Scalar Quantization - Tools for Signal Compression - page 16

k P are known as the partial correlation coefficients
(PARCOR). The final equation in the above algorithm shows that all coefficients have
a magnitude less than one because the variances are always positive. This property is
what makes them particularly interesting in coding.

The coefficients k 1 ···

1.3.3. Prediction gain

1.3.3.1. Definition
The prediction gain (gain which arises from prediction) is the ratio of the
quantization error powers, obtained through using the optimum quantization without
prediction, with those obtained when using prediction with a constant resolution b .It
is equal to:

G p = cσ 2 X 2 − 2 b

cσ Y 2 − 2 b = σ 2 X

[1.11]

σ Y

Making use of [1.7], the prediction gain can be written as:

r X (0)
r X (0) + P

G p ( P )=

a op i r X ( i )

i =1

It depends on the prediction order P . The prediction gain can also be written as

a function of the PAR-COR. Since σ Y = σ 2 X i =1 (1 k i ), the prediction gain is

−

equal to:

1
i =1 (1

G p ( P )=

k i ) −

This function increases with P . We can show that it tends toward the limit G p ( ∞ )

which is known as the asymptotic value of the prediction gain.

1.3.4. Asymptotic value of the prediction gain

This asymptotic value can be expressed in different ways, for example, as a
function of the autocovariance matrix determinant. By taking the determinant of the
two parts of equation [1.10], we find 7 :

P

σ j det R ( P +1)=

j =0

7. Specifying explicitly the dimension of the matrix R in this section.