Efficient LPC Quantization Methods - Digital Speech: Coding for Low Bit Rate Communication Systems - page 92

and,

Q p + 1 (z)
( 1

Q (z) = (5.19)

+ z)

B 0 z p B 1 z (p − 1 ) = + + ... + B p

where,

A 0 = 1 (5.20)

B 0 = 1 (5.21)

A k = (α k − α p + 1 − k ) + A k − 1 (5.22)

B k = (α k + α p + 1 − k ) − B k − 1 (5.23)

=

for k 1 , ... ,p

The LSFs are the angular positions of the roots of P (z) and Q (z) with

0 π . The roots occur in complex conjugate pairs and have the

following properties:

≤ ω i ≤

1. All roots of P (z) and Q (z) lie on the unit circle.
2. The roots of Q (z) and P (z) alternate with each other on the unit circle, i.e.
the following is always satisfied, 0 ≤ ω q, 0 <ω p, 0 <ω q, 1 <ω p, 1 ... , ≤ π .

5.3.1 ComplexRootMethod
The roots of equation (5.18) can be solved using complex arithmetic. This
will give complex conjugate roots on the unit circle and the frequencies are
then given by the inverse tangent of the roots. This method is obviously very
complex as it involves solving two polynomials of p th order using complex
arithmetic. Also, as it uses an iteration procedure for determining the roots,
the time required for this method is not deterministic which is undesirable
for real-time implementations.

5.3.2 RealRootMethod
As the coefficients of P (z) and Q (z) are symmetrical the order of equation
(5.18) can be reduced to p/ 2.

A p − 1
1

P (z) = A 0 z p A 1 z 1 + + ... + + A 0 (5.24)

z p/ 2 [ A 0 (z p/ 2 z − p/ 2 ) A 1 (z (p/ 2 − 1 ) z − (p/ 2 − 1 ) ) = + + + + ... + A p/ 2 ]