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there is error in the theory of line spectrum pair?

Started by tms3...@yahoo.com.cn May 29, 2009
hello,everyone
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)
P(z)=A(z)+A(Z^(-1))*Z^(-m-1)
Q(z)=A(z)-A(Z^(-1))*Z^(-m-1)

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?