Speech Signal Analysis and Modelling - Digital Speech: Coding for Low Bit Rate Communication Systems - page 79

The β j coefficients can now be solved by inverting V(i, j) , e.g. using Cholesky
decomposition. In the above formulation, a 'fix-up' may be used to ensure
that the filter so formed is stable, e.g. by adding a small noise source into
the formulation, the matrix inversion to obtain [ V(i, j) ] − 1 can be made more
reliably. However, a stable pitch filter is not a pre-condition for the pitch
analysis as rapid transitions are sometimes desired.
In the above formulation it is assumed that the pitch lag, τ , has already been
found and that β j

β j,τ . In order to determine τ , various pitch measurement
algorithms can be used. These include the Autocorrelation [12], average
magnitude difference function (AMDF) [13], Cepstrum [14] and Maximum
Likelihood [15]. These methods exhibit different characteristics especially
with a noisy input signal.
As the preceding analysis to determine β j has shown, pitch analysis is
performed on a block containing N samples. However, the size of our
window in which the block is taken is required to be considerably longer
than the analysis frame length, N . This is because our pitch value, τ ,can
vary between a minimum, τ min , of around 16 samples to a maximum, τ max ,
of around 160 samples. Therefore, our ideal analysis window is significantly
greater in length ( N

=

+ τ max ≥ 200 samples) such that it contains more than

one complete pitch period.
For simplicity, consider a 1-tap pitch filter, i.e. ( I

= 0),

1

P 1 (z) = (4.62)

βz − τ 1 −

Thus,

R(τ )
V( 0 , 0 )

β = (4.63)

N − 1

r(m)r(m − τ)

=

m 0

= τ min ≤ τ ≤ τ max (4.64)

N − 1

r 2 (m − τ)

m = 0

Substituting this into equation (4.58),

N − 1

− τ)

2

r(m)r(m

N − 1

m = 0

r 2 (m) E = − (4.65)

−

N 1

m = 0

r 2 (m − τ)

m = 0