there is error in the theory of line spectrum pair?

Started by May 29, 2009
I'm studying the IEEE article "LINE SPECTRUM PAIR (LSP) AND SPEECH DATA COMPRESSION" by author FRANK K. SOONG and BIING-H WANG JUANG.they proved that the LSP is practical.but I doubt about the theory.

LPC inverse filter A(Z)=1+a1*Z^(-1)+a2*Z^(-2)+...+am*Z^(-m)

they proved that P(z) and Q(z) only have zeros on the unit circle if A(z) only have zeros inside the unit circle. they use a formula to prove the point,that is |Zi*Z-1|^2-|Z-Zi|^2=(1-|z|^2)*(1-|Zi|^2) Zi is constant and |Zi|<1
I think the formula in not correct,so i doubt about the theory of LSP.I know LSP is widely used in speech coding.anybody can help me?