Linear-Phase fr 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. e, of order N H()=∑20小n=n 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 fr Transfer Functions The above transfer function has a linear phase, if its impulse response hn] is either symmetric, 1.e h{n]=h{N-n],0≤n≤N or is antisymmetric, 1.e h{]=-h[N-nl20≤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 fr Transfer Functions Since the length of the impulse response can be either even or odd, we can define four types oflinear phase FIR transfer functions For an antisymmetric fir filter of odd length, i.e. Neven 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 hInI n 013:4578 3:4 Center of Center of symmetry symmetry Type 1: N=8 Type 2: N=7 h[n] hn] 6 6 Center of Center of symmetr ry symmetty Type 3: N=& 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 fr 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()is given by H(二)=0]+hu]-1+h212+3]z3 +44+h5]5+66+h7=-7+h8]8 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