A CLASS OF NONLINEAR PREDICTOR FUNCTIONS FOR THE SPEECH SIGNAL

yn
= H2 (y )
n
=
XX
P P i
hi;j yn
=1
j
=i
?y ?
i n
j
(1)
This requires a P -sample memory and N = =1 i lter coe cients. The lter coe cients h are represented as
t P i i;j
ABSTRACT
2.1. H2 Filter A Volterra H2 prediction lter with sample memory may
be written as 1] ^
P
1. INTRODUCTION
A primary objective in low bitrate speech coding is to reduce the representation of the time-varying signal to some parametric form which may be e ciently quantized to utilize the available bandwidth. Due to the `quasi-stationary' nature of speech, the channel parameters to be transmitted are normally updated at frequent intervals, typically 50 times per second. Nearly all low-rate coders employ a twostage prediction process: the long-term or pitch prediction, and the short-term or spectral prediction. These models are usually of order one for the pitch predictor, and order 8 to 10 for the spectral (or formant) predictor. If the prediction process is prefect, the residual after the prediction lter will approximate white Gaussian noise. Until recently, the formant predictor section of the coder was invariably a linear predictor, on the assumption of stationarity of the input. Recently, some non-linear approaches have been investigated 1]. A particular problem with nonlinear prediction is the selection of the type and order of nonlinearity to be used. Unlike linear prediction, there is no convenient, mathematically tractable solution to the problems of nonlinearity type, order and parameter estimation. We have previously investigated a neural network approach based on a `soft' nonlinear function. It was found that the gain resulting from nonlinear prediction was somewhat signal-dependent, required excessive training times for the network, and was quite sensitive to the training parameters.
A CLASS OF NONLINEAR PREDICTOR FUNCTIONS FOR THE SPEECH SIGNAL
John Leis
Faculty of Engineering University of Southern Queensland Toowoomba, AUSTRALIA email: leis@.au
The method of linear prediction has found considerable application in coding speech e ciently. The objective of the linear predictor is to remove all \predictable" redundancy from the source, and thus the residual may be e ciently coded. In the spectral domain, the linear prediction may be thought of as a whitening lter, with the residual having a noiselike characteristic. In practice, this all-pole model is not a perfect representation. As the linear all-pole model is only an approximation over a short interval to the true speech signal, it is reasonable to ask whether there are any nonlinear components which may be predicted and hence removed in the coding stage { this would lead to a smaller average prediction residual. In this paper, we report some results using the Volterra series both as a direct-form preditor and as a predictor for the linear prediction (LP) residual.
h = ?2m + 2Mh hopt = M?1 m
0 B B B B B B B B B B B B B B B B B B B B B B B B B @
yn yn
yn
?
. . .
?1 ?2
P
::::::::: yn
n
yn
?1 y ?1 ?1 y ?2
yn
yn
yn
?1 y ? ?2 y ?2 ?2 y ?3
y
n
=
0 B B B B B B B B B B B B B B B @
yn
yn
?1 y ?1 ?1 y ?2
n
yn
yn
yn
?1 y ? ?2 y ?2 ?2 y ?3
n n
. . .
n
P
yn
?2 y ?
. . .
n
. . .
n
P
1 C C C C C C C C C C C C C C C A
2.2. H12 Filter A Volterra H12 lter with sample memory may be written
h1;P h2;2 h2;3
. . .
. . .
h2;P
. . .
1 C C C C C C C C C C C C C C C C C C C C C C C C C C C A
(8)
hP;P
The optimal solution is found by solving
@e @
Using the identities
as
P yn
(3)
^
= H1 (y ) + H2 (y ) =
X
P i
n
hi yn
=1
? +
i
XX
P P i
n
hi;j yn
=1
j
=i
?y ?
i n
j
(7)
yP yP
The lter coe cients are now represented by the augmented vector
The predicton error is thus
2 1.5 1
x 10
4
Speech Test Signal
Linear Prediction Gain 24
0.5
23
Amplitude 0
22
−0.5
21
−1 −1.5 −2 −2.5 0
en
n
e
=
E
? 2 ? 2h m + h Mh
yn
T T
h =
(4)
with
m = M =
2
E E
(y y )
?y y
n n
n
T n
(5)
0 B B B B B B B B B B B B B B B B B B B B B B B B B B B @
h1 h2
. . .
hP :::: h1;1 h1;2 h1;3