Speech Signal Analysis and Modelling - Digital Speech: Coding for Low Bit Rate Communication Systems - page 78

in setting a suitable threshold for the termination of the iteration run. Overall,
it is substantially more complicated. However, the iterative method has been
reported to give a better prediction gain and better perceptual performance
[5]. This is usually achieved with a shifting of the LPC prediction gain to
the pitch prediction gain. Here, only the one-shot method is considered
as follows:
By removing the LPC effect in equation (4.55), we obtain,

I

e(n) = r(n) − β j r(n − τ − j) (4.56)

j =− I

The estimates can now be determined by minimizing the mean squared
error, i.e.

2

I

e 2 (n)

r(n)

E = E { }= E − β j r(n − τ − j) (4.57)

j =− I

Replacing the expectation with finite summations, we get

r n (m)

2

I

e n (m)

= = − − −

E β j r n (m τ j) (4.58)

m m

j =− I

By setting ∂E/∂β j to zero, we obtain

I

β j V(i, j) = R(τ + i) − I ≤ i ≤ I (4.59)

=−

j I

which can be written in matrix form as,

=

V( − I, − I) ··· V( − I, I) β − I

.
β I

R(τ − I)

. . . .

V(I, − I) ··· V(I, I) R(τ + I)

where,

N − 1

+ = − −

R(τ i) r(m τ i)r(m) (4.60)

m = 0

N − 1

V(i, j) = r(m − τ − i)r(m − τ − j), − I ≤ i ≤ I, − I ≤ j ≤ I (4.61)

m = 0