5.2.4 ISP to LP conversion

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Once the ISPs are quantized and interpolated, they are converted back to the LP coefficient domain . The conversion to the LP domain is done as follows. The coefficients of F₁(z) and F₂(z) are found by expanding Equations (14) and (15) knowing the quantized and interpolated ISPs q_i=,i=0,…,m-1, where m=16. The following recursive relation is used to compute f₁(z)

with initial values f₁(0)=1 and f₁(1)=-2q₀. The coefficients f₂(i) are computed similarly by replacing q_2i-2 by q_2i-1 and m/2 by m/2-1, and with intial conditions f₂(0)=1 and f₂(1)=-2q₁.

Once the coefficients f₁(z) and f₂(z) are found, F₂(z) is multiplied by 1-z^-2, to obtain F’₂(z); that is

( 19 )

Then F’₁(z) and F’₂(z) are multiplied by 1+q_m-1 and 1-q_m-1, respectively. That is

Finally the LP coefficients are found by

( 20 )

This is directly derived from the relation , and considering the fact that F’₁(z) and F’₂(z) are symmetric and antisymmetric polynomials, respectively.