Z Transform (2)

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

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

3. 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 Z Transform (1) - Hany Ferdinando

4. 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 Z Transform (1) - Hany Ferdinando

5. 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 Z Transform (1) - Hany Ferdinando

6. 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 Z Transform (1) - Hany Ferdinando

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

8. Convolution and Deconvolution • h = 2k, 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 Z Transform (1) - Hany Ferdinando

9. 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 Z Transform (1) - Hany Ferdinando

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

11. 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) Z Transform (1) - Hany Ferdinando

12. Next… We have finished to discuss the z transform. No other way to understand the z transform well unless you exercise yourself. • 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 Z Transform (1) - Hany Ferdinando