Efficient LPC Quantization Methods - Digital Speech: Coding for Low Bit Rate Communication Systems - page 85

The α i coefficients are the direct form of LPC. The filter H(z) is stable if it
is minimum phase, i.e. all the roots of the equation (5.1) are within the unit
circle. If α i were quantized directly, the stability of the filter H(z) is not easily
guaranteed as the roots of equation (5.1) are not usually computed to check
for stability. Thus a more useful parameter, the PARCOR (partial correlation)
coefficients, k i , are usually used for quantization. The distribution plots of
PARCOR parameters for a 10 th -order LPC filter are shown in Figure 5.1. The
forward and backward transformation are given below [3].
LPC to PARCOR:

a p = α j 1 ≤ j ≤ p

j

For i = p, p − 1 , ... , 1 (5.2)

a i − 1
j

= (a j + a i a i − j )/( 1 k i ), − 1 ≤ j ≤ i − 1

a i − 1 k i − 1 =

i − 1

PARCOR to LPC:

For i = 1 , 2 , ... ,p

a i = k i (5.3)

a j = a i − 1 k i a i − 1

− ≤ ≤ −

j ,

1 j i 1

j i −

a p

α j =

≤ ≤

j ,

1 j p

The LPC filter is stable if 1 . 0. Although k i can easily be checked for

stability, they are not suitable for quantization because they possess a nonflat
spectral sensitivity, i.e. values of k i near unity require more quantization
accuracy than those away from unity. Thus, nonlinear functions of k i are
required, with the Log-Area Ratio (LAR) and inverse sine (IS) functions
being the most widely used [4]. For LAR and IS, the forward and backward
transformation are given below:
PARCOR to LAR:

| k i |≤

log 1

,

− k i

g i = 1 ≤ i ≤ p (5.4)

1 + k i

LAR to PARCOR:

1 ,

10 g i −

= ≤ ≤

k i 1 i p (5.5)

10 g i

1 +