## 5.2.3 LP to ISP conversion

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The LP filter coefficients *a _{k}, k=1,…,16*, are converted to the ISP representation for quantization and interpolation purposes. For a 16th order LP filter, the ISPs are defined as the roots of the sum and difference polynomials

( 10 )

and

( 11 )

respectively. (The polynomials *f’ _{1}(z)* and

*f’*are symmetric and antisymmetric, respectively). It can be proven that all roots of these polynomials are on the unit circle and they alternate each other [5].

_{2}(z)*f’*has two roots at

_{2}(z)*z*= 1

*(*

**=0)

*and*

*z*= -1 (

**=

**). To eliminate these two roots, we define the new polynomials

( 12 )

and

( 13 )

Polynomials *f _{1}(z)* and

*f*have 8 and 7 conjugate roots on the unit circle respectively. Therefore, the polynomials can be written as

_{2}(z)( 14 )

and

( 15 )

where *q _{i}=cos(_{i})* with

**being the immittance spectral frequencies (ISF) and

_{i}*a*[16] is the last predictor coefficient. ISFs satisfy the ordering property . We refer to as the ISPs in the cosine domain.

Since both polynomials *f*_{1}*(z)* and *f*_{2}*(z)* are symmetric only the first 8 and 7 coefficients of each polynomial, respectively, and the last predictor coefficient need to be computed.

The coefficients of these polynomials are found by the recursive relations

for *i*=0 to 7

( 16 )

where *m*=16 is the predictor order, and .

The ISPs are found by evaluating the polynomials *F*_{1}*(z)* and *F*_{2}*(z)* at 100 points equally spaced between 0 and and checking for sign changes. A sign change signifies the existence of a root and the sign change interval is then divided 4 times to better track the root. The Chebyshev polynomials are used to evaluate *F _{1}(z)* and

*F*[6]. In this method the roots are found directly in the cosine domain {

_{2}(z)*q*}. The polynomials

_{i}*F*and

_{1}(z)*F*evaluated atcan be written as

_{2}(z)and ( 17 )

with

and ( 18 )

where *T _{m}*=cos(

*m*) is the

*m*th order Chebyshev polynomial,

*f(i)*are the coefficients of either

*F*

_{1}

*(z)*or

*F*

_{2}

*(z)*, computed using the equations in (16). The polynomial

*C(x)*is evaluated at a certain value of

*x*= cos(

**) using the recursive relation:

where *n _{f}*=8 in case of

*C*

_{1}(

*x*) and

*n*=7 in case of

_{f}*C*

_{2}(

*x*), with initial values

*b*

_{nf}=

*f*(0) and

*b*

_{nf+1}=0. The details of the Chebyshev polynomial evaluation method are found in [6].