Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

1-18 Passive, Active, and Digital Filtersjω – z ijωs planez iψ ziσjωjω – z iz iψ ziσFIGURE 1.5Zero-pole plots of minimum-phase and corresponding nonminimum-phase filter.M zi ¼ M zi and c zi ¼ p c zirespectively. The amplitude response of the modified filter is obtained from Equation 1.16 asjM(v) ¼ H 0j Q Mi¼1 M z i QNi¼1 Mmij¼ H 0p ij Q Mi¼1 M z iQNi¼1 Mmi p i¼ M(v)Therefore, replacing the zeros of the transfer function by their mirror images leaves the amplituderesponse unchanged.The phase response of the original filter is given by Equation 1.17 asu(v) ¼ arg H 0 þ XMc zii¼1X Ni¼1m i c pi(1:21)and since c zi ¼ p c zi, the phase response of the modifier filter is given byu(v) ¼ arg H 0 þ XM¼ arg H 0 þ XMc zii¼1i¼1X Ni¼1(p c zi)m i c piX Ni¼1m i c pi(1:22)that is, the phase response of the modified filter is different from that of the original filter. Furthermore,from Equations 1.21 and 1.22u(v)u(v) ¼ XMi¼1(p 2c zi)and since p=2 c zi p=2, we haveu(v) u(v) 0