1. CE 40763 Digital Signal Processing Fall 1992 Z Transform
Hossein Sameti
Department of Computer Engineering
Sharif University of Technology

2. Definition of Z-transform
Vs.
Complex function
Complex variable
Q: Why do we need Z-transform?
A: For some signals, Fourier transform does not converge.
e.g.:
Relationship between and :
Z-transform evaluated at unit circle corresponds to DTFT.
 The role of Z-transform in DT is similar to Laplace transform in CT.

3. Convergence of Z-transform
For what values of z, does the z-transform converge?
DTFT of
DTFT of
exists if
is absolutely summable.
Z-transform exists and converges.
Convergence only depends on r.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

4. Definition of ROC
Region of Convergence (ROC): ROC is a region in “z” domain that converges.
ROC is radially symmetric.
It also converges for
If Z-transform converges for
ROC does not depend on the angle.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

5. ROC Possibilities There are the following possibilities:
All Z-domain except one or two points (origin / infinity)
Inside a circle
Outside a circle
Between two circles (ring)
ROC does not exist
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

6. Example: ROC for a finite-length Signal
ROC: everywhere except at the origin:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

7. Example: ROC for another finite-length Signal
ROC: everywhere except at infinity:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

8. Example: ROC for RHS
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

9. Example: ROC for LHS Same Z-transform as the
previous example. However,
the ROC is different.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

10. Importance of ROC
We have shown that X(z) needs the ROC to uniquely specify a sequence.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

11. Example: ROC for RHS What is the ROC? Union? Intersection?
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

12. Example: ROC for BHS What is the ROC? Union? Intersection?
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

13. Example: ROC for BHS Z-transform does not exist.
What is the ROC? Union? Intersection?
Z-transform does not exist.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

14. Properties of the ROCs of Z-Transforms
(1) The ROC of X(z) consists of a ring in the z-plane centered about the origin (equivalent to a vertical strip in the s-plane)
(2) The ROC does not contain any poles (same as in LT).
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

15. Book Chapter10: Section 1
More ROC Properties
(3) If x[n] is of finite duration, then the ROC is the entire z-plane, except possibly at z = 0 and/or z = ∞.
Why?
Examples:
CT counterpart
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

16. ROC Properties Continued
Book Chapter10: Section 1
ROC Properties Continued
(4) If x[n] is a right-sided sequence, and if |z| = ro is in the ROC, then all finite values of z for which |z| > ro are also in the ROC.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

17. Book Chapter10: Section 1
Side by Side
If x[n] is a left-sided sequence, and if |z| = ro is in the ROC, then all finite values of z for which 0 < |z| < ro are also in the ROC.
(6) If x[n] is two-sided, and if |z| = ro is in the ROC, then the ROC consists of a ring in the z-plane including the circle |z| = ro.
What types of signals do the following ROC correspond to?
right-sided left-sided two-sided
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

18. Example: Book Chapter10: Section 1
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

19. Book Chapter10: Section 1
Example, continued
Clearly, ROC does not exist if b > 1 ⇒ No z-transform for b|n|.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

20. Summary of Observations
ROC is only a function of r.
FLS ROC is everywhere except at the origin or infinity.
RHS ROC is outside some circle.
LHS ROC is inside some circle.
BHS ROC is a ring or it does not exist.
FT exists if ROC includes unit circle.
Let’s prove one of these properties.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

21. Properties of the Z-Transform
Linearity:
ROC: at least the intersection of ROC1 and ROC2.
- Linearity helps us to find the z-transform of a signal by expressing the signal as a sum of elementary signals.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

22. Linearity Example: Solution: Using linearity property:
Earlier we showed that:
If we set:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

23. Time Shifting ROC: Similar ROC with some exceptions at 0 and infinity
Example:
ROC?
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

24. Scaling in the z-domain
Example:
Earlier:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

25. Time Reversal Example: Earlier:
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26. Differentiation in the z-domain
Example:
Earlier:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

27. Convolution of Two Sequences
ROC: at least the intersection of ROC1 and ROC2
Example:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

28. Inverse Z-Transform
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

29. Inverse Z-Transform Methods of calculating the inverse Z-transform:
Cauchy theorem for calculating the inverse Z-transform (not covered)
Series expansion
Partial fraction expansion and using look-up tables
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

30. Power Series Expansion
Basic concept: expand X(z) in terms of powers of z-1
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

31. Power Series Expansion
Example:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

32. Partial Fraction Expansion and Lookup Table
This method is applicable if we have a ratio of two polynomials in the form of
We would like to rewrite X(z) in the following manner:
This expansion is performed in a way that the inverse Z-transform of can be easily found using a lookup table and using the linearity property, we can find x(n):
Example:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

33. Polynomial Factorization
Any non-constant single-variable polynomial of degree N can be factored into N terms:
If the coefficients of the polynomial are real, then the roots are either real or complex conjugates.
If is a complex root, then is also a root.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

34. Partial Fraction Expansion and Lookup Table
Example:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

35. Partial Fraction Expansion and Lookup Table
Example:
x(n) causal
In the case of complex conjugate roots, the coefficients in the partial fraction expansion will be complex conjugates.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

36. Partial Fraction Expansion Algebra: A = 1, B = 2
Example:
Partial Fraction Expansion Algebra: A = 1, B = 2
Note, particular to z-transforms:
1) When finding poles and zeros, express X(z) as a function of z.
2) When doing inverse z-transform using PFE, express X(z) as a function of z-1.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

37. ROC III:
ROC II:
ROC I:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

38. Linear Constant- Coefficient Difference Equation (LCCDE)
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

39. Classification of LTI Systems
h(n),H (z)
x(n),X(z)
y(n),Y(z)
LTI Systems
h(n)
Transfer function
FIR
IIR
FIR has a finite number of non-zero values. It is easy to implement.
IIR has infinite number of non-zero values.
- why polynomials are good? They are continuous and differentiable
With rational transfer function
No rational transfer function
Use difference equations for implementation

40. Example- LCCDE Cumulative Average System:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

41. General form of L.C.C.D.E.
The general form for a linear constant-coefficient difference equation is given by:
Given the difference equation of an IIR system and the initial condition of y(n), we can compute the output y(n) through the difference equation more efficiently using finite number of computations.
Order of the system
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

42. LCCDE and LTI Systems
Question: How can an LCCDE correspond to an LTI system?
E.g.:
It is not even a system, since a unique input does not correspond to a unique output.
Q: How can we show this?
A: If is a solution, is also a solution.
Q: How can we resolve this problem?
A: We need to add an initial condition (I.C.).
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

43. Calculating the Output using the IC
Causal implementation
Anti-causal implementation
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

44. LCCDE and Linearity By Combining the two answers, we get:
Q: Does this system correspond to a linear system?
A: No, zero input does not result in zero output.
An LCCDE corresponds to a linear system, when IC=0.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

45. LCCDE and Time-invariance
Q: Does this system correspond to a TI system?
A: Consider
. Now let’s shift the input:
Output:
Shift variant
In order for an LCCDE to correspond to a TI system, IC=0.
(for 𝑛 0 ≥0)
In order for an LCCDE to correspond to an LTI system, IC=0.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

46. LCCDE and Causal LTI Systems
In order an LCCDE to correspond to a causal LTI system :
Initial rest condition (IRC)=0
E.g., if , for a first order LCCDE, the IRC should be as follows:
E.g., if , for a second order LCCDE, the IRC should be as follows:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

47. Example – Finding the impulse response
Solution1:

48. Same Example with Z-Transform
h(n),H (z)
x(n),X(z)
y(n),Y(z)
Remember:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

49. Poles/ Zeros Z-transform with rational transfer function: Zero: X(z)=0
Pole: X(z)=∞
Getting rid of the negative powers:
Poles and zeros?
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

50. Pole-Zero plot A pole-zero plot can represent X(z) graphically.
Example:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

51. Pole-Zero plot
If a polynomial has real coefficients, its roots are either real or occur in complex conjugates.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

52. Pole-Zero Location and Time-Domain Behavior of Causal Signals
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

53. Pole-Zero Location and Time-Domain Behavior of Causal Signals
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

54. Pole-Zero Location and Time-Domain Behavior of Causal Signals (Double poles)
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

55. Pole-Zero Location and Time-Domain Behavior of Causal Signals (Double poles)
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

56. Pole-Zero Location and Time-Domain Behavior of Causal Signals (Two Complex-conjugate poles)
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

57. LTI System Analysis in the z-domain: Stability
For BIBO stability of an LTI system, we shall have
Now, let’s see what happens in the z-domain:
CONCLUSION: an LTI system is BIBO stable iff the ROC of its impulse response includes the unit circle.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

58. LTI System Analysis in the z-Domain: Causality
Causal LTI systems have h(n) = 0 for n<0.
ROC of the z-transform of a causal sequence is the exterior of a circle.
CONCLUSION: an LTI system is causal iff the ROC of its system function is the exterior of a circle of radius r<∞ including the point z=∞.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

59. LTI System Analysis in the z-Domain: Causality and Stability
Conditions for stability and causality are different and one does not imply the other.
A causal system might be stable or unstable just as a non- causal system can be stable or unstable.
For causal systems though, the condition of stability can be narrowed since:
A causal system has z-transform with ROC outside a circle of radius r.
The ROC of a stable system must contain the unit circle.
Then a causal and stable system ROC must be 𝑧 >𝑟 with 𝑟<1.
This means that a causal LTI system is BIBO
stable iff all the poles of H(z) are inside the
unit circle.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

60. Causality and Stability of LTI Systems
Example:
Specify ROC and h(n) for the following cases:
The system is stable:
The system is causal:
The system is anti-causal:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

61. Summary
Discussed z-transform, region of convergence, properties of z-transform and how the inverse z- transform can be calculated.
Discussed LCCDEs and the concept of the pole-zero for their z-transform
Next: we will analyze the properties of ideal filters
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology