Speech Signal Analysis and Modelling - Digital Speech: Coding for Low Bit Rate Communication Systems - page 69

or in matrix form,

=

φ n ( 1 , 1 )φ n ( 1 , 2 ) · φ n ( 1 ,p) α 1

α 2
.
α p

φ n ( 1 , 0 )
φ n ( 2 , 0 )
.
φ n (p, 0 )

φ n ( 2 , 1 ) · · φ n ( 2 ,p)

. . . .

φ n (p, 1 ) · · φ n (p, p)

A solution to equation (4.33) is not as straightforward as for the equivalent
AM. This is because the covariance matrix, φ n (i, j)

p
matrix φ is not Toeplitz. However, efficient matrix inversion solutions such
as Cholesky decomposition can be applied where φ is expressed as [1]:

= φ n (j, i) ,butthe p ×

VDV T φ = (4.34)

V is a lower triangular matrix whose main diagonal elements are 1s and D
is a diagonal matrix. The elements of the V and D matrices are determined
from equation (4.34) as follows:

j

φ n (i, j) = V im d m V jm 1 ≤ j ≤ i − 1 (4.35)

m = 1

or equivalently,

j − 1

V ij d j = φ n (i, j) − V im d m V jm 1 ≤ j ≤ i − 1 (4.36)

m = 1

and for the diagonal elements of D ,

i

φ n (i, i) = V im d m V im (4.37)

m = 1

or,

i − 1

V im d m for i d i = φ n (i, i) − ≥ 2 (4.38)

m = 1

and,

d 1 = φ n ( 1 , 1 ) (4.39)