1.Design of Equiripple Linear-Phase FIR Filters Chapter 10B Part B The linear-phase FIR filter obtained by minimizing the peak absolute value of ●●● ●●● s=max E(@) FIR Digital Filter Design Computer-Aided Design of FIR which is usually called the equiripple FIR Digital Filters filter After is minimized.the weighted error function E(@)exhibits an equiripple behavior in the frequency range R 1.Design of Equiripple 1.Design of Equiripple 1.Design of Equiripple Linear-Phase FIR Filters Linear-Phase FIR Filters Linear-Phase FIR Filters The general form of frequency response of a causal linear-phase FIR filter of length 2M+1: Parks-McClellan Algorithm H(e)=eoreiBH(@) Based on iteratively adjusting the coefficients ●For filter design, of H(@)until the peak absolute value of E(@) [1,in the passband where the amplitude response H()is a real D()= is minimized 0.in the stopband function of If peak absolute value of E(@)in a band .(@is required to satisfy the above desired Weighted error function is given by o.≤o≤aiso,then the absolute error response with a ripple of t in the passband E(@)=W(@)H(@)-D(@) satisfies and a ripple of in the stopband where D(@)is the desired amplitude response (o)-D(a)≤ ,0.≤ù≤ and W(@)is a positive weighting function 4
1 Chapter 10B FIR Digital Filter Design 2 Part B Computer-Aided Design of FIR Digital Filters 3 1. Design of Equiripple Linear-Phase FIR Filters The linear-phase FIR filter obtained by minimizing the peak absolute value of which is usually called the equiripple equiripple FIR filter After is minimized, the weighted error function exhibits an equiripple behavior in the frequency range R max ( ) E R E( ) 4 1. Design of Equiripple Linear-Phase FIR Filters The general form of frequency response of a causal linear-phase FIR filter of length 2M+1: where the amplitude response is a real function of Weighted error function is given by where is the desired amplitude response and is a positive weighting function / 2 ( ) () j jN j He e e H H( ) E WH D () () () () D( ) W ( ) 5 1. Design of Equiripple Linear-Phase FIR Filters Parks-McClellan Algorithm Based on iteratively adjusting the coefficients of until the peak absolute value of is minimized If peak absolute value of in a band is , then the absolute error satisfies H( ) E( ) E( ) a b 0 0 () () , ( ) H D a b W 6 1. Design of Equiripple Linear-Phase FIR Filters For filter design, is required to satisfy the above desired response with a ripple of in the passband and a ripple of in the stopband 1, in the passband ( ) 0, in the stopband D H( ) p s
1.Design of Equiripple Linear-Phase FIR Filters Thus,weighting function can be chosen either as in the passband W()= in the stopband or W()= 6,/.in the passband in the stopband
7 1. Design of Equiripple Linear-Phase FIR Filters Thus, weighting function can be chosen either as or 1, in the passband ( ) / , in the stopband p s W / , in the passband ( ) 1, in the stopband s p W