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 mirrorimage 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 − = ~