Linear-Phase FIR Transfer Functions a plot of the magnitude response of Ho(z) along with that of the 7-point moving average filter is shown below nodified filter 0.8 moving-average 号0.6 ∑0 0.2 0.2 0.4 0.6 0.8 /π 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 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 -3,-4 z 21+2-+2+z z which is seen to be a cascade of a 2-point Ma filter with a 6-point Ma filter Thus, Ho() has a double zero atz=-1 12 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 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 H(=)=硕0]+Mk1+h22+h3]3 +[41-4+5=-5+616+列7-7 13 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 FIR Transfer Functions Making use of the symmetry of the impulse response coefficients the transfer function can be written as H(z)=h0](1+x-)+h(二+) +2|(2+5)+h3(3+z-4) =2720(=72+2-72)+小[1(=32+-512) +h21(-3/21--3/2 )+h3(=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 FIR Transfer Functions The corresponding frequency response is given by H(e)=e /o12h[0]cos(o)+2h[1]cos(5o) +2h2]co(39)+23]cos(9)} As before, the quantity inside the braces is a real function of O, and can assume positive or negative values in the range0≤0≤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