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’₂(z) 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]. f’₂(z) has two roots at z = 1 (=0) and z = -1 ( = ). To eliminate these two roots, we define the new polynomials

( 12 )

and

( 13 )

Polynomials f₁(z) and f₂(z) have 8 and 7 conjugate roots on the unit circle respectively. Therefore, the polynomials can be written as

( 14 )

and

( 15 )

where q_i=cos(_i) with _i being the immittance spectral frequencies (ISF) and 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 f₁(z) and f₂(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 F₁(z) and F₂(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 F₁(z) and F₂(z) [6]. In this method the roots are found directly in the cosine domain {q_i}. The polynomials F₁(z) and F₂(z) evaluated atcan be written as

and ( 17 )

with

and ( 18 )

where T_m=cos(m) is the mth order Chebyshev polynomial, f(i) are the coefficients of either F₁(z) or F₂(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 C₁(x) and n_f=7 in case of C₂(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].