1. The z-Transform Prof. Siripong Potisuk

2. LTI System description Previous basis function: unit sample or DT impulse  The input sequence is represented as a linear combination of shifted DT impulses.  The response is given by a convolution sum of the input and the reflected and shifted impulse response Now, use eigenfunctions of all LTI systems as basis function

3. Eigenfunctions of LTI Systems An eigenfunction of a system is an input signal that, when applied to a system, results in the output being the scaled version of itself. The scaling factor is known as the system’s eigenvalue. Complex exponentials are eigenfunctions of LTI systems, i.e., the response of an LTI system to a complex exponential input is the same complex exponential with only a change in amplitude.

4. s =  + j  z = re j 

5. If the input is linear combination of complex exponentials, then the output is also a linear combination of the same set of complex exponentials (both CT and DT). z = re j 

6. Z-transform of DT Signals The bilateral z-transform of a general DT signal x[n] is defined as

7. If the Fourier transform of converges, then the z-transform converges  The z-transform of a sequence has associated with it a region of convergence (ROC) on the complex z-plane defined as a range of values of z for which X(z) converges. Convergence Issue Ex. For r = 1, the z-transform reduces to the DTFT on the unit circle contour in the complex z-plane.

8. Region of Convergence

9. Ex.1 Find the z-transform of the following right-sided sequence This form to find inverse ZTusing PFE

10. Ex.2 Find the z-transform of the following left-sided sequence

11. Properties of the ROCs of the z-transform

14. The Unilateral Z-transform - The unilateral z-transform of a causal DT signal x[n] is defined as - Equivalent to the bilateral z-transform of x[n]u[n ] - Since x[n]u[n] is always a right-sided sequence, ROC of X(z) is always the exterior of a circle. - Useful for solving difference equations with initial conditions

15. Ex. 1 Find the z-transform of the following sequence x = {2, -3, 7, 4, 0, 0, ……..} Ex. 2 Find the z-transform of  [n]

16. Ex. 3 Find the z-transform of  [n -1] Ex. 4 Find the z-transform of  [n +1]

17. Ex. 5 Find the z-transform of u [n ] Ex. 6 Find the z-transform of

18. Rewriting x[n] as a sum of left-sided and right-sided sequences and finding the corresponding z-transforms, Notice from the ROC that the z-transform doesn’t exist for b > 1 where

19. Properties of z-Transform

21. Rational z-Transform For most practical signals, the z-transform can be expressed as a ratio of two polynomials

22. Rational z-Transform It is customary to normalize the denominator polynomial to make its leading coefficients one, i.e., Also, it x[n] is a causal signal, then X(z) will be a proper rational polynomial with

23. where where, for a fixed r, IDTFT DTFT (A contour integral) Inverse z-Transform

24. Synthetic Division Method Perform long division of the numerator polynomial by the denominator polynomial to produce the quotient polynomial q(z  1 ) Write X(z) as a normalized rational polynomial in z  1 by multiplying the numerator and denominator by z  N Identify coefficients in the power series definition of X (z) where

25. Ex. Find the inverse z-transform of Equating coefficients, Remarks: This method doesn’t produce a closed-form expression for x[n]

26. Ex. Find the inverse z-transform of

27. Partial Fraction Method Produce a closed-form expression for x[n] Write X(z) as a sum of terms, each of which can be inverse z-transformed by using a z-transform table Suitable for rational X(z) whose denominator polynomial has distinct real poles (multiplicity of one or simple real poles)

28. Partial Fraction Expansion Given X (z) with N distinct poles in factored form as express X(z)/z in terms of partial fraction expansion: The inverse z-transform of X(z) can be written as

29. Ex. Find the inverse z-transform of