine as shown in Figure 5.1.  The positive number of r determine the damping factor and  is the sinusoidal frequency.

5. 5 Figure 5.1: Real and imaginary parts of the signal z n. Cont’d…

6. 6  The transfer function,  The z-transform of an arbitrary signal x[n] is,  The inverse z-transform is,Cont’d…

7. 7  The region of converges (ROC) is the range of r for which the below equation is satisfied: 5.2.1 Convergence.

8. 8 5.2.2 z-Plane.  It is convenience to represent the complex frequency z as a location in z-plane as shown in Figure 5.2. Figure 5.2: The z-plane. A point z = re j  is located at a distance r– from the origin and an angle  relative to the real axis.  The point z=re j   is located at a distance r from the origin and the angle  from the positive real axis. unit circlez=re j   Figure 5.3 is a unit circle in the z-plane. z=re j   describes a circle of unit radius centered on the origin in the z-plane. Figure 5.3: The unit circle, z = ej , in the z-plane.

9. 9 5.2.3 Poles and Zeros.  The z-transform form, a ratio of two polynomial in z -1,  The X(z) can be rewrite as a product of terms involving the roots of the numerator and denominator polynomial,  Where, c k = the roots of the numerator polynomial and the zeros (o) of X(z). d k = the roots of the denominator polynomial and the poles (x) of X(z)

10. 10 Figure 5.4: The relationship between the ROC and the time extent of a signal. (a) A right-sided signal has an ROC of the form |z| > r +. (b) A left-sided signal has an ROC of the form |z| < r –. (c) A two-sided signal has an ROC of the form r + < |z| < r –. 5.3 The Properties of ROC.

11. 11 Figure 5.5: ROCs for Example 7.5 (text). (a) Two-sided signal x[n] has ROC in between the poles. (b) Right-sided signal y[n] has ROC outside of the circle containing the pole of largest magnitude. (c) Left-sided signal w[n] has ROC inside the circle containing the pole of smallest magnitude. Pg 565 TextCont’d…

12. 12  Taking a path analogous to that used the development of the  Laplace transform, the z transform of the causal DT signal is and the series converges if |z| > |a|. This defines the ROC as the exterior of a circle in the z plane centered at the origin, of radius|a|.  The z transform is CausalCont’d…

13. 13  By similar reasoning, the z transform and region of convergence of the anti-causal signal below, are Anti-CausalCont’d…

14. 14 Re Im o o xx Example 5.0: Two-Sided Exponential Sequence Example Solution: Time Domain -> s Domain

15. 15 5.4 Properties of z-Transform.  Most properties of z-transform are similar to the DTFT. We assumed that,  The effect of an operation on the ROC is described by a change in the radii of the ROC boundaries. (1) Linearity,

16. 16 (2) Time Reversal. (3) Time Shift. with ROC R x, except possibly z=0 or |z|= infinity.Cont’d…

17. 17 (4) Multiplication by an Exponential Sequence. (5) Convolution. (6) Differentiation in the z-Domain. Cont’d…

18. 18  There are two common methods; 5.5.1 Partial-Fraction Expression. 5.5.2 Power-Series Expansion. 5.5 The Inverse Z-Transform.

19. 19 5.5.1 Partial-Fraction Expansion. Example 5.1: Inversion by Partial-Fraction Expansion. Find the inverse z-transform of, with ROC 1<|z|<2. Figure 5.6: Locations of poles and ROC. Solution: Step 1: Use the partial fraction expansion of Z(s) to write Solving the A, B and C will give

20. 20 Step 2: Find the Inverse z-Transform for each Terms. - The ROC has a radius greater than the pole at z=1/2, it is the right- sided inverse z-transform. - The ROC has a radius less than the pole at z=2, it is the left-sided inverse z-transform. - Finally the ROC has a radius greater than the pole at z=1, it is the right-sided inverse z-transform.Cont’d…

21. 21 Step 3: Combining the Terms. .Cont’d…

22. 22 Example 5.2: Inversion of Improper Rational Function. Find the inverse z-transform of, with ROC |z|<1. Figure 5.7: Locations of poles and ROC. Solution: Step 1: Convert X(z) into Ratio of Polynomial in z -1. Factor z 3 from numerator and 2z 2 from denominator.

23. 23 Step 2: Use long division to reduce order of numerator polynomial. Factor z 3 from numerator and 2z 2 from denominator.

24. 24 Factor z 3 from numerator and 2z 2 from denominator. We define, Where, With ROC|z|<1 Step 3: Find the Inverse z-Transform for each Terms. .

25. 25 5.6 Transfer Function.  The transfer function is defined as the z-transform of the impulse response. y[n]= h[n]*x[n]  Take the z-transform of both sides of the equation and use the convolution properties result in,  Rearrange the above equation result in the ratio of the z-transform of the output signal to the z-transform of the input signal.  The definition applies at all z in the ROC of X(z) and Y(z) for which X(z) is nonzero.

26. 26 Example 5.3: Find the Transfer Function. Find the transfer function and the impulse response of a causal LTI system if the input to the system isSolution: Step 1: Find the z-Transform of the input X(z) and output Y(z). With ROC |z|>1/3 With ROC |z|>1.

27. 27 Step 2: Solve for H(z)., with ROC |z|>1. Solve for impulse response h[n],, with ROC |z|>1. So the impulse response h[n] is, .

28. 28 5.7 Causality and Stability.  The impulse response of a causal system is zero for n<0.  The impulse response of a casual LTI system is determined from the transfer function by using right-sided inverse transform.  The poles inside the unit circle, contributes an exponentially decaying term to the impulse response.  The poles outside the unit circle, contributes an exponentially increasing term. Figure 5.8: Pole and impulse response characteristic of a causal system. (a) A pole inside the unit circle contributes an exponentially decaying term to the impulse response. (b) A pole outside the unit circle contributes an exponentially increasing term to the impulse response.

29. 29  Stable system; the impulse response is absolute summable and the DTFT of impulse response exist.  The impulse response of a casual LTI system is determined from the transfer function by using right-sided inverse transform.  The poles inside the unit circle, contributes a right-sided decaying exponential term to the impulse response.  The poles outside the unit circle, contributes a left-sided decaying exponentially term to the impulse response.  Refer to Figure 5.9. Figure 5.9: Pole and impulse response characteristics for a stable system. (a) A pole inside the unit circle contributes a right-sided term to the impulse response. (b) A pole outside the unit circle contributes a left-sided term to the impulse response. Cont’d…

30. 30  From the ROC below the system is stable, because all the poles within the unit circle and causal because the right-sided decaying exponential in terms of impulse response. Figure 7.16: A system that is both stable and causal must have all its poles inside the unit circle in the z-plane, as illustrated here. Stable/Causal ? Cont’d…

31. 31 5.8 Implementing Discrete-Time LTI System.  The system is represented by the differential equation.  Taking the z-transform of difference equation gives,  The transfer function of the system,

32. 32 Example 5.4 : Causality and Stability. Can this system be both stable and causal?Solution: Step 1: Find the characteristic equation of the system. From the plot, the system is unstable because the pole at z = 1.2 is outside the unit circle. .

33. 33 Figure 7.26: Block diagram of the transfer function. Cont’d…

34. 34 Figure 7.27: Development of the direct form II representation of an LTI system. (a) Representation of the transfer function H(z) as H 2 (z)H 1 (z). (b) Direct form II implementation of the transfer function H(z) obtained from (a) by collapsing the two sets of z –1 blocks. Cont’d…

35. 35 5.9 The Unilateral z-Transform.  The unilateral z-Transform of a signal x[n] is defined as, Properties.  If two causal DT signals form these transform pairs, (1) Linearity. (2) Time Shifting.