5.Computer-Aided Design of lIR 5.Computer-Aided Design of IIR 5.Computer-Aided Design of IIR Digital Filters Digital Filters Digital Filters Objective--Determine iteratively the Chebyshev or minimax criterion Least-p Criterion coefficients of (z)so that the difference between D(ei)and H(e)over closed Minimizes the peak absolute value of the ·Minimizes weighted error: subintervals of 0sos is minimized s=max E(@) W(e)[H(e)-D(e)Tdo This difference usually specified as a where R is the set of disjoint frequency bands over the specified frequency range R withp a weighted error function in the range 0ss,on which D(e)is positive integer E(@)=W(e)[H(ei)-D(e) defined p=2 yields the least-squares criterion where W(e)is some user-specified For example,for a lowpass filter design,R is ·Asp→oo,the least p-th solution approaches weighting function the disjoint union of (0,@)and ( the minimax solution 5.Computer-Aided Design of lIR Digital Filters In practice,the p-th power error measure is approximated as e)-D where ,1sisK,is a suitably chosen dense grid of digital angular frequencies For linear-phase FIR filter design,H(ei)and D(e)are zero-phase frequency responses .For IIR filter design,H(e)and D(e)are magnitude functions37 5. Computer-Aided Design of IIR Digital Filters Objective Objective -- Determine iteratively the coefficients of H(z) so that the difference between and over closed subintervals of is minimized This difference usually specified as a weighted error function where is some user-specified weighting function ( ) j H e  ( ) j D e  0     () ( ) ( ) ( ) jj j E We He De    ! " # ( ) j W e  38 5. Computer-Aided Design of IIR Digital Filters Chebyshev or minimax criterion Minimizes the peak absolute value of the weighted error: where R is the set of disjoint frequency bands in the range , on which is defined For example, for a lowpass filter design, R is the disjoint union of and max ( ) E  $  % R 0     ( ) j D e  (0, ) p ( ,)  s 39 5. Computer-Aided Design of IIR Digital Filters Least-p Criterion Minimizes over the specified frequency range R with p a positive integer p=2 yields the least-squares criterion As pėĞ, the least p-th solution approaches the minimax solution () () () P jj j We He De d    $  % !  R " # 40 5. Computer-Aided Design of IIR Digital Filters In practice, the p-th power error measure is approximated as where , , is a suitably chosen dense grid of digital angular frequencies For linear-phase FIR filter design, and are zero-phase frequency responses For IIR filter design, and are magnitude functions & ' 1 ()() () ii i K P jj j i We He De   $ !  " # i 1 i K ( ) j H e  ( ) j D e  ( ) j H e  ( ) j D e