Speech Signal Analysis and Modelling - Digital Speech: Coding for Low Bit Rate Communication Systems - page 66

achieved by replacing the expectations of equation (4.17) by summations over
finite limits, i.e.

φ n (i, j) = E { s(n − i)s(n − j) }

= s n (m − i)s n (m − j), for i = 1 , ... ,p, j = 0 , ... ,p (4.19)

m

There are two approaches to interpret equation (4.19), and these lead to two
methods, namely the Autocorrelation and Covariance methods [1, 7].

The Autocorrelation Method

For the Autocorrelation Method (AM), the waveform segment, s n (m) ,is
assumed to be zero outside the interval 0 ≤ m ≤ N − 1where N is the length

of the sample sequence. Since for N p we are trying to predict

zero sample values (which are not actually zero) the prediction error for these
samples will not be zero. Similarly, the beginning of the current frame will
be affected by the same inaccuracy incurred in the previous frame. The limits
for equation (4.19) can be expressed as,

≤ m ≤ N +

N − 1 −| (i − j) |

= +| − | ≤ ≤ ≤ ≤

φ n (i, j) s n (m)s n (m i j ), 1 i p, 0 j p (4.20)

m = 0

Equation (4.20) can be reduced to the short-time autocorrelation function
given by,

φ n (i, j) = R n ( | i − j | ), for i = 1 , ... ,p j = 0 , ... ,p (4.21)

where,

N − 1 − j

R n (j) = s n (m)s n (m + j) (4.22)

m = 0

Using the AM, equation (4.17) can be expressed as
p

| − | = ≤ ≤

α j R n ( i j ) R n (i), 1 i p (4.23)

j = 1

or in matrix form by,

=

−

R n ( 0 ) R n ( 1 ).R n (p 1 ) α 1

α 2
.
α p

R n ( 1 )
R n ( 2 )
.
R n (p)

−

R n ( 1 ) . . R n (p 2 )

. . . .

R n (p − 1 ) . . R n ( 0 )