Linear-Phase FIR Transfer Functions It is nearly impossible to design a linear- phase iir transfer function It is al ways possible to design an Fir transfer function with an exact linear-phase response Consider a causal Fir transfer function H(z) of length n+1. 1. e. of order w N 0 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 1 Linear-Phase FIR Transfer Functions • It is nearly impossible to design a linearphase IIR transfer function • It is always possible to design an FIR transfer function with an exact linear-phase response • Consider a causal FIR transfer function H(z) of length N+1, i.e., of order N: = − = N n n H z h n z 0 ( ) [ ]
Linear-Phase FIR Transfer Functions The above transfer function has a linear phase, if its impulse response hn] is either symmetric, 1.e hn]=hN-n],0≤n≤N or is antisymmetric, 1.e hn]=-h[N-n],0≤n≤N Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 2 Linear-Phase FIR Transfer Functions • The above transfer function has a linear phase, if its impulse response h[n] is either symmetric, i.e., or is antisymmetric, i.e., h[n] = h[N − n], 0 n N h[n] = −h[N − n], 0 n N
Linear-Phase FIR Transfer Functions Since the length of the impulse response can be either even or odd, we can define four types of linear-phase FIr transfer f unctions For an antisymmetric fir filter of odd length, 1. e..N even h[N2]=0 We examine next the each of the 4 cases Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 3 Linear-Phase FIR Transfer Functions • Since the length of the impulse response can be either even or odd, we can define four types of linear-phase FIR transfer functions • For an antisymmetric FIR filter of odd length, i.e., N even h[N/2] = 0 • We examine next the each of the 4 cases
Linear-Phase fir Transfer Functions hin] hnI 4 0 3:4 Center of Center of symmetry symmet Type 1: N=8 Type 2: N=7 h[nI hnl 3 6 Center of symmetty Type 3: N=8 Type 4: N=7 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 4 Linear-Phase FIR Transfer Functions Type 1: N = 8 Type 2: N = 7 Type 3: N = 8 Type 4: N = 7
Linear-Phase FIR Transfer Functions Type 1: Symmetric Impulse response with Odd length In this case, the degree N is even Assume n=8 for simplicity The transfer function H(z)is given by H()=h[0]+1-1+h2|=2+h3}23 +小[414+h55+小6]6+h77+h88 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 5 Linear-Phase FIR Transfer Functions Type 1: Symmetric Impulse Response with Odd Length • In this case, the degree N is even • Assume N = 8 for simplicity • The transfer function H(z) is given by 1 2 3 H z h h z h z h z ( ) [0] [1] [2] [3] − − − = + + + 4 5 6 7 8 4 5 6 7 8 − − − − − + h[ ]z + h[ ]z + h[ ]z + h[ ]z + h[ ]z
Linear-Phase fir Transfer Functions Because of symmetry, we have h[o]h[8 h[l]=h7,h2]=h6],andh[3]=h[5 Thus we can write H(x)=h0](+3)+(x1+7) +和2](=2+6)+3(3+25)+414 z4{40(=4+4)+1(3+23) +和2(2+2-2)+h3(+21)+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 FIR Transfer Functions The corresponding frequency response is then given by H(e0)=e/+0{2h0cos(4o)+2h1]os(3o) +2h2]cos(20)+2h[3]cos(0)+h4 The quantity inside the braces is a real unction of 0, and can assume positive or negative values in the range0≤0)≤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 fir 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 τ(0) de(o) 4 indicating a constant group delay of 4 samples 8 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 FIR Transfer Functions In the general case for Type 1 FIR filters the frequency response is of the form H(e10)=e j/2 H(0) where the amplitude response h(o), also called the zero-phase response, is of the form N/2 H(0)=,]+2∑h少-n]cos(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 H10()=[+z1+z2+z3+z4+x5+l=6 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 10 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
