Analysis by Synthesis LPC Coding - Digital Speech: Coding for Low Bit Rate Communication Systems - page 239

Thus, substituting equation (7.72) into equation (7.69) and rearranging, the a i
can be calculated as,

a i = φ(i)/R(i, i), i = 1 , 2 , ... ,M (7.73)

˜

The best estimate for s(n) is then given by,

M

ˆ s opt (n) = a i s i (n), n = 0 , 1 , ... ,L − 1 (7.74)

=

i 1

The above procedure comprises aminimummean square error approximation
inwhich the optimum solution can be derived only if the set of sequences

ˆ

s i (n)
are linearly independent (i.e. form a basis) and are orthogonal to each other.
A popular method for achieving this orthogonalization is the Gram-Schmidt
procedure, summarized below.
Consider a set of m

1vectors p i (n) each of length L which form a basis.
The objective is to construct orthogonal vectors q i (n) so that

+

=

L − 1

0for i = j

q i (n)q j (n) i, j = 0 , 1 , ... ,m (7.75)

= 0for i = j,

n = 0

Let q 0 (n) = p 0 (n) and define q 1 (n) as a linear combination of q 0 (n) and p 1 (n) ,

then

q 1 (n) = p 1 (n) − α 01 q 0 (n) (7.76)

Then for q 1 (n) to be orthogonal to q 0 (n) ,

L − 1

q 1 (n)q 0 (n) = 0 (7.77)

n = 0

i.e.

L − 1 L − 1

α 01 q 0 (n) p 1 (n)q 0 (n) − = 0 (7.78)

n = 0 n = 0

and the correlation α 01 is given by,

L − 1

p 1 (n)q 0 (n)

n = 0

α 01 = (7.79)

−

L 1

q 0 (n)

n = 0