2.2 Linear-Phase Transfer Functions 2.2 Linear-Phase Transfer Functions 2.2 Linear-Phase Transfer Functions Consider first an FIR filter with a symmetric .A real-coefficient polynomial H(z)satisfying 。It follows the relation H(z)=±z-w-Hz' impulse response:h(n)=h(N-1-n) the above condition is called a mirror-image that if z=z is a zero of H(2),so is=1/z Its transfer function can be written as polynomial (MIP) V-I N-I Moreover,for an FIR filter with a real impulse H(e)=∑m)z=∑h(N-1-n)z In the case of anti-symmetric impulse response,the zeros of H(z)occur in complex =0 m-0 response,the corresponding expression is conjugate pairs By making a change of variable m=N-1- n,we can write H(2)=-2-N-DH(2-) Hence,a zero at z=z is associated with a zero He=艺Mm2-=2me which is called an antimirror-image at z=z* polynomial(AIP) =3-WH() 44 2.2 Linear-Phase Transfer Functions 2.2 Linear-Phase Transfer Functions 2.2 Linear-Phase Transfer Functions Am ●Since a zero at=士1 is its own reciprocal,,it can appear only singly Likewise,for a Type 3 or 4 filter, H1)=-H(1) Now a Type 2 FIR filter satisfies H(z)=z-w-H(2- implying H(z)must have a zero at z=1 Ret with degree N-1 odd .On the other hand,only the Type 3 FIR filter is restricted to have a zero at z=-1 since here ·Hence,H(-1)=(-l)w-H(-1)=-H(-1) the degree N-I is even and hence, implying (-1)=0,i.e.,H(z)must have a H(-1)=-(-1)w-H(-1)=-H(-1) Zero at z=-1 443 2.2 Linear-Phase Transfer Functions Consider first an FIR filter with a symmetric impulse response: Its transfer function can be written as By making a change of variable m=Nˉ1ˉ n ,we can write hn hN n () ( 1 )   1 1 0 0 () () ( 1 ) N N n n n n H z hnz hN nz          1 1 ( 1 ) ( 1) 0 0 ( 1) 1 () ( ) ( ) ( ) N N Nm N m m m N H z hmz z hmz z Hz              44 2.2 Linear-Phase Transfer Functions A real-coefficient polynomial H(z) satisfying the above condition is called a mirror-image polynomial (MIP) polynomial In the case of anti-symmetric impulse response, the corresponding expression is which is called an antimirror antimirror-image polynomial (AIP) polynomial ( 1) 1 () ( ) N Hz z Hz     45 2.2 Linear-Phase Transfer Functions It follows the relation that if z=zi is a zero of H(z), so is z=1/zi Moreover, for an FIR filter with a real impulse response, the zeros of H(z) occur in complex conjugate pairs Hence, a zero at z=zi is associated with a zero at z=zi* ( 1) 1 () ( ) N Hz z Hz     46 2.2 Linear-Phase Transfer Functions Re z jIm z 1z * 1z 1 1 z * 1 z 2 z 2 1 z 3 z * 3 z 4 z 47 Since a zero at z=f1 is its own reciprocal, it can appear only singly Now a Type 2 FIR filter satisfies with degree Nˉ1 odd Hence, implying H(ˉ1)=0 ,i.e., H(z) must have a zero at z=ˉ1 2.2 Linear-Phase Transfer Functions ( 1) 1 () ( ) N Hz z Hz    ( 1) ( 1) ( 1) ( 1) ( 1) N H HH         48 Likewise, for a Type 3 or 4 filter, implying H(z) must have a zero at z=1 On the other hand, only the Type 3 FIR filter is restricted to have a zero at z=ˉ1 since here the degree Nˉ1 is even and hence, 2.2 Linear-Phase Transfer Functions H H (1) (1)   ( 1) ( 1) ( 1) ( 1) ( 1) N H HH