Linear-Phase fr Transfer Functions Because of symmetry, we have ho]=h 8 h1]=h[7],h[2]=h6],andh[3]=h[5 Thus we can write H()=h0(+23)+1(x+z7) +2(2+6)+h33+25)+4-4 =z4{h0(=4+24)+h](=3+x3) +h2](z2+2-2)+h3](z+2-)+h4]} Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 6 Linear-Phase FIR Transfer Functions • Because of symmetry, we have h[0] = h[8], h[1] = h[7], h[2] = h[6], and h[3] = h[5] • Thus, we can write 8 1 7 H z h z h z z ( ) [0](1 ) [1]( ) − − − = + + + 2 6 3 5 4 2 3 4 − − − − − + h[ ](z + z ) + h[ ](z + z ) + h[ ]z { [ ]( ) [ ]( ) 4 4 4 3 3 0 1 − − − = z h z + z + h z + z [2]( ) [3]( ) [4]} 2 2 1 + h z + z + h z + z + h − −
Linear-Phase fr Transfer Functions The corresponding frequency response is then given by H(e/0)=e40(2h[0]cos(40)+2h[]cos(30) +2{2]cos(20)+2h[3]cos(0)+h4]} The quantity inside the braces is a real function of @, and can assume positive or negative values in the range0≤o)≤π Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 7 Linear-Phase FIR Transfer Functions • The corresponding frequency response is then given by • The quantity inside the braces is a real function of w, and can assume positive or negative values in the range 0 w ( ) {2 [0]cos(4 ) 2 [1]cos(3 ) 4 = w + w w − w H e e h h j j + 2h[2]cos(2w) + 2h[3]cos(w) + h[4]}
Linear-Phase fr Transfer Functions The phase function here is given by 6(0)=-40+β Whereβ is either0orπ, and hence, It is a linear function of o in the generalized sense The group delay is given by dO(0) τ0 4 indicating a constant group delay of 4 samples Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 8 Linear-Phase FIR Transfer Functions • The phase function here is given by where b is either 0 or , and hence, it is a linear function of w in the generalized sense • The group delay is given by indicating a constant group delay of 4 samples (w) = −4w+b ( ) 4 ( ) w = − = w w d d
Linear-Phase fr Transfer Functions In the general case for Type 1 FiR filters the frequency response is of the form H(e/0)=e JNo/2 H(0) where the amplitude response H(o), also called the zero-phase response, is of the orm N/2 H(0)=]+2∑2-nlos(on) Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 9 Linear-Phase FIR Transfer Functions • In the general case for Type 1 FIR filters, the frequency response is of the form where the amplitude response , also called the zero-phase response, is of the form ( ) ( ) / 2 = w w − w H e e H j jN ~ H (w) ~ H (w) ~ = + − w = / 2 1 2 2 [ ] 2 [ ]cos( ) N n N N h h n n
Linear-Phase FR Transfer Functions Example- Consider 0(-)= 2 3,-4 5 z+2+2+2+2ˇ+12 2 which is seen to be a slightly modified version of a length-7 moving-average FIR filter The above transfer function has a symmetric impulse response and therefore a linear phase response Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 10 Linear-Phase FIR Transfer Functions • Example - Consider which is seen to be a slightly modified version of a length-7 moving-average FIR filter • The above transfer function has a symmetric impulse response and therefore a linear phase response ( ) [ ] 6 2 1 2 3 4 5 1 2 1 6 1 0 − − − − − − H z = + z + z + z + z + z + z