#### Transcript Fourier (1) - Petra Christian University

Z Transform (2) Hany Ferdinando Dept. of Electrical Eng. Petra Christian University Overview Unilateral Z transform Z transform in LTI system Convolution and deconvolution Frequency response analysis Applications Z Transform (1) - Hany Ferdinando 2 Unilateral Z Transform The general formula of z transform is X(z) x(n)z n n This is bilateral z transform. Consider that the range of n is from –∞ to ∞. For Unilateral z transform, the formula becomes X(z) x(n)zn n 0 Z Transform (1) - Hany Ferdinando 3 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 4 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 5 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 6 Convolution and Deconvolution y = h * u in the time domain becomes Y = HU in the z-domain Therefore, we can write it as y Z Hz Uz 1 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 7 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 8 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 9 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 10 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 11 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 12