延安大学：《数字信号处理》课程PPT教学课件（DigitalSignal Processing，DSP）Some special filters

Some special Filters Copyright C 2001, S K Mitra

1 Copyright © 2001, S. K. Mitra Some Special Filters

Some Special Filters Allpass Filter · Comb filter Minimum-Phase and Maximum-Phase Copyright C 2001, S K Mitra

2 Copyright © 2001, S. K. Mitra Some Special Filters • Allpass Filter • Comb Filter • Minimum-Phase and Maximum-Phase

Allpass Transfer Function Definition An iir transfer function A(=)with unity magnitude response for all frequencies, 1.e A(e/@ )/2 or all o is called an allpass transfer function An m-th order causal real-coefficient allpass transfer function is of the form Ay()=±aM+a+…+1 M+1 M z 1+d1z-+…+dM-12 M+1 M Copyright C 2001, S K Mitra

3 Copyright © 2001, S. K. Mitra Allpass Transfer Function Definition • An IIR transfer function A(z) with unity magnitude response for all frequencies, i.e., is called an allpass transfer function • An M-th order causal real-coefficient allpass transfer function is of the form =   | ( )| 1, for all j 2 A e M M M M M M M M M d z d z d z d d z d z z A z − + − − − − − + − − + + + + + + + + =  1 1 1 1 1 1 1 1 1 ... ... ( )

Allpass Transfer Function If we denote the denominator polynomial of M(2) as DA/(二) DM(z)=1+d12+…+dM-12 M+1 then it follows that AM(z)can be written as -M AM(z)=± 2 Note from the above that ()jD is a pole of a real coefficient allpass transfer function, then it has a zero at z=le- jp Copyright C 2001, S K Mitra

4 Copyright © 2001, S. K. Mitra Allpass Transfer Function • If we denote the denominator polynomial of as : then it follows that can be written as: • Note from the above that if is a pole of a real coefficient allpass transfer function, then it has a zero at AM (z) DM (z) M M M DM z d z dM z d z − + − − − = + + + + 1 1 1 1 1 ... ( ) AM (z) ( ) ( ) ( ) D z z D z M M M M A z − −1 =   = j z re −  = j r z e 1

Allpass Transfer Function The numerator of a real-coefficient allpass transfer function is said to be the mirror- image polynomial of the denominator, and vice versa We shall use the notation DM(z)to denote the mirror-image polynomial of a degree-M polynomial DM(z),i.e DM(2=2DM(2 Copyright C 2001, S K Mitra

5 Copyright © 2001, S. K. Mitra Allpass Transfer Function • The numerator of a real-coefficient allpass transfer function is said to be the mirror￾image polynomial of the denominator, and vice versa • We shall use the notation to denote the mirror-image polynomial of a degree-M polynomial , i.e., DM (z) ~ DM (z) D (z) z DM (z) M M − = ~

Allpass Transfer Function · The expression M AM/(=)=土 2 D M(2 2 implies that the poles and zeros of a real coefficient allpass function exhibit mirror image symmetry in the z-plane 02+0.18z1+0.42+2÷0s A(二)= 1+0.4z-1+0.18z-2-0.2z 0 Real part Copyright C 2001, S K Mitra

6 Copyright © 2001, S. K. Mitra Allpass Transfer Function • The expression implies that the poles and zeros of a real￾coefficient allpass function exhibit mirror￾image symmetry in the z-plane ( ) ( ) ( ) D z z D z M M M M A z − −1 =  1 2 3 1 2 3 3 1 0.4 0.18 0.2 0.2 0.18 0.4 ( ) − − − − − − + + − − + + + = z z z z z z A z -1 0 1 2 3 -1.5 -1 -0.5 0 0.5 1 1.5 Real Part Imaginary Part

Allpass Transfer Function To show that AM(e)=1 we observe that Av(2-1)=±=MDv(=) v/( Therefore AM/(二)AM(二-)= - MDM 2 MDM 2 2 DM(=) 2 Hence Ar(e0)2=AM(=)A1(x1 Iz=e Copyright C 2001, S K Mitra

7 Copyright © 2001, S. K. Mitra Allpass Transfer Function • To show that we observe that • Therefore • Hence ( ) 1 ( ) 1 ( ) − =  − D z z D z M M M M A z ( ) ( ) ( ) 1 ( ) 1 1 ( ) ( ) − − − = − D z z D z D z z D z M M M M M M M M A z A z | ( )|=1 j AM e | ( )| ( ) ( ) 1 2 1 = =  =  − j M M z e j AM e A z A z

Allpass Transfer Function Now. the poles of a causal stable transfer function must lie inside the unit circle in the z-plane Hence, all zeros of a causal stable allpass transfer function must lie outside the unit circle in a mirror-image symmetry with its poles situated inside the unit circle Copyright C 2001, S K Mitra

8 Copyright © 2001, S. K. Mitra Allpass Transfer Function • Now, the poles of a causal stable transfer function must lie inside the unit circle in the z-plane • Hence, all zeros of a causal stable allpass transfer function must lie outside the unit circle in a mirror-image symmetry with its poles situated inside the unit circle

Allpass Transfer Function Figure below shows the principal value of the phase of the 3rd-order allpass function 0.2+0.18z-1+0.4z-2+ A3(=) 1+0.4x-1+0.18z-2-0.2z note the discontinuity by the amount of 2t in the phase e(o) Principal value of phase -2 0.8 o/π Copyright C 2001, S K Mitra

9 Copyright © 2001, S. K. Mitra Allpass Transfer Function • Figure below shows the principal value of the phase of the 3rd-order allpass function • Note the discontinuity by the amount of 2p in the phase q() 1 2 3 1 2 3 3 1 0.4 0.18 0.2 0.2 0.18 0.4 ( ) − − − − − − + + − − + + + = z z z z z z A z 0 0.2 0.4 0.6 0.8 1 -4 -2 0 2 4 /p Phase, degrees Principal value of phase

Allpass Transfer Function If we unwrap the phase by removing the discontinuity, we arrive at the unwrapped phase function 0c(o)indicated below Note: The unwrapped phase function is a continuous function of o Unwrapped phase 10 0.2 0.4 0.6 0.8 Copyright C 2001, S K Mitra

10 Copyright © 2001, S. K. Mitra Allpass Transfer Function • If we unwrap the phase by removing the discontinuity, we arrive at the unwrapped phase function indicated below • Note: The unwrapped phase function is a continuous function of  q () c 0 0.2 0.4 0.6 0.8 1 -10 -8 -6 -4 -2 0 /p Phase, degrees Unwrapped phase