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