EE5410 Signal Processing -- linear-phase systems -- symmetry condition
1. In linear-phase systems, the symmetry conditions cause the zeros of H(z) to occur in mirror-image pairs. That is, if z0 is a zero of H(z), then 1/z0 is also a zero of H(z).
2. if h[n] is real, then the zeros of H(z) occur in complex-conjugate pairs.
3. therefore, a) real zeros NOT on the unit circle occur in reciprocal pairs.
b) complex zeros NOT on the unit circle occur in group of FOUR, corresponding to the complex conjugates and reciprocals.
c) if a zero is ON the unit circle, its reciprocal is also its conjugate. consequently, complex zeros on the unit circle are conveniently grouped into pairs.
d) Zeros at z=+-1 are their OWN reciprocal and complex conjugate.
4. Then, H(z) can be FACTORED into a product of first-, second-, and fourth-order factors.
5. Each of these factors is a polynomial whose coefficients have the SAME symmetry as the coefficients of H(z); ie., each factor is a linear-phase polynomial in z^-1.