Z Transform 2 Hany Ferdinando Dept of Electrical

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

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

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 3 Z Transform (1) - Hany Ferdinando

Unilateral Z Transform n n n 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 4 Z Transform (1) - Hany Ferdinando

Z Transform in LTI System n n n 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 5 Z Transform (1) - Hany Ferdinando

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

Convolution and Deconvolution n n 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 7 Z Transform (1) - Hany Ferdinando

Convolution and Deconvolution n 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 Ro. C Multiply H(z) and U(z) Combine the Ro. Cs Find the inverse of their multiplication result 8 Z Transform (1) - Hany Ferdinando

Convolution and Deconvolution n 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 9 Z Transform (1) - Hany Ferdinando

Frequency Response n n 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) 10 Z Transform (1) - Hany Ferdinando

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

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