Digital Signal Processing Lecture-2 Spring 2010 Z Transform Chap.-3

1. Digital Signal Processing Lecture-2 Spring 2010Z Transform Chap.-3 Hassan Bhatti Hassan Bhatti, DSP, Spring 2011

2. z-Transform • The z-transform is the most general concept for the transformation of discrete-time series. • The Laplace transform is the more general concept for the transformation of continuous time processes. • For example, the Laplace transform allows you to transform a differential equation, and its corresponding initial and boundary value problems, into a space in which the equation can be solved by ordinary algebra. • The switching of spaces to transform calculus problems into algebraic operations on transforms is called operational calculus. The Laplace and z transforms are the most important methods for this purpose. Hassan Bhatti, DSP, Spring 2010

3. The Transforms The Laplace transform of a function f(t): The one-sided z-transform of a function x(n): The two-sided z-transform of a function x(n): Hassan Bhatti, DSP, Spring 2010

4. Relationship to Fourier Transform Note that expressing the complex variable z in polar form reveals the relationship to the Fourier transform: which is the Fourier transform of x(n). Hassan Bhatti, DSP, Spring 2010

5. Relation ship of Transformations Hassan Bhatti, DSP, Spring 2010

6. Region of Convergence The z-transform of x(n) can be viewed as the Fourier transform of x(n) multiplied by an exponential sequence r-n, and the z-transform may converge even when the Fourier transform does not. By redefining convergence, it is possible that the Fourier transform may converge when the z-transform does not. For the Fourier transform to converge, the sequence must have finite energy, or: Hassan Bhatti, DSP, Spring 2010

7. Convergence, continued The power series for the z-transform is called a Laurent series: The Laurent series, and therefore the z-transform, represents an analytic function at every point inside the region of convergence, and therefore the z-transform and all its derivatives must be continuous functions of z inside the region of convergence. In general, the Laurent series will converge in an annular region of the z-plane. Hassan Bhatti, DSP, Spring 2010

8. Some Special Functions First we introduce the Dirac delta function (or unit sample function): or This allows an arbitrary sequence x(n) or continuous-time function f(t) to be expressed as: Hassan Bhatti, DSP, Spring 2010

9. Convolution, Unit Step These are referred to as discrete-time or continuous-time convolution, and are denoted by: We also introduce the unit step function: Note also: Hassan Bhatti, DSP, Spring 2010

10. Poles and Zeros • When X(z) is a rational function, i.e., a ration of polynomials in z, then: • The roots of the numerator polynomial are referred to as the zeros of X(z), and • The roots of the denominator polynomial are referred to as the poles of X(z). Note that no poles of X(z) can occur within the region of convergence since the z-transform does not converge at a pole. Furthermore, the region of convergence is bounded by poles. Hassan Bhatti, DSP, Spring 2010

11. Example Region of convergence  The z-transform is given by: a Which converges to: Clearly, X(z) has a zero at z = 0 and a pole at z = a. Hassan Bhatti, DSP, Spring 2010

12. Convergence of Finite Sequences Suppose that only a finite number of sequence values are nonzero, so that: Where n1 and n2 are finite integers. Convergence requires So that finite-length sequences have a region of convergence that is at least 0 < |z| < , and may include either z = 0 or z = . Hassan Bhatti, DSP, Spring 2010

13. Z-Transform of sin/cos Z-Transform Time Domain Hassan Bhatti, DSP, Spring 2010

14. Exponentially Modulated sin/cos A damped oscillating signal – a typical output of a second order system Hassan Bhatti, DSP, Spring 2010

15. Inverse z-Transform The inverse z-transform can be derived by using Cauchy’s integral theorem. Start with the z-transform Multiply both sides by zk-1 and integrate with a contour integral for which the contour of integration encloses the origin and lies entirely within the region of convergence of X(z): Hassan Bhatti, DSP, Spring 2010

16. Z Z-1 Z Z-Transform/Inverse Z-Transform LTI: yimpuse(k)=0.3k-1 u(k)=0.7k y(k)? = * (convolution) Transfer Function · (multiplication) = Hassan Bhatti, DSP, Spring 2010

17. Inverse Z-Transform • Table Lookup – if the Z-Transform looks familiar, look it up in the Z-Transform table! • Long Division • Partial Fraction Expansion Z u(k) U(z) Z-1? Z-1? Hassan Bhatti, DSP, Spring 2010

18. Long Division • Sort both nominator and denominator with descending order of z first • u(0)=3, u(1)=5, u(2)=7, u(3)=9, …, guess: u(k)=3ustep(k)+2uramp(k) Hassan Bhatti, DSP, Spring 2010

19. where k>0 Partial Fraction Expansion • Many Z-transforms of interest can be expressed as division of polynomials of z May be trickier: complex root duplicate root Hassan Bhatti, DSP, Spring 2010

20. Real Unique Poles Hassan Bhatti, DSP, Spring 2010

21. Repeat Real Poles(1) Hassan Bhatti, DSP, Spring 2010

22. Repeat Real Poles(2) Hassan Bhatti, DSP, Spring 2010

23. Complex Poles(1) Hassan Bhatti, DSP, Spring 2010

24. Complex Poles(2) Hassan Bhatti, DSP, Spring 2010

25. An Example Z-1 U1(z)=c0 u1(k)=c0*uimpulse(k) Z-1 u2(k)=c1*2k-1, k>0 (z-2)(z-4) Z-1 u2(k)=c2*4k-1, k>0 c0? c1? c2? Hassan Bhatti, DSP, Spring 2010

26. Get The Constants! (z-2)(z-4) Hassan Bhatti, DSP, Spring 2010

27. Partial Fraction Expansion – cont’d How to get c0 and cj’s ? define Hassan Bhatti, DSP, Spring 2010

28. An Example – Complete Solution Hassan Bhatti, DSP, Spring 2010

29. Z Z-1 Solving Difference Equations Transfer Function Hassan Bhatti, DSP, Spring 2010

30. A Difference Equation Example Exponentially Weighted Moving Average y(k)=cy(k-1)+(1-c)u(k-1) Hassan Bhatti, DSP, Spring 2010

31. Z Z Z-1 Solve it! LTI: y(k)=0.4y(k-1)+0.6u(k-1) u(k)=0.8k y(k)? Hassan Bhatti, DSP, Spring 2010

32. Properties • z-transforms are linear: • The transform of a shifted sequence: • Multiplication: But multiplication will affect the region of convergence and all the pole-zero locations will be scaled by a factor of a. Hassan Bhatti, DSP, Spring 2010

33. Convolution of Sequences Hassan Bhatti, DSP, Spring 2010

34. More Definitions Definition. Periodic. A sequence x(n) is periodic with period  if and only if x(n) = x(n + ) for all n. Definition. Shift invariant or time-invariant. Consider a sequence y(n) as the result of a transformation T of x(n). Another interpretation is that T is a system that responds to an input or stimulusx(n):y(n) = T[x(n)]. The transformation T is said to be shift-invariant or time-invariant if: y(n) = T [x(n)] implies that y(n - k) = T [x(n – k)] For all k. “Shift invariant” is the same thing as “time invariant” when n is time (t). Hassan Bhatti, DSP, Spring 2010

35. This implies that the system can be completely characterized by its impulse response h(n). This obviously hinges on the stationarity of the series. Hassan Bhatti, DSP, Spring 2010

36. Definition. Stable System. A system is stable if Which means that a bounded input will not yield an unbounded output. Definition. Causal System. A causal system is one in which changes in output do not precede changes in input. In other words, Linear, shift-invariant systems are causal iff h(n) = 0 for n < 0. Hassan Bhatti, DSP, Spring 2010

37. Here H(ei) is called the frequency response of the system whose impulse response is h(n). Note that H(ei) is the Fourier transform of h(n). Hassan Bhatti, DSP, Spring 2010

38. We can generalize this state that: These are the Fourier transform pair. This implies that the frequency response of a stable system always converges, and the Fourier transform exists. Hassan Bhatti, DSP, Spring 2010

39. Hassan Bhatti, DSP, Spring 2010

40. Z Transformation Pairs Hassan Bhatti, DSP, Spring 2010

41. Z Transformation Pairs Hassan Bhatti, DSP, Spring 2010

42. Properties of Z Transform, Linearity Hassan Bhatti, DSP, Spring 2010

43. Time Shifting Hassan Bhatti, DSP, Spring 2010

44. Multiplication by Exponential Sequence Hassan Bhatti, DSP, Spring 2010

45. Time Reversal Hassan Bhatti, DSP, Spring 2010

46. Convolution of Sequences Hassan Bhatti, DSP, Spring 2010

47. Properties of Z Transformation Hassan Bhatti, DSP, Spring 2010