Efficient LPC Quantization Methods - Digital Speech: Coding for Low Bit Rate Communication Systems - page 98

Squares adaptive method [7] the LSF parameters can be computed directly
from the speech samples themselves. The LMS algorithm aims to minimize
the mean-square value of the PARCOR lattice filter output, and thus flatten
its frequency spectrum by a 'noisy steepest-descent' procedure which uses
the squared value of a single output sample to approximate the mean-square
value. Thus the algorithm begins the sequential estimation using evenly-
distributed estimated LSFs and, as each sample of speech is processed, a
new LSF vector estimate is obtained. Depending on the adaptation rate
required, the algorithm converges to the correct value after around 100
samples of input.
The LMS method is very attractive because it requires no LPC analysis.
However, as it is a 'learning' type algorithm, it is susceptible to 'out-lier'
input samples, i.e. samples which are different in character to the majority of
speech samples. The effect of these unusual inputs is to throw the algorithm
off its convergence curve; if this occurs at the end of a frame there will be no
time for correction before the final values are used.

5.4 LSF to LPC Transformation

There are two methods for the inverse transformation, neither of which
is as computationally intensive as the forward transformation. The two
methods are equivalent but the LPC synthesis method is perhaps more easily
visualized.

5.4.1 DirectExpansionMethod
In all of the LPC to LSF methods above the aim is to find the roots of
equation (5.16), i.e. a i and b i . Having found these roots using any of the
methods, the LPC, α i , can be simply found by multiplying out the product
terms of equation (5.16), i.e.

z − (p + 1 ) [ P (z)( 1

P p + 1 (z) = − z) ] (5.36)

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

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

S 1 z − 1 S p z − p S p + 1 z − (p + 1 )

= + + + +

S 0 ... (5.37)

Similarly,

T 1 z − 1 T p z − p T p + 1 z − (p + 1 ) Q p + 1 (z) = T 0 + + ... + + (5.38)