1. Z Transform (2) Hany Ferdinando Dept. of Electrical Eng. Petra Christian University

2. Z Transform (1) - Hany Ferdinando2 Overview Unilateral Z transform Z transform in LTI system Convolution and deconvolution Frequency response analysis Applications

3. Z Transform (1) - Hany Ferdinando3 Unilateral Z Transform The general formula of z transform is This is bilateral z transform. Consider that the range of n is from –∞ to ∞. For Unilateral z transform, the formula becomes

4. Z Transform (1) - Hany Ferdinando4 Unilateral Z Transform All properties of bilateral z transform can be used in unilateral z transform, except the shifting property For this, one can derived it from the formula This property is important in solving difference equation

5. Z Transform (1) - Hany Ferdinando5 Z Transform in LTI System The analysis of discrete-time LTI system cannot be separated from z transform. If X(z) is input, H(z) is impulse response of a system and Y(z) is output of that system, then Y(z) = H(z)X(z) (see convolution property) H(z) is referred to as the transfer function of the system

6. Z Transform (1) - Hany Ferdinando6 Z Transform in LTI System The stability and causality can be associated with constraints on the pole-zero pattern and RoC of the H(z)  If the system is causal, then the RoC of H(z) will be outside the outermost pole  If the system is stable, then the RoC of H(z) must include the unit circle  If the system is stable and causal, then both consequences above are fulfilled

7. Z Transform (1) - Hany Ferdinando7 Convolution and Deconvolution y = h * u in the time domain becomes Y = HU in the z-domain Therefore, we can write it as  H z is h in the z-domain and U z is u in the z-domain  Z -1 [ ] is inverse Z transform

8. Z Transform (1) - Hany Ferdinando8 Convolution and Deconvolution h = 2 k, k ≥ 0 and u = 2 -k, k ≥ 0. Convolve h and u  Find H(z) and U(z), don’t forget the RoC  Multiply H(z) and U(z)  Combine the RoCs  Find the inverse of their multiplication result

9. Z Transform (1) - Hany Ferdinando9 Convolution and Deconvolution h = {1,2,3} and y = {1,1,2,-1.3}. Find u if y = h*u  Find H(z) and Y(z)  it’s easy  Find U(z) from Y(z)/H(z)  Then take inverse Z transform from U(z) to get u

10. Z Transform (1) - Hany Ferdinando10 Frequency Response It is used to evaluate the digital filter The procedures:  Substitute z with e j   Separate real and imaginary part  Calculate the magnitude and the phase angle  Draw both results (for test, it is not necessary)

11. Z Transform (1) - Hany Ferdinando11 Application To solve linear difference equation To characterize the transfer function of discrete-time LTI system To design digital filter (it is in DSP course)

12. Z Transform (1) - Hany Ferdinando12 Next… Signals and Linear System by Robert A. Gabel, chapter 6, p 349-363 Signals and Systems by Alan V. Oppenheim, chapter 9, p 573-603 We have finished to discuss the z transform. No other way to understand the z transform well unless you exercise yourself.