1. CHAPTER 5
Z-Transform.
EKT 230

2. 5.0 Z-Transform. 5.1 Introduction. 5.2 The z-Transform.
5.2.1 Convergence.
5.2.2 z-Plane.
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.

3. Relation to the Laplace Transform
The z transform is to discrete-time signals and systems what the Laplace transform is to continuous-time signals and systems
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4. Definition
The definition of the z transform of a discrete-time function x is
and x and X form a “z-transform pair”
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18. Figure 5.3: The unit circle, z = ej, in the 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 = rej is located at a distance r– from the origin and an angle  relative to the real axis.
The point z=rej W is located at a distance r from the origin and the angle W from the positive real axis.
Figure 5.3 is a unit circle in the z-plane. z=rej W 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.

19. 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,

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

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

22. 5.5 The Inverse Z-Transform.
There are two common methods;
Partial-Fraction Expression.
Power-Series Expansion.

23. 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

24. Cont’d…
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.

25. Cont’d…
Step 3: Combining the Terms. .

26. 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 z3 from numerator and 2z2 from denominator.

27. Step 2: Use long division to reduce order of numerator polynomial.
Factor z3 from numerator and 2z2 from denominator.

28. Factor z3 from numerator and 2z2 from denominator.
We define,
Where,
With ROC|z|<1
Step 3: Find the Inverse z-Transform for each Terms. .

29. 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.

30. 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 is
Solution:
Step 1: Find the z-Transform of the input X(z) and output Y(z).
With ROC |z|>1/3
With ROC |z|>1.

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

32. 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.

33. Cont’d…
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.

34. Cont’d… Stable/Causal ?
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.
Stable/Causal ?
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.

35. 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,

36. 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.

38. Practice Questions