1.3 Allpass Transfer Function 1.3 Allpass Transfer Function Part A Types of Transfer Functions A Simple Application 1.Based on Magnitude Charncteristics .A simple but often used application of an G 1.1 Ideal Filters allpass filter is as a delay equalizer 12 Bounded Real Transfer Functions .Let G(z)be the transfer function of a digital Sincee)=1,we have 1.3 Allpass Transfer Functions filter designed to meet a prescribed magnitude response G(e)(e=G(ei) 2.Based on Phase Characteristics The nonlinear phase response of G(z)can be .Overall group delay is the given by the sum of 2.1 Zero-Phase Transfer Fanctioms 22 Linear-Phase Tramsfer Functions corrected by cascading it with an allpass filter the group delays of G(z)and A(z) 之3 linicum-Phase and》aximui-Phas A(z)so that the overall cascade has a constant Transfer Functions group delay in the band of interest 2 2.1 Zero-Phase Transfer Functions 2.1 Zero-Phase Transfer Functions 2.1 Zero-Phase Transfer Functions A second classification of a transfer function One zero-phase filtering scheme is sketched is with respect to its phase characteristics However,it is not possible to design a causal below In many applications,it is necessary that the digital filter with a zero phase(pp.287-288) digital filter designed does not distort the For non-real-time processing of real-valued x) V(e)U)) e-】 phase of the input signal components with input signals of finite length,zero-phase frequencies in the passband From the figure,we can arrive at filtering can be very simply implemented by .One way to avoid any phase distortion is to relaxing the causality requirement Y(ei)=W'(e)=H(e(e)=H(ei(e) make the frequency response of the filter real =H'(e)H(e)X(e)H(eX(e) and nonnegative,i.e.,to design the filter with a zero phase characteristic 29 Real and Zero-Phase25 1.3 Allpass Transfer Function A Simple Application A simple but often used application of an allpass filter is as a delay equalizer delay equalizer Let G(z) be the transfer function of a digital filter designed to meet a prescribed magnitude response The nonlinear phase response of G(z) can be corrected by cascading it with an allpass filter A(z) so that the overall cascade has a constant group delay in the band of interest 26 1.3 Allpass Transfer Function Since , we have Overall group delay is the given by the sum of the group delays of G(z) and A(z) G(z) A(z) 2 () 1 j A e  ( )( ) ( ) jj j Ge Ae Ge  27 Part A Types of Transfer Functions 1. Based on Magnitude Characteristics 1. Based on Magnitude Characteristics 1.1 Ideal Filters 1.1 Ideal Filters 1.2 Bounded Real Transfer Functions 1.2 Bounded Real Transfer Functions 1.3 Allpass Allpass Transfer Functions Transfer Functions 2. Based on Phase Characteristics 2. Based on Phase Characteristics 2.1 Zero-Phase Transfer Functions Phase Transfer Functions 2.2 Linear-Phase Transfer Functions Phase Transfer Functions 2.3 Minimum-Phase and Maximum-Phase Transfer Functions Transfer Functions 28 2.1 Zero-Phase Transfer Functions A second classification of a transfer function is with respect to its phase characteristics In many applications, it is necessary that the digital filter designed does not distort the phase of the input signal components with frequencies in the passband One way to avoid any phase distortion is to make the frequency response of the filter real and nonnegative, i.e., to design the filter with a zero phase zero phase characteristic 29 2.1 Zero-Phase Transfer Functions However, it is not possible to design a causal digital filter with a zero phase (pp. 287-288) For non-real-time processing of real-valued input signals of finite length, zero-phase filtering can be very simply implemented by relaxing the causality requirement 30 2.1 Zero-Phase Transfer Functions One zero-phase filtering scheme is sketched below From the figure, we can arrive at x(n) H(z) Folding H(z) Folding v(n) u(n) w(n) y(n) ( ) j X e ( ) j V e ( ) j U e ( ) j W e ( ) j Y e * ** * 2 * ( ) ( ) ( ) ( ) ( )( ) ( )( )( ) ( ) ( ) j j j j jj jjj j j Ye W e H e U e H e Ve H e He Xe He Xe     Real and Zero-Phase