Linear-Phase fr Transfer Functions a plot of the magnitude response of Ho(z) along with that of the 7-point moving average filter is shown below modified filter moving-average 0.8 30.6 0.4 0.2 0.2 0.4 0.6 0.8 0/兀 Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 11 Linear-Phase FIR Transfer Functions • A plot of the magnitude response of along with that of the 7-point movingaverage filter is shown below H (z) 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 w/ Magnitude modified filter moving-average
Linear-Phase FR Transfer Functions Note the improved magnitude response obtained by simply changing the first and the ast impulse response coefficients of a moving-average(MA)filter It can be shown that we an express H10(2)=(+z)(1+x1+22+3+z4+x35) which is seen to be a cascade of a 2-point ma filter with a 6-point ma filter Thus, Ho(z)has a double zero atz=-l 0=几 Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 12 Linear-Phase FIR Transfer Functions • Note the improved magnitude response obtained by simply changing the first and the last impulse response coefficients of a moving-average (MA) filter • It can be shown that we an express which is seen to be a cascade of a 2-point MA filter with a 6-point MA filter • Thus, has a double zero at , i.e., (w = ) ( ) ( ) ( ) 1 2 3 4 5 6 1 1 2 1 0 1 1 − − − − − − H z = + z + z + z + z + z + z H0 (z) z = −1
Linear-Phase fr Transfer Functions Type 2: Symmetric Impulse response with Even lengt gth In this case the degree n is odd Assume N=7 for simplicity The transfer function is of the form H(=)=0]+和]-1+h2]-2+h3]z3 +44+5=5+h61z-6+h77 Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 13 Linear-Phase FIR Transfer Functions Type 2: Symmetric Impulse Response with Even Length • In this case, the degree N is odd • Assume N = 7 for simplicity • The transfer function is of the form 1 2 3 0 1 2 3 − − − H(z) = h[ ]+ h[ ]z + h[ ]z + h[ ]z 4 5 6 7 4 5 6 7 − − − − + h[ ]z + h[ ]z + h[ ]z + h[ ]z
Linear-Phase fr Transfer Functions Making use of the symmetry of the impulse response coefficients. the transfer function can be written as H()=小0(+z)+hu](+z6 +h[2](二+二)+h{31 3,-4 z°+z 20(=2+72)+M(=512+-512) +h2(=32+2312)+小3(=12+2-12) Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 14 Linear-Phase FIR Transfer Functions • Making use of the symmetry of the impulse response coefficients, the transfer function can be written as ( ) [ ]( ) [ ]( ) 7 1 6 0 1 1 − − − H z = h + z + h z + z [ ]( ) [ ]( ) 2 5 3 4 2 3 − − − − + h z + z + h z + z { [ ]( ) [ ]( ) 7/ 2 7/ 2 7/ 2 5/ 2 5/ 2 0 1 − − − = z h z + z + h z + z [ ]( ) [ ]( )} 3/ 2 3/ 2 1/ 2 1/ 2 2 3 − − + h z + z + h z + z
Linear-Phase fr Transfer Functions The corresponding frequency response is given by H(ev=e j70/2 CO(70 20]co(2②)+2h[lcos( +2h2]cos(39)+2[3]co(9) As before the quantity inside the braces is a real function of@, and can assume positive or negative values in the range0≤o≤兀 15 Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 15 Linear-Phase FIR Transfer Functions • The corresponding frequency response is given by • As before, the quantity inside the braces is a real function of w, and can assume positive or negative values in the range ( ) {2 [0]cos( ) 2 [1]cos( ) 2 5 2 w − 7w/ 2 7w w H e = e h + h j j 2 [2]cos( ) 2 [3]cos( )} 2 2 3w w + h + h 0 w