5.2.3 LP to LSP conversion (all modes)
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The LP filter coefficients 
, are converted to the line spectral pair (LSP) representation for quantization and interpolation purposes. For a 10th order LP filter, the LSPs are defined as the roots of the sum and difference polynomials:
(10)
and
, (11)
respectively. The polynomial 
and 
are symmetric and anti‑symmetric, respectively. It can be proven that all roots of these polynomials are on the unit circle and they alternate each other. 
has a root 
(
) and 
has a root 
(
). To eliminate these two roots, we define the new polynomials:
(12)
and
(13)
Each polynomial has 5 conjugate roots on the unit circle 
, therefore, the polynomials can be written as
(14)
and
, (15)
where 
with 
being the line spectral frequencies (LSF) and they satisfy the ordering property 
. We refer to 
as the LSPs in the cosine domain.
Since both polynomials 
and 
are symmetric only the first 5 coefficients of each polynomial need to be computed. The coefficients of these polynomials are found by the recursive relations (for 
to 4):
(16)
where 
is the predictor order.
The LSPs are found by evaluating the polynomials 
and 
at 60 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 
and 
. In this method the roots are found directly in the cosine domain 
. The polynomials 
or 
evaluated at 
can be written as:
,
with:
, (17)
where 
is the 
th order Chebyshev polynomial, and 
are the coefficients of either 
or 
, computed using the equations in (16). The polynomial 
is evaluated at a certain value of 
using the recursive relation:
with initial values 
and 
The details of the Chebyshev polynomial evaluation method are found in P. Kabal and R.P. Ramachandran [4].