In this problem we consider the effect of complex-conjugate pair of poles and zeros on the frequency response of a system. Let Hz (?) = (1 ? re^j0 e^-j? ) (1- re^-j0e^-j?)

(a) Show that the magnitude response in decibels is |H.()an = 10 loge(1+r-2r cos(w- f)] - 10log,f1 +r - 2r cosu + e) (b) Show that the phase response is given as r sin(w - A) 1-r cos(w -A) sintw + A) + tan- ), (w) = tan 1-r cos(a) + e (c) Show that the group delay is given as - r cos(w - ) 1+r - 2r cos(w - 0) -r cos(w + 4) 1 +r - 2r cos(w - 0) (w) = (d) If H, tw) = 1/H.(@), show that (H,(w)leH = -|H.(a)lan %3D O, (w) = -0,(w) T (w) = -1, tw) %3D (e) Plot |H,(u)!, O,(w) and t(w) for r = 0.9, and 6 0.T/2. %3D