Finite Impulse Response Filters - Introduction to Digital Signal Processing and Filter Design - page 259

Similarly, the FIR filters with antisymmetric coefficients satisfy the property

z − N H(z − 1 )

H(z) =− (5.26)

A polynomial H(z) satisfying (5.25) is called a mirror image polynomial ,and
the polynomial that satisfies (5.26) is called an anti-mirror image polynomial .
We see that a polynomial H(z) that has symmetric coefficients is a mirror image
polynomial and one with antisymmetric coefficients is an anti-mirror image
polynomial. The reverse statement is also true and can be proved, namely, that
a mirror image polynomial has symmetric coefficients and an anti-mirror image
polynomial has antisymmetric coefficients.
From (5.25) and (5.26), it is easy to note that in a mirror image polynomial
as well as an anti-mirror image polynomial, if z

z 1 is a zero of H(z) ,then
1 /z is also a zero of H(z) . If the zero z 1 is a complex number r 1 e jφ

=

; | r | < 1,

r 1 e − jφ is also a zero. Their reciprocals ( 1 /r 1 )e − jφ and ( 1 /r 1 )e jφ are
also zeros of H(z) , which lie outside the unit circle |

then z 1 =

| = 1. Therefore complex
zeros of mirror image polynomials and anti-mirror image polynomials appear
with quadrantal symmetry as shown in Figure 5.4. If there is a zero on the unit
circle (e.g., at z 0 =

z

e − jφ is already located on the unit
circle, as the complex conjugate of z 0 , and therefore zeros on the unit circle do
not have quadrantal symmetry. Obviously a zero on the real axis at z r

e jφ ) , its reciprocal z − 1 =

r inside
the unit circle will be paired with one outside the unit circle on the real axis at
z − 1
r

=

= 1 /r .

1

0.5

6

0

− 0.5

− 1

− 1.5 − 1 − 0.5 0 0.5 1 1.5

Real Part

Figure 5.4 Zero and pole locations of a mirror image polynomial.