Question

7.) The transfer function of a transmission channel is given by H(z) = (2+1.4)(2+ + 2z + 4) (z + 0.8) (2 -0.6) In order to correct for the magnitude distortion introduced by the channel on a signal passing through it, we wish to connect a stable and causal digital filter characterized by a system func- tion G(2) at the receiving end. Determine G(z) such that the H(2) G(%) = 1 for | <. A good starting point would be to plot the pole-zero pattern of Hz). Feel free to ver- ify your analysis using Python and a Jupyter notebook.

Answer 1

7) Given lH(es) Gleju) = 1. | A(z)GC2)] =1 (17=evio). 1 GCZ) = - | H(25). Consider Gi(z)= Hitz (210-8)(2-06) (2+i)(23722+4) (Z+0.8) (Z-0.6) (z+14(z-(-1453:)) (2-61-v3i) Given that G (2) is a stable and Causal digital filter- ie. G(2) must have au of its poles inside the unit Grcie. Since the poles of Gicz) at 2=-131 are out Side the unit circie, Gica) is not both causal and stable

To make Gicz) equal to G/z), we multiply Gilz) with the transfer function of Alipags filter because for an All pass filter, the magnitude response Ap (2)) = 1 - so it Can not change the magnitude spectrum of Gilz) but it can cancel but the poles in lying outside the unit circle G(2)= G (2) Apca) G(z) = (2+0-8)(2-0.6) (2+14)(z-61tv3i)(2-67-v3i)) -.. * Aplz) Apce) _+-2)(-6-1+03:) C#+23:)2-(-1-vai)) (2-(#:)) (2-(+- :)) Aplz) - (2-4-1+v3i)(2-6-1-v3i) (2-6*+;)(z-( Z.:)) How G[2) = G'(z) Ap(z). G(z) - 0.25 (2+0.8)(2–0.6) (2+1.4)(2-6*+2:))(2-( -:))