Speech Signal Analysis and Modelling - Digital Speech: Coding for Low Bit Rate Communication Systems - page 77

s(n)

Pitch
Synthesis
Filter

r(n) LPC

Synthesis
Filter

Gx(n)

Figure 4.12 A typical pitch-LPC formulation model

to Figure 4.12, the combined analysis model can be represented by a time
domain difference equation,

p

I

s(n) = Gx(n) + b j r(n − T − j) + a j s(n − j) (4.54)

j =− I j = 1

where r(n) is the past excitation (LPC residual) signal. Following a similar
procedure to that of LPC analysis, our goal is to determine estimates (β j , τ , α j )
of the model parameters (b j ,T,a j ) . Then, the prediction error is given by
( e(n)

= Gx(n) ),

p

I

e(n) = s(n) − β j r(n − τ − j) − α j s(n − j) (4.55)

j =− I j = 1

The mean squared error solution to equation (4.55) is not as straightforward
as for the LPC analysis due to the presence of the delay factor τ .Inorderto
overcome this hurdle two suboptimal approaches can be taken:

• One-Shot Optimization: If one assumes that the pitch spectrum infor-

mation of the residual r(n) is close to the pitch spectrum information of
the input speech s(n) ,thenwecansolvefor α j as before and use the
residual from the LPC inverse filter to determine (β j , τ) . Thus during the
first iteration, the LPC coefficients are estimated to minimize the interme-
diate residual energy. The pitch filter coefficients are then found using this
intermediate residual signal. This procedure can be considered to be near
optimal provided the long-term lag, τ , is greater than the analysis frame
size, i.e. τ>N .

• IterativeSequentialApproach: An analysis similar to theOne-Shotmethod

described above is first performed. During subsequent iterations, the LPC
is re-optimized with the previously-determined pitch filter coefficients
[11]. Also, the pitch filter parameters are recalculated based on the newly-
formed intermediate residual. This iteration process can be continued until
a certain threshold is reached or for a fixed number of iterations.

For practical reasons, the one-shot method is usually preferred as it only
requires one iteration. In the iterative sequential method the main difficulty is