Lecture 2 : Z-Transform

1. Lecture 2: Z-Transform Xiliang Luo 2014/9

2. Fourier Transform

3. Convergence • A sufficient condition: absolutely summable • it can be shown the DTFT of absolutely summable sequence converge uniformly to a continuous function

4. Square Summable • A sequence is square summable if: • For square summable sequence, we have mean-square convergence:

5. Z-Transform a function of the complex variable: z If we replace the complex variable z by , we have the Fourier Transform!

6. Z-Transform & Fourier Transform

7. Complex z-plane

8. Region of Convergence • The set of z for which the z-transform converges is called ROC of the z-transform. • Absolutely summable criterion:

9. ROC • ROC consists of a ring in the z-plane

10. Closed-Form in ROC • When X(z) is a rational function inside ROC, i.e. • P(z), Q(z) are polynomials in z • Zeros: values of z such that X(z) = 0 • Poles: values of z such that X(z) = infinity

11. Z-Transform Example: Right-Sided

12. Z-Transform Example:Left-Sided

13. Diff. Sum, Same Z-Transform? • One is right-sided exponential sequence • One is left-sided exponential sequence • But they share the same algebraic expressions for their Z-Transforms • This emphasizes the importance of the region of convergence!!

14. ROC Properties

15. ROC Properties

16. ROC Properties

17. Inverse z-Transform • From the z-Transform, we can recover the original sequence using the following complex contour integral: C is a closed contour within the ROC of the z-transform

18. Inverse z-Transform Methods • Inspection • familiar with the common transform pairs • Partial Fraction Expansion • Power Series Expansion

19. z-Transform Properties • 1. Linearity • 2. Time Shifting • 3. Multiplication by an Exponential Sequence

20. z-Transform Properties • 4. Differentiation of X(z) • 5. Conjugation of a Complex Sequence • 7. Time Reversal

21. z-Transform Properties • 7. Convolution of Sequences

22. z-Transform and LTI Systems • LTI system is characterized by its impulse response h[n] h[n] y[n] x[n] H(z) is called the system function of this LTI system!

23. Cauchy-Riemann Equations • If function f(z) is differentiable at z0=x0+y0, then its component functions must satisfy the following conditions:

24. Analytic Functions • A function f(z) is analytic at a point z0 if it has a derivative at each point in some neighborhood of z0. • So, If f(z) is analytic at a point z0, it must be analytic at each point in some neighborhood of z0.

25. Taylor Series • Theorem: Suppose that a function f is analytic throughout a disk: |z-z0|<R0, centered at z0 and with radius R0, then f(z) has the power series representation:

26. Laurent Series • If a function is not analytic at a point z0, one cannot apply Taylor’s theorem at that point! • Laurent’s Theorem: Suppose a function f is analytic throughout an annular domain centered at z0: Let C denote any positively oriented simple closed contour around z0 and lying in the domain, then, at each point in the domain, f(z) has the series representation:

27. Laurent Series

28. Homework Problems 3.52: 3.56: 3.57: 3.59:

29. Next • Sampling of Continuous-Time Signals • Please read the textbook Chapter 4 in advance!