1. AMI 4622 Digital Signal Processing
Unit 5 The Z Transform

2. Introduction What is the z-transform?
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The Z Transform is an invaluable tool for:
Representing
Analysing Discrete Time Signals and Systems
Designing
It plays a similar role in discrete time systems to that which the Laplace transform plays in continuous time systems
Infer the degree of system stability
Visualise the system frequency response

3. Discrete Time Signals and Systems
A discrete-time system is represented by a sequence of numbers x(n), x(nT), xn
A discrete-time system is essentially a mathematical algorithm which takes an input sequence x(n) and produces an output sequence y(n)
A discrete-time system may be linear or nonlinear, time invariant or time varying. Linear time invariant (LTI) systems form an important class of systems used in DSP.
A discrete-time system is linear if it:
Has homogeneity (scaling)
Has additivity
Obeys the rules of superposition

4. The z-transform Laplace – analyse the behaviour of electrical circuits
z-transform – analyse the behaviour of discrete systems
Fourier and z-transforms are related through

5. The z-transform
The z-transform of a sequence x(n) which is valid for all n, is defined as:
where z is a complex variable
In causal systems x(n) is only non-zero for n >= 0, so the above equation reduces to:
……(power series)
The z-transform is a power series with an infinite number of terms and therefore may not converge.
The region where the z-transform converges is known as the Region of Convergence (ROC)
It is in this region where the value of X(z) is finite (in this case valid).

6. The z-transform
ROC

7. The z-transform
ROC

8. The z-transform
ROC

9. The z-transform
ROC

10. The z-transform ROC Region of convergence
(everywhere outside of the unit circle)
|z| > 1

11. The z-transform Region of Convergence
Causal sequences of finite duration the z-transform converges everywhere except at z = 0.
Causal infinite duration sequences the z-transform converges everywhere outside a circle bounded by the radius of the pole with the largest radius.
For stable causal systems the ROC always encloses the circle of unit radius (important for the systems to have a frequency response)
Table of common z-transforms

12. The z-transform
Plot the signal, determine the z-transform, and the region of convergence for the given expressions

13. The z-transform
And one last one!!!

14. The inverse z-transform
Power Series
First expand the series in either descending powers of z, or ascending powers of z-1
Then perform long division
Partial fractions
Expand into a sum of simple partial fractions
The inverse z-transform of each partial fraction is then obtained from tables

15. Power series method
Example X(z) = z / (z – 0.5)
Self Test Solution…

16. Power series method
Try this yourself

17. Power series method
Solution: (error 0.5 -> 0.5z) (try impz() in Matlab)

18. Partial fractions method
Example 1 – X(z) contains simple first order poles
Solution……..

19. Partial fractions method
Example 1 – X(z) contains simple first order poles

20. Partial fractions method
Example 2 – X(z) contains first order conjugate poles
Lets try this
Solution……….

21. Partial fractions method
Example 2 – X(z) contains first order conjugate poles

22. Residue method
The IZT is obtained by evaluating the contour integral.

23. Residue method
Example
Solution…..

24. Residue method
Example

25. Comparison of the inverse z-transforms
Power Series
Does not lead to a closed form solution
Simple and lends itself to computer implementation
Because of its recursive nature care is required to minimise possible build up of numerical error when the number of data points in the IZT is large.

26. Comparison of the inverse z-transforms
Partial Fractions
Advantage is it leads to a closed form solution
Disadvantage there is a need to factorise the denominator polynomial
Preferred over residue method for generating the coefficients of parallel structures

27. Comparison of the inverse z-transforms
Residue method
Advantage is it leads to a closed form solution
Disadvantage there is a need to factorise the denominator polynomial
Widely used in the analysis of quantisation errors in discrete-time systems

28. Properties of the z-transform
Linearity
Delays or Shifts
Convolution
Differentiation
Relationship with the Laplace transform

29. Properties of the z-transform
Relationship with the Laplace transform
Laplace variable
let
then
and so

30. Properties of the z-transform
Show Movie – Laplace z-transform relationship

31. Properties of the z-transform
Relationship with the Laplace transform
example for zeros at -0.5+j j j …….exp(-0.5)exp(j0.7071) etc

32. Applications of the z-transform in signal processing
Pole-zero description of discrete time systems
Frequency response estimation
Geometric evaluation of frequency response
Direct computer evaluation of frequency response
Frequency response estimation via the FFT
Determine the difference equations
Obtain the impulse response estimation

33. Pole-zero description of discrete time systemsif
then

34. Pole-zero description of discrete time systems
Express the following transfer function in terms of its poles and zeros and sketch the pole zero diagram.

35. Pole-zero description of discrete time systems
Solution:
try help zplane and help roots
in Matlab

36. Pole-zero description of discrete time systems
Determine the transfer function H(z) of a discrete time filter with the pole-zero diagram shown:
0.25,j0.7071
-0.5
0.5
0.25,j

37. Pole-zero description of discrete time systems
Determine the transfer function H(z) of a discrete time filter with the pole-zero diagram shown: Solution:
0.25,j0.7071
-0.5
0.5
0.25,j

38. Frequency response estimation
We often need to evaluate the frequency response of a discrete-time system
For example, in the design of a discrete filter we need to examine the spectrum of the filter to ensure that the filter specification is satisfied…………..

39. Frequency response estimation
FIR Filter frequency response
Pass Band
Pass Band
Stop Band

40. Frequency response estimation
The frequency response can be readily obtained from the z-transform
To do this we set
This evaluates the z-transform around the unit circle
From this we obtain the Fourier transform of the system

41. Frequency response estimation
Let
zplane_freq_resp.m

42. Geometric evaluation of frequency response
This method allows us to estimate the frequency response of a system based on it’s pole-zero diagram
Therefore we have to express the z-transform in terms of the poles and zeros of the system and let
The magnitude response is obtained from
The phase response is obtained from
Example yn = xn + xn ExampleSlide42.m

43. Geometric evaluation of frequency response
Example Determine using the geometric method, the frequency response at dc, the sampling frequency, of the causal discrete-time system with the following difference equation. yn = xn + xn yn-1

44. Geometric evaluation of frequency response
Solution:
zplane_freq_resp_solution.m
and
ExampleSlide43.m

45. Geometric evaluation of frequency response
Determine using the geometric method, the frequency response at dc, Fs/4 and Fs/2, of the causal discrete-time system with the following z-transform

46. Geometric evaluation of frequency response
Do exercise 5.5

47. Direct computer evaluation of frequency response
The Geometric method gives a feel of the frequency response
But can be tedious if the precise response is required at many frequencies
Difficult to find the pole and zero locations for high order filters
The direct method shown next, makes the substitution directly into the transfer function and then to evaluate the resulting expression.

48. Direct computer evaluation of frequency response
Class: Do given example using Matlab

49. Frequency response estimation via the FFT
The FFT can be used to evaluate the frequency response of a discrete-time sequence
This can be done directly for an FIR filter
FrequencyResponseEstimationViaFFT.m
For IIR systems it is necessary to first obtain the impulse response of the system for example, using the power series method

50. Frequency units used in discrete time systems
Frequency units in the z-plane

51. Stability considerations
One stability criterion for a LTI system is that all bounded inputs produce bounded outputs - bounded input bounded output (BIBO)
A LTI system said to be BIBO satisfies:
where h(k) is the impulse response of the system
This only applies to systems with an impulse response of infinite duration
If the impulse response is of finite length then it is obvious that the system will always be stable
For an unstable system the impulse response will increase indefinitely with time
StabilityTest.m

52. Difference equations
The difference equation specifies the operations that must be performed on the input data.
The z-transform delay operator
We can easily move between discrete time and the transfer function....
discrete-time transfer function

53. Difference equations The transfer function H(z) can then be obtained
If the denominator coefficients are zero then this is……
referred to an finite impulse response (FIR) since the length of the expression is finite
If at least one of the denominator coefficients are non-zero (at least one of the poles will be non-zero) then this is……
referred to as an infinite impulse response (IIR) system

54. Impulse response estimation
We often need to determine the impulse response of a discrete-time system
The impulse response of a FIR is required to implement the system
The impulse response of a IIR is required to for stability analysis
The impulse response of a discrete time system is defined as the inverse z-transform of the systems transfer function, H(z)
If the z-transform is available as a power series………
then the coefficients of the z-transform give directly the impulse response
If the z-transform is expressed as a ratio of polynomials (IIR) then the IZT methods shown earlier must be used.

55. Impulse response estimation
The impulse response may also be viewed as the response of a discrete-time system to a unit impulse.
….this provides a simple method of computing h(n) (as well as another method of obtaining the IZT)

56. Impulse response estimation
Example
Find the impulse response of the discrete-time filter characterised by the following transfer function.
1) by using the power series method
2) by applying a unit impulse to the system