Speech Signal Analysis and Modelling - Digital Speech: Coding for Low Bit Rate Communication Systems - page 68

and for i = 2,

R n ( 1 )

R n ( 2 )R n ( 0 ) −

α ( 1 )

R n ( 1 ) ] /E ( 1 ) k 2 = [ R n ( 2 ) − =

n

1

R n ( 0 ) R n ( 1 )

−

α ( 2 ) = k 2

2

R n ( 1 )R n ( 0 ) − R n ( 1 )R n ( 2 )

α ( 2 )
1

= α ( 1 )
1

k 2 α ( 1 )

− =

1

R n ( 0 ) − R n ( 1 )

and, from this,

α ( 2 ) α ( 2 2 . α 1 = , and α 2 =

1

The Covariance Method

For the Covariance Method (CM), the opposite approach to the AM is taken.
Here the interval over which the mean squared error is computed is fixed, i.e.

N − 1

e n (m) E = (4.30)

=

m 0

Equation (4.19) can be written as,

N − 1

φ n (i, j) = s n (m − i)s n (m − j), 1 ≤ i ≤ p, 0 ≤ j ≤ p (4.31)

m = 0

Changing the summation index,

N − i − 1

φ n (i, j) = s n (m)s n (m + i − j), 1 ≤ i ≤ p, 0 ≤ j ≤ p (4.32)

m =− i

The expression given by equation (4.32) is slightly different to equation (4.20)
used in the AMas it requires the use of samples in the interval 1.

In effect, equation (4.31) is not a true autocorrelation function, but rather the
cross-correlation between two very similar but not identical, finite-length
sampled sequences. Using equation (4.31), our original LPC equation (4.17)
can be expressed as,

− p ≤ m ≤ N −

p

α j φ n (i, j) = φ n (i, 0 ), 1 ≤ i ≤ p (4.33)

j = 1