=Paper=
{{Paper
|id=Vol-2604/paper43
|storemode=property
|title=Model of Adaptive Language Synthesis Based On Cosine Conversion Furies with the Use of Continuous Fractions
|pdfUrl=https://ceur-ws.org/Vol-2604/paper43.pdf
|volume=Vol-2604
|authors=Lyubomyr Chyrun
|dblpUrl=https://dblp.org/rec/conf/colins/Chyrun20
}}
==Model of Adaptive Language Synthesis Based On Cosine Conversion Furies with the Use of Continuous Fractions==
https://ceur-ws.org/Vol-2604/paper43.pdf
Model of Adaptive Language Synthesis Based On Cosine
Conversion Furies with the Use of Continuous Fractions
Lyubomyr Chyrun[0000-0002-9448-1751]
Ivan Franko National University of Lviv, Lviv, Ukraine
Lyubomyr.Chyrun@lnu.edu.ua
Abstract. The proposed article describes an adaptive model for the synthesis of
voice signals in a digital signal processor. The use of continuous fractions in a
digital signal processor is suggested. The realization of continuous fractions
with the help of multicellular structures is given. This procedure is used to im-
plement the model of the human vocal tract.
Keywords. Language Synthesis, Adaptive Synthesis, digital signal processor,
Cosine Conversion Furies, Continuous Fractions, Speech synthesis system
1 Introduction
Modern voice signals recognition systems integrate technologies from such fields of
modern science as signal processing, pattern recognition, natural language, and lin-
guistics. Such systems that are widely used in signal processing have created a real
boom in digital signal processing (DSP). Previously, the field was dominated by vec-
tor-oriented processors and algebraic mathematical apparatus, while the current gen-
eration of DSP relies on sophisticated statistical models and uses complex software
for practical implementation. Modern voice signals recognition models are able to
understand the continuous input language for dictionaries, consisting of hundreds of
thousands of words in operating environments. Linear predictive analysis of voice
signals is historically the most important in voice analysis technologies. The basis of
this is the filter source model, which is an ideal linear filter.
2 Analytical Review of Literary and Other Sources
Linear predictive coding is most commonly used in speech analysis and synthesis, or
in transmitting or storing speech signals. For this purpose, ideal cell structures are
typically used to model the human vocal tract. For the first time, these structures with
reflection coefficients were formulated by Markel, Gray [1], and Makhoul [2]. The
model in the state space of a non-ideal cell structure with two and four factors per
section for digital signal processors was analyzed in [3]. The general system of voice
synthesis given in [4] is presented in Fig. 1.
Copyright © 2020 for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
КV
Oscillations Difference signal
Frequency periodic amplitude Synthesized
activation language
model of the
voice tract
white noise
obstacles Model coefficients
КU
Voiced / not voiced
Fig. 1. Speech synthesis system
In the general case, the problem of linear prediction is as follows [9-12]. Let us have a
voice signal s n , and let
p
~
s n sn k
k 1
k
be predicted magnitude. Inaccuracy of prediction in this case is given as follows:
p
e n s n ~
s n s n sn k .
k
k 1
We usually want to minimize the error to find the best, or optimal, values k . De-
termine the short-term average error:
2
p p
2 2
E e n
s n
k s n k s n 2 s n k s n k
n n k 1 n n k 1
2 2
p p p
k s n k s 2
n 2 k s n s n k k s n k
n k 1 n k 1 n n k 1
We can minimize the error l for everyone 1 l p by differentiating E and equat-
ing the result to zero
E p
0 2 s n s n l 2 k sn k s n l
l n
n k 1
In the case of the covariance method, we will start by slightly redefining the terms
p p
s n s n l k s n k s n l or c l ,0 k c k , l
n k 1 n k 1
This equation is also known as linear prediction equation (Yule-Volcker equation).
k are called linear prediction coefficients, or predictor coefficients. When calculat-
ing the equations for all values l , we can write them in a matrix form
c C
where
1 c 1,1 c 1,2 ... c 1, p c 1,0
c 2,1 c 2,2
2 ... c2, p c 2,0
C c
... ... ... ...
p c p,1 c p,2 ... c p, p c p,0
To solve this equation, you need to find the inverted matrix:
c C 1
This method is called the covariance method. Note that the covariance matrix is
symmetric. The fastest way to find the solution to this equation is the Holetsky
method (the covariance matrix is divided into lower and upper triangular matrices).
Using a slightly different approach to minimize the error, we can find a solution to the
linear prediction equation using the autocorrelation method
R 1 r
where
1 r 0 r 1 ... r p 1 r 1
r 1
2 r 0 ... r p 2 r 2
R r
... ... ... ...
p r p 1 r p 2 ... r 0 r p
The matrix of the system is symmetrical and all diagonal elements are equal, which
means that the inverted matrix always exists and the solutions of the system are in the
left half plane.
Autoregressive modeling using least squares prediction, or linear prediction, forms
the basis of a wide range goals of signal processing and communication systems, that
include adaptive filtering and control, modeling of speech and coding systems, adap-
tive channel alignment, parametric spectrum estimation, and identification systems.
To implement linear prediction of data or model goals, it is necessary to determine
the values of linear prediction coefficients, as well as the order. Some commonly used
practice model selection methods include the Akayke information criterion method,
the Schwarz minimum description length method, and the Risenan prediction of least
squares principle. In the original form, the first two criteria include a clear balance
between the similarity of model input and the notion of fine for model complexity.
Intuitively in the information criterion method, the primary purpose is to minimize the
number of bits that will be required to describe the data [13-18]. When it is already
possible to model the data parametrically and then encode the blocks, use the ap-
proach of allocating blocks of similar data, and then the model is fined with the addi-
tional number of bits required to encode its parameters [19-24].
However, the voice model based on cosine Fourier transform for language synthe-
sis has better properties and use, less sensitivity to quantization effects and, as a re-
sult, produces more natural synthesized language [25-29]. The parameters of this
model are the coefficients of cosine Fourier transform. This model is based on the
cosine decomposition of the logarithmic short-term voice range, and the synthesis is
implemented by approximate inverse Fourier cosine transformations using continuous
chain fractions [30-32]. This approach is parametric and is not based on any simplify-
ing assumptions about the voice model, because the poles as well as the zeros of the
voice model are justified [33-44].
3 The Voice Model Based on a Cosine Fourier Transform
Suppose that we have the logarithmic range ln S e jT of the voice data segment
sn , where Т – is the sampling interval, f s 1 - the sampling frequency, and
T
- the angular frequency. This function can be expressed using the true Fourier cosine
conversion coefficients Фур’є cn
ln S e jT c e n
jn T
. (1)
n
The complex coefficients of the cosine Fourier transform of a discrete system with
minimum phase stability are random and may be related to the following relations
g n cn , n 0, N F 2 ,
g n 2 cn , 0 n N F 2 , (2)
g n 0, n0
where N F - the dimension of the applied FFT. A digital filter whose logarithmic
correspondence approximates a function ln S e jT is determined by the transfer
function system
N 0 1 N 0 1
~
S z e c0 exp 2c z
n 1
n
n
e c0
exp2c z
n 0
n
n
(3)
c
where 0 N 0 N F 2 . The coefficient e 0 is equal to the value of the RMS of the
cosine Fourier transform model for the multiple signal. In our experiments on the
voice model we used f s 8kHz , N F 512 , the voice segment length is 25 ms with
12 ms overlap and N 0 25 .
~
It follows from (3) that the system of transfer functions S z is the product of tran-
scendental transfer functions
n
H n z e 2 cn z , 0 n N 0 1 (4)
The corresponding impulse feature is given
2cn i
, m ni , i 0, 1, 2,
hn m i! (5)
0, m ni
~
This means that the system of transfer functions S z has the following form (Fig. 2)
e c0
1 2 3
x(m) e 2 c1z e 2 c2 z e 2 c3 z
2c z N 0 1
e N 0 1 y(m)
Fig. 2. Voice model of cosine Fourier transform
N 0 1
~
S z e c 0 H z n (6)
n0
To implement a transfer function H n z using a digital filter, it is necessary to find an
approximation H n z , that can be practically implemented. One option for approxi-
mating an exponential function in (4) is continuous chain fractions [4]. Another pos-
sibility of implementing an exponential function approximation is to use a Pade ap-
proximation. Then the system of transfer functions in a practical voice model based
on a cosine Fourier transform will look like
N 0 1
~
~ c0 ~
S z e H n z . (7)
n0
4 Decomposition Approximation Using Continuous Chain
Fractions
The exponential function expressed by a decomposition into a continuous fraction can
be represented as the following decomposition [5], [6], [7]:
1 x x x x x
ex , (8)
1 1 2 3 2 2s 1
where the parameter is x 2cn z n . The accuracy of the approximation of the voice
model depends not only on the number of cosine Fourier transform coefficients in (3),
but also on the number of members of a continuous fraction in (8), that is, on the
length of a continuous fraction to be determined with s . A finite chain fraction for a
function e x can also be expressed by a set of real functions that approximate an ex-
ponential function with increasing accuracy
1 1 2 x 6 2x 12 6 x x 2
ex , , , , , (9)
1 1 x 2 x 6 4 x x 12 6 x x 2
2
These functions are known as Pade approximations of an exponential function. It is
recommended to use an odd number of elements of a continuous fraction in (8). This
leads to an approximation of an exponential function by a rational function with equal
degrees of polynomials in the numerator and denominator in (9). These are the ap-
proximations chosen
~ 2 x ~ 12 6 x x 2
H 1 z , H 2 z ,
2 x 12 6 x x 2
~ 120 60 x 12 x 2 x 3
H 3 z , (10)
120 60 x 12 x 2 x 3
~ 1680 840 x 180 x 2 20 x 3 x 4
H 4 z
1680 840 x 180 x 2 20 x 3 x 4
where z is the variable z-transformation and x 2cn z n . In the general case, to
achieve a better approximation, we can use decompositions of rational functions by
taking more suitable fractions.
cn
2 s 1 m 2 s m n ,
2s 1
c
2 s m n 2 s 1 m n 2 s 1 m ,
2s 1
c
5 m n 4 m n 6 m ,
3
cn
4 m 3 m n 5 m ,
3
3 m c n 2 m n 4 m ,
2 m 2c n 1 m n 3 m ,
(11)
1 m x m 2 m ,
y m 1 m .
5 Structure of adaptive synthesis
As noted above, the approximation error for e x is determined by the number of ele-
ments of a continuous fraction to decompose an exponential function into a continu-
ous fraction. This error further depends on the magnitudes of the modules of the true
Fourier cosine transform coefficients cn . On the basis of a statistical analysis of the
Fourier cosine coefficients for the description of the voice model of a male loud-
speaker, the following estimation was made in relation to the stability of the system
and the well-defined safety limit for transfer functions H n z in equation (5). From
the above it follows that the functions H n z can be approximated as follows:
~
n 1, 2, 3 H 3 z
~
n 4, 5 H 2 z
~
n 6, 7, , 25 H1 z
It is more effective in relation to the total error of approximation and in relation to
saving the number of arithmetic operations required for the practical implementation
of voice modeling to use the adaptive structure of a continuous fraction. The number
of corresponding cells (Fig. 3) can be selected according to the magnitudes of the
cosine Fourier transform coefficients. The following adaptive empirical rule can be
used:
~
for cn 0.3 two cells - match H1 z ,
~
for cn 0.5 four cells - match H 2 z ,
~
for cn 1 six cells - match H 3 z ,
~
for cn 1 eight cells - match H 4 z .
For example, the voice model of the stationary part (24 ms) of the “е” vowel sound is
used
c0 0.491 - logarithm of the value of the difference signal
c1 0.700 c 2 0.354 c3 0.026 c4 0.205
c5 0.159 c6 0.027 c7 0.310 c825 0.3
Using the empirical rule indicated, the voice model of the cosine Fourier transform is
presented in Fig. 3 can be built:
c0 c1 c2 c3 c4
p(n) ~ ~ ~ ~
e c0 H 3 z H 2 z H 1 z H 1 z
~ ~ ~ ~
H 1 z H 1 z H 2 z … H 1 z
s(n)
…
c5 c6 c7 c8 … c25
Fig. 3. Voice model of cosine Fourier transform of the stationary part of the vowel “е”: p n
is activated signal; s n is synthesized voice signal
In the practical implementation of transcendental transfer functions H n z the follow-
ing numerical results were obtained:
1
Table 1. The value of the transcendental transfer function H 1 z e z
п H n z - exact values H n z - approximate values
1 1.00000000000000 0.99993896484375
2 1.00000000000000 0.99993896484375
3 0.50000000000000 0.49999648242188
4 0.16666666666667 0.16666549414062
5 0.08333333333333 0.06092749023438
6 0.00138888888889 0.00003051757812
п H n z - exact values H n z - approximate values
7 0.00019841269841 0.00030517578125
8 0.00002480158730 -0.0012207031250
9 0.00000271636432 -0.00003051757812
We will also present our numerical results in the following diagram (Fig. 4).
1 .20
Exact value
Appr oxima te value
1 .00
0 .80
0 .60
0 .40
0 .20
0 .00
1 2 3 4 5
1
Fig. 4. The value of the transcendental transfer function H1 z e z
6 Conclusions
Voice modeling based on cosine Fourier transform is in fact related to spectral syn-
thesis of voice signals, and is not based on any simplifying a priori considerations
about the language reproduction system. It also contains information about the range
of the activated voice path.
The voice modeling procedure based on Fourier cosine transforms requires more
arithmetic operations than approaches based on linear predictive coding, but the struc-
ture of the digital filter can be optimized.
Continuous fractions offer an interesting tool not only in language synthesis. A
high-order approximation of algebraic transcendental functions can be used in bio-
logical and industrial modeling systems. The direct implementation of continuous
fractions further enables the implementation of multi-chamber structures.
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