华南理工大学电子与信息学院：《数字信号与处理》（英文版）Lecture 12 Linear-Phase fr Transfer Functions

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 linear￾phase 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

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