Application of Laplace Transformation (cuts topic)
1. 1
THISIS
TOPIC
Application of Laplace Transformation
in the Vocal Tract Modeling
SUBMITTED TO:
SirTahir Mushtaq Qurashi Sb
SUBMITTED BY:
Muhammad Faisal Ejaz
NCBA&E
MULTAN CAMPUS
2. 2
Abstract
The Laplace transform is a widely used Integral Transform in mathematics and
electrical engineering named after Pierre-Simon Laplace that transforms a function
of time into a function of complex frequency. The inverse Laplace transform takes a
complex frequency domain function and yields a function defined in the time
domain. The Laplace transform is related to the Fourier transform, but whereas the
Fourier transform expresses a function or signal as a superposition of sinusoids, the
Laplace transform expresses a function, more generally, as a superposition of
Moments. Given a simple mathematical or functional description of an input or
output to a system, the Laplace transform provides an alternative functional
description that often simplifies the process of analyzing the behavior of the system,
or in synthesizing a new system based on a set of specifications. So, for example,
Laplace transformation from the time domain to the frequency domain transforms
differential equations into algebraic equations and convolution into multiplication.
This topic considers a generalized acoustic tube model of the vocal tract, related
it to the pole-zero type linear prediction .The generalization done by vocal tract
model. The transform function is obtained from the generalized model by
conglomerating one of the three branching to the branch section at the junction
of three branches .It is also discuss how to find coefficient for the pole-zero type
linear prediction from the voiced sounds . Also discussed is how to evaluate the
reflection coefficients by connecting the pole-zero type linear prediction
algorithms to the transfer function of the generalized model.
3. 3
CHAPTER #1
Introduction:
The Laplace transform is named after mathematician and astronomer Pierre-Simon
Laplace, who used a similar transform (now called z transform) in his work on
probability theory. The current widespread use of the transform came about soon
after World War II although it had been used in the 19th century by Abel, Larch,
Heaviside, and Bromwich.
What is Laplace Transformation?
The Laplace transform is a widely used integral transform in mathematics and
electrical engineering named after Pierre-Simon Laplace that transforms a function
of time into a function of complex frequency.
What Does the Laplace Transform Do?
The main idea behind the Laplace Transformation is that we can solve an equation
(or system of equations) containing differential and integral terms by transforming
the equation in "t-space" to one in "s-space". This makes the problem much easier to
solve. The kinds of problems where the Laplace Transform is invaluable occur in
electronics. You can take a sneak preview in the Applications of Laplace section.
Definition of the Transform:
The Laplace transform converts a function of real variable f (t) into a function of
complex variable F(s).The Laplace transform is defined as
4. 4
The variable s is a complex variable that is commonly known as the Laplace
operator.
OR
Starting with a given function of t, we can define a new function the
variable s.
This new function will have several properties which will turn out to be convenient
for purposes of solving linear constant coefficient ODE’s and PDE’s.
The definition of is as follows:
Definition:
Let be defined for t 0 and let the Laplace transform of be defined by,
For example:
The Laplace transform is defined for all functions of exponential type. That is, any
function
.which is
(a) Piecewise continuous = has at most finitely many finite jump discontinuities on any
interval of finite length.
(b) Has exponential growth: for some positive constants M and k.
5. 5
History:
The Laplace transform is named after mathematician and astronomer Pierre-Simon
Laplace, who used a similar transform (now called z transform) in his work on
probability theory. The current widespread use of the transform came about soon
after World War II although it had been used in the 19th century by Abel, Larch,
Heaviside, and Bromwich. Leonhard Euler investigated integrals of the form
And
As solutions of differential equations but did not pursue the matter very far. Joseph
Louis Lagrange was an admirer of Euler and, in his work on integrating probability
density functions, investigated expressions of the form
Which some modern historians have interpreted within modern Laplace transform
theory.
These types of integrals seem first to have attracted Laplace's attention in 1782
where he was following in the spirit of Euler in using the integrals themselves as
solutions of equations. However, in 1785, Laplace took the critical step forward
when, rather than just looking for a solution in the form of an integral, he started to
apply the transforms in the sense that was later to become popular. He used an
integral of the form:
6. 6
akin to a Mellin transform, to transform the whole of a difference equation, in order
to look for solutions of the transformed equation. He then went on to apply the
Laplace transform in the same way and started to derive some of its properties,
beginning to appreciate its potential power.
Laplace also recognized that Joseph Fourier's method of Fourier series for solving
the diffusion equation could only apply to a limited region of space as the solutions
were periodic. In 1809, Laplace applied his transform to find solutions that diffused
indefinitely in space.
Formal definition
The Laplace transform is a frequency domain approach for continuous time signals
irrespective of whether the system is stable or unstable. Laplace transform approach
is also known as S-domain approach. The Laplace transform of a function f (t),
defined for all real numbers t ≥ 0, is the function F(s), which is a unilateral transform
defined by:
The parameter s is the complex number frequency:
with real numbers and ω.
7. 7
Other notations for the Laplace transform include or alternatively
instead of F.
The meaning of the integral depends on types of functions of interest. A necessary
condition for existence of the integral is that f must be locally integrable on [0, ∞).
For locally integrable functions that decay at infinity or are of exponential type, the
integral can be understood as a (proper) Lebesgue integral. However, for many
applications it is necessary to regard it as a conditionally convergent improper
integral at ∞. Still more generally, the integral can be understood in a weak sense,
and this is dealt with below.
One can define the Laplace transform of a finite Boral measure μ by the Lebesgue
integral
An important special case is where μ is a probability measure or, even more
specifically, the Dirac delta function. In operational calculus, the Laplace transform
of a measure is often treated as though the measure came from a distribution
function f. In that case, to avoid potential confusion, one often writes
Where the lower limit of 0−
is shorthand notation for
8. 8
This limit emphasizes that any point mass located at 0 is entirely captured by the
Laplace transform. Although with the Lebesgue integral, it is not necessary to take
such a limit, it does appear more naturally in connection with the Laplace–Stieltjes
transform.
Bilateral Laplace transform (Two-sided Laplace Transform):
When one says "the Laplace transform" without qualification, the unilateral or one-
sided transform is normally intended. The Laplace transform can be alternatively
defined as the bilateral Laplace transform or two-sided Laplace transform by
extending the limits of integration to be the entire real axis. If that is done the
common unilateral transform simply becomes a special case of the bilateral
transform where the definition of the function being transformed is multiplied by the
Heaviside step function.
The bilateral Laplace transform is defined as follows:
Properties of the Laplace Transform
The Laplace transform has the following General properties:
1. Linearity:
2. Homogeneity:
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3. Transform of the Derivative:
4. Derivative of the Transform:
5. Some Special Transforms:
There are some transform pairs that are useful in solving problems involving the heat
equation .The derivations are given in an appendix.
The Linear Property:
Let and be functions whose Laplace transforms exist for s > and s >
respectively. Then, for s > max { , } and and any constants.
This means that the Laplace transform is a linear operator.
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Example:
1)
2)
=
LAPLACE TRANSFORMATION PROPERTIES
By building up some basic properties of the Laplace transform, we can expand the
list of functions .we know the transform of, thus increasing the number of IVP’s we
can solve by this method.
Property 1:
In words, multiplying by -x in our usual function space is the same as differentiation
in transform space.
Example 1:
Find a function whose Laplace transform is
Solution:
We have that
Furthermore, we know that
So by Property 1 we have
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Property 2:
In words, multiplying by in our usual function space is the same as translation to
the right by as in transform space.
Example 3:
Find a function whose Laplace transform is
Solution:
From an example in the text, we have
To make the given function look more like this one (and avoid using partial
fractions) we
Can complete the square to get
=
Now we’ve completed the square.
=
To get in the Numerator.
By property 2.
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By the linearity of L.
Thus the given function
Is the Laplace transform of
Some Additional Examples
In addition to the Fourier transform and Eigen function expansions, it is sometimes
convenient to have the use of the Laplace transform for solving certain problems in
partial differential equations. We will quickly develop a few properties of the
Laplace transform and use them in solving some example problems.
Additional Properties of the Transform:
Let be a function of exponential type and suppose that for some b > 0,
Then is just the function , delayed by the amount b .Then
Let z = t - b so that
If we define
Then
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And we find
Transform of a Delay:
A related results is the following
Delay of a Transform:
These result (Transform of a Delay) and (Delay of a Transform) assert that a
delay in the function induces an exponential multiplier in the transform and,
conversely, a delay in the transform is associated with an exponential multiplier for
the function.
A final property of the Laplace transform asserts that
Inverse of a Product:
Where
The product is called the convolution product of f and g. Life would be
simpler if the inverse Laplace transform of was the point wise
product , but it isn’t, it is the convolution product. The convolution product
has some of the same properties as the point wise product, namely
And
14. 14
We will not give the proof of the result 7 but will make use of it nevertheless.
Chapter #2
Applications in Electronics
(Circuit Equations)
There are two (related) approaches:
1. Derive the circuit (differential) equations in the time domain, then transform these
ODEs to the s-domain;
2. Transform the circuit to the s-domain, and then derive the circuit equations in the s-
domain (using the concept of "impedance").
We will use the first approach. We will derive the system equations(s) in the t-plane,
and then transform the equations to the s-plane. We will usually then transform back
to the t-plane.
Example 1:
Consider the circuit when the switch is closed at t=0, VC(0) =1.0 V. Solve for the
current i (t) in the circuit.
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Therefore:
Collecting I terms and subtracting from both sides:
Multiply throughout by s:
Solve for I:
Finding the inverse Laplace transform gives us the current at time t:
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Example 2
In the circuit shown below, the capacitor is uncharged at time t = 0. If the switch is
then closed, find the currents i1 and i2, and the charge on C at time t greater than
zero.
Answer
We could either:
Set up the equations, take Laplace of each, then solve simultaneously
18. 18
Set up the equations, solve simultaneously, and then take Laplace.
For the first loop, we have:
Divide by 5 on both sides
For the second loop, we have:
Dividing 5 on both sides
Substituting (2) into (1) gives:
Simplifying:
19. 19
Multiply throughout by 5:
Next we take the Laplace Transform of both sides.
Note:
In this example,
So,
Now taking Inverse Laplace:
20. 20
And using result (2) from above, we have:
For charge on the capacitor, we first need voltage across the capacitor:
So, since , we have:
Graph of
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Example 3
A rectangular pulse vR(t) is applied to the RC circuit shown. Find the response, v (t).
Graph of vR(t):
Note: for all t < 0 s implies v (0–
) = 0 V. (We'll use this in the solution.
It means we take , the voltage right up until the current is turned on, to be zero.)
Answer:
Now
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To solve this, we need to work in voltages, not current.
We start with
The voltage across a capacitor is given by
It follows that
So for this example we have:
Substituting known values:
Then
Taking Laplace Transform of both sides:
Since , we have:
24. 24
Solution Using Scientific Notebook
1. To find the Inverse Laplace:
2. To solve the original DE:
Exact solution for v (t):
To see what this means, we could write it as follows:
To get an even better idea what our expression for means, we graph it as
follows:
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Chapter No.3:Application in Acoustics
INTRODUCTION:
The linear prediction, which had been widely used as a tool for speech signal
recognition, speaker recognition and speech synthesis, is closely related with the
acoustic modeling of the vocal tract. In fact, it is possible to drive the all-pole type
linear prediction algorithm directly from the acoustic tube modeling on the oral
cavity and the main vocal tract [H.Wakita, Direct estimation of the vocal tract shape
by inverse filtering of acoustic speech waveform , “IEEE Trans. AU, vol-21 , pp.
417-427, 1973.]. The mismatched spectral shaping or the marginal performance of
the all-pole linear prediction may be therefore regarded as stemming from the
imperfection of the corresponding vocal tract tube model , s point of views. The
most significant imperfection of the existing vocal tract modeling is in that it leaves
out the nasal cavity, thus sacrificing the effects of the nasal sounds (refer to [3] for
schematic diagrams of the vocal tract). Therefore it is of importance to generalize the
26. 26
existing vocal tract model to include the nasal cavity as well as the oral cavity. The
resulting linear prediction counterpart will than become a pole-zero type.
In this topic, we present a generalized acoustic tube model of the vocal tract which
consists of the oral cavity, the nasal cavity and the main vocal tract. The generalized
new model will thus consist of three branches. The main difference b/w the existing
two-branch model and the new model lie in the branch section at which the three
branches meet.
The transfer function obtained from the proposed model will be than used for the
formulation of a pole-zero type linear prediction algorithms. The prediction
coefficients in both the denominator and the numerator as well as the reflection
coefficients of the generalized model will finally be evaluated by analyzing the
voiced and the nasal sounds.
Generalized tube modeling of vocal tract:
The generalized model we consider in this topic consists of three branches
corresponding to the main vocal tract, the oral cavity and the nasal cavity as shown
as figure. (3.27a).
27. 27
(Rabiner & Schafer, Fig. 3.27a, p. 78)
When sectionalizing the branches, we obtain four different type of section: the
glottis section, the radiation section, the mid section and the branch section. The first
three section are essentially the same as for the existing two branch model, and are
well established in the literature (see for example [J. D .Markel and A.H.Grey, linear
prediction of speech, Springer - Verlag, New York, 1976.])
The fourth section, the branches section, represents the junction where the three
branches meet, and is thus unique to the generalized vocal tract model.
Modeling of the Branch Section:
we assume, as is indicated in diagram that the main vocal tract branch consists of L
section, section M through section M+N-1; the oral cavity branch consists of M
section, section 0 through section M-1 ; and the nasal cavity branch consists of N
sections; section 0 through section N-1. For convenience, we differentiate the oral
and the nasal cavity branches by superscripting “n” on the notation for the nasal
cavity branch whenever necessary.
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We assume that the cross section area is constant over each section, indicating the
area of the section by .So, at the branch section, three sections of area
OR ( meet.
We denote by and respectively the volume velocity and the
pressure at time t at a point in the section. Solve the momentum equation and
the continuity of mass equation [L.E. Kinsler and A.R. Frey, Fundamental of
acoustics, John Wiley & Sons, New York, 1982.] for the section, we obtain
Where c denotes the speed of sound of air, the air density, and the + and – signs
denote the forward and the backward travelling components respectively. Assuming
that the length `of each section is same so that the propagation time through each
section is , then we have the following boundary conditions as the junction of the
three branches:
Apply the boundary conditions to the above solutions, we obtain
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We define the reflection coefficient to be
(4)
Then the equations can be rewritten as
(5a)
(5b)
Or as
(6a)
(6b)
(6c)
Therefore the model for the branch section takes the form as shown in figure.
Modeling of the other sections:
The junction of the mid sections is a simplified version of the junction for the branch
section. That is, the model of the mid section “m” can be obtained by removing the
branch for “ (or by setting to zero) with M replaced by m on equations
(6a), (6b), and (6c). Thus we obtain
(7a)
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(7b)
Where the reflection coefficient “ ” can be expressed as
(8)
Note that equations (7a) and (7b) satisfy the form of Kelly-Lochbaum structures [J.
D .Markel and A.H.Grey, linear prediction of speech, Springer - Verlag, New York,
1976.]
For the glottis section we put an artificial matching section M+L along with the
corresponding reflection coefficient as is usually done in the literature (see [J.
D .Markel and A.H.Grey, linear prediction of speech, Springer - Verlag, New York,
1976.]). Then we obtain the relations
(9a)
(9b)
(9c)
The mathematical model for the glottis section is shown in diagram 3(b), where G
indicates the glottis. For the radiation section, we denote the radiation impedance by
(or for the nasal cavity ), That is
(10)
Where indicate the radiation point. Then we obtain
(11a)
(11b)
The mathematical model for the radiation is thus as shown in Diagram 3(c).
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The overall pictorial representation of the generalized vocal tract model can be
obtained by combining the four types of sections Diagrams 2 and 3 back to Diagram
1(b), as is done a Diagram 3.
Transfer Function Of The New Model:
We now consider the z-domain representation of the generalized model to find the
transfer function for the vocal tract. While it is possible to get the z-domain, we
rather consider the z-domain expression from the equation of each section so that we
can obtain a more convenient expression to handle. We denote the z-domain
variables with capital letters.
For the branch section, we obtain
(12a)
(12b)
By taking the z-transform on equations (5a) and (5b) under the assumption that the
sampling period is .
In a similar manner, we obtain
(13)
For the mid section m; and
(14a)
(14b)
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(14c)
For the glottis section; and
(15a)
(15b)
For the radiation section
In order to evaluate the transfer function H (z) { } connecting the
glottis section to one of the radiation sections, we conglomerate the other radiation
section down to the branch section. For the convenience, we assume that the sections
in the nasal cavity branch are conglomerated. Cascading all the mid sections in the
nasal cavity branch, we obtain the relation
(16)
We define and G (z) respectively by
(17a)
(17b)
Then G (z) can be evaluated by combining equations (15a), and (16) and (17). From
equations (12b) and (17b), we obtain
(18)
33. 33
Where q=
Putting equation (18) back to (12a), we finally obtain
(19)
Where
(20a)
(20b)
(20c)
(20d)
Hence, the two branched version of the generalized vocal model takes the shape
shown in Diagram 4 in the z-domain. Notice that the existing vocal tract model can
be deduced from the model by Setting , which implies that , that is,
the nasal cavity section is not considered.
Therefore, the transfer function can be evaluated from the two-section model,
or by combining equation (13), (14), (15) and (19). The transfer function thus
obtained takes the form
(21)
Pole-Zero Type Linear Prediction:
34. 34
As the transfer function for the generalized vocal tract model has both poles and
zero, it is necessary to consider the formulation of the pole-zero type linear
prediction method. A considerable amount of works are reported in the literature on
the pole-zero type linear prediction, but their major interest is in improved spectral
shaping (see [5]-[7]). Our main concern, however, is in considering the pole-zero
type linear prediction that can be related to the generalized vocal tract model.
Recalling that the generalized meaningful for the nasal sound and consonants, it is
necessary to make the pole-zero type algorithm compatible with those sounds. Since
the excitation of the sounds is assumed to consist of pitch pulse train and white
Gaussian noise, we must remove the effects of the pitch component from the sounds
to obtain a smoothed transfer response. It can be done by applying a homomorphism
signal processing to the sounds. The processed signal corresponds to the white
Gaussian noise response of the generalized vocal tract model, and its frequency
response corresponds to the excitation-to-sound transfer function “ ”. Let
(22a)
(22b)
(22c)
Then, we have the relation
(23)
Where denotes the white Gaussian noise input; the autocorrelation of ;
and the cross-correlation of and . Since is white Gaussian,
, for , and therefore
35. 35
(24)
Taking p terms, through , of equation (24),
We obtain
(25)
Thus is obtained by solving equation (25). Knowing the denominator
term , we pass the signal through a filter whose system function is .
Then the resulting signal corresponds to the output of the system whose transfer
function is . If we set , then can be obtained in a similar
fashion, and B (z) can be evaluated from C (z). Given any desired orders And ,
we can therefore come up with the pole-zero type linear prediction from the excited
sounds.
References
[1] H.Wakita: direct estimation of the vocal tract shape by inverse
f er ng of a o pee h wavefor “IEEE ran for ation AU,
vol-21, pp.417-427, 1973.
[2] J.D.Markel and A.H.Grey, Linear prediction of speech, Springer-
Verlag, New York, 1976.
[3] J.L.Flangan, Speech Analysis, Synthesis and perception, Spinger-
Verlag,, New York, 1972.
36. 36
[4] L.E.Kinsler and A.R.Frey, Fundamentals of acoustics, John Wiley
& sons, New York, 1982.
[5] K H Song and C K Un “Po e-zero modeling of speech based on
high-order po e ode f ng and de o po on e hod ” IEEE
Trans. ASSP, vol-31, pp.1556-1565, 1983.
[6] S.Marple, Jr., Digital Spectral Analysis with applications, Prentice
Hall, Englewood Cliffs, New Jersey, 1987.
[7] J Cadzow “Overe a ed ra ona ode eq a on approa h ”
Proc. IEEE, vol-70, pp.907-938, 1982
Two Dimensional Featured One Dimensional Digital Waveguide
Model for the Vocal Tract
Introduction:
A vocal tract model based on a digital waveguide is presented in which the vocal
tract has been decomposed into a number of convergent and divergent ducts. The
divergent duct is modeled by a 2D-featured 1D digital waveguide and the convergent
duct by a one dimensional waveguide. The modeling of the divergent duct is based
on splitting the volume velocity into axial and radial components. The combination
of separate modeling of the divergent and convergent ducts forms the foundation of
the current approach. The advantage of this approach is the ability to get a transfer
function in zero-pole form that eliminates the need to perform numerical calculations
on a discrete 2D mesh. In this way the present model named as a 2D-featured 1D
digital waveguide model has been found to be more efficient than the standard 2D
37. 37
waveguide model and in very good comparison with it in the formant frequency
patterns of the vowels /a/, /e/, /i/, /o/ and /u/. The model has two control parameters,
the wall and glottal reflection coefficients that can be effectively employed for
bandwidth tuning. The model also shows its ability to generate smooth dynamic
changes in the vocal tract during the transition of vowels.
Human speech production system consists of three main components like lungs,
vocal folds and vocal tract. The coordination of these three components results into
voiced sound, unvoiced sound or combination of these two. For voiced sound
production like that of vowel, the air is pushed out from the lungs into the larynx. In
the larynx, there are two identical vocal folds which are initially closed. The closure
of the vocal folds causes a sub-glottal pressure. When this pressure rises above the
resistance of the vocal folds, the vocal folds open themselves and air is passed
through it. As the pressure decreases with the release of airflow, the vocal folds then
close themselves quickly. The quasi-periodic opening and closing of the vocal folds
continues due to constant supply of the air pressure from the lungs. Thus the
vibration of the vocal folds forms a train of periodic pulses that acts as an excitation
signal for the vocal tract. A non-uniform acoustic tube which extends from the
glottis to the lips is called a vocal tract. The position of the vocal articulators like
larynx, velum, jaw, tongue, and lips, forms a particular shape of the vocal tract. The
shape of the vocal tract modifies spectral characteristics of the quasi-periodic air
flow passing through it, which leads to the generation of voiced speech. In this way
different shapes of the vocal tract generate different voiced speeches. Several
approaches have been employed to model the voiced speech system on the basis of
physical models such as cylindrical segments (Kelly and Lochbaum, 1962; Mullen et
al., 2003) and conical segments (Välimäki and Karjalainen, 1994; Strube,
38. 38
2003;Makarov, 2009) for the vocal tract modeling. In cylindrical approach, each
tube segment of the vocal tract is modeled by the forward- and backward-traveling
wave components of the solution of the wave equation (Morse, 1981; Smith, 1998)
known as one-dimensional waveguide model. It was firstly used in Kelly–Lochbaum
model of the human vocal tract for speech synthesis (Kelly and Lochbaum, 1962).
However, the digital waveguide modeling (DWM), which is an extension of a one-
dimensional waveguide, is recently being used in the modeling of the vocal tract
(Van Duyne and Smith, 1993a, b; Cooper et al., 2006; Mullen et al., 2006, 2007;
Speed et al., 2013).Digital waveguides are very popular for realistic and high quality
sound generation in real time, and are successfully employed in physical modeling
of sound synthesis.
The greatest advantage of a 1-D digital waveguide model is that it has complete
solution to the wave equation which is also computationally efficient for sound
synthesis applications. Moving to higher dimensions leads to a number of limitations
imposed on DWM models for an optimal solution to all sound synthesis systems.
The most important tone is the dispersion error, where the velocity of a propagating
wave depends upon both its frequency and direction of traveling, leading to wave
propagation errors and mistuning of the expected resonant modes. The dispersion
error is highly dependent upon mesh topology and has been investigated in (Van
Duyne and Smith, 1996; Fontana and Rocchesso, 2001; Campos and Howard, 2005).
Another limitation is the restriction on sampling frequency. High sampling rates
require high mesh density which corresponds to high computational cost.
A 1D waveguide model is computationally efficient while the standard 2D and 3D
waveguide models have better accuracy but heavy computational cost (Murphy and
Howard, 2000; Campos and Howard, 2000; Beeson and Murphy, 2004; Murphy et
39. 39
al., 2007). In the present work we propose an efficient two-dimensional waveguide
model of the vocal tract that has comparable formant frequencies with the standard
2D waveguide but has efficiency comparable to that of a 1D waveguide model. In
the present model we approximate only the divergent part of the vocal tract by
divergent ducts and consider two-dimensional volume velocity in it while in the
convergent duct that represents convergent part of the vocal tract, we employ
conventional one-dimensional approximation of the volume velocity. In this way the
accuracy of the current model can never be better than the standard 2D waveguide
model which considers two-dimensional volume velocity in the whole of the vocal
tract. Therefore, we make it as a reference model for the comparison.
The present results of the formant frequencies from the numerical simulation using
area functions for specific vowels (Juszkiewicz, 2014) exhibit good comparison with
the standard 2D waveguide model. The computational cost of the standard 2D
waveguide is very high while the current approach is much more efficient. The
present section is followed by five more sections. In Section 2, we describe our
proposed vocal tract model. In this section, we also develop its mathematical
formulation. Section 3 describes how to find a transfer function of the vocal tract.
Section 4 is reserved for the numerical simulation of the model. Section 5 is
dedicated for the results and discussion and Section 6 is for the conclusions.
Vocal tract model:
We derive a new model of vocal tract with a new transfer function relating it to pole-
zero type linear prediction developed on the basis of the procedure given in (Kang
and Lee, 1988). Current approach is to propose an efficient two-dimensional
waveguide that has formant frequencies comparable with those of the standard 2D
40. 40
waveguide. We consider the vocal tract consisting of concatenated cylindrical
acoustic tubes of same lengths but different cross-sectional areas. We define a
convergent duct by the concatenation of two cylinders, where a cylinder with larger
radius is followed by the one with the smaller radius. The connection of two
cylinders in which a narrow cylinder is followed by a wider cylinder in the direction
of flow is called a divergent duct. A serial combination of these two types of ducts
constitutes the vocal tract. For example, in Fig. 1, the concatenation of the
cylinders and forms a divergent duct while that of
Fig.1. Vocal tract decomposition into cylindrical tubes of different diameters
41. 41
Fig.2.Model divergence duct with imaginary tube and splitting of volume
velocity
The cylinders and constitutes a convergent duct. Similarly concatenations of
with , with , with and with are labeled as divergent ducts while those of
with , with and with define convergent ducts. In the divergent duct, we
assume that the volume velocity splits into its axial and radial components as shown
in Fig. 2. The modeling of such ducts in the form of axial and radial components
may improve the formant patterns of a 1D digital waveguide which are comparable
with a 2D digital waveguide. The convergent duct may be represented by the usual
1D waveguide model as there is no 2D splitting of volume velocity at the entrance
from a wider cylinder to the narrow one. The vocal tract is divided into cylindrical
segments of same length so that the propagating time of sound wave through each
cylindrical segment in an axial direction is same, say, τ. However, each of the
uniform cylindrical segments may have a different cross-section area or diameter, so
42. 42
that the time taken for the sound wave to propagate through a cylindrical segment in
a radial direction may not be an integer multiple of τ. In such a case, the delay in a
radial direction will necessarily be a fractional delay (Laaksoet al., 1996; Välimäki,
1995; Samadi et al., 2004). In the current model, it may be noted that in the
divergent duct reflection of wave occurs at two different places, one is where
impedance changes and the other is at the wall of the cylindrical tube. This leads to
the presence of two different types of delays in the modeling of divergent duct. The
delay in a transverse direction is formulated as the absolute difference of the radii of
the two concatenated cylindrical tubes, which will necessarily be a fractional delay
and has been approximated by the Lagrange interpolator (Laakso et al., 1996;
Välimäki, 1995; Samadi et al., 2004).For the formulation of the model, we consider
a divergent duct consisting of two cylindrical tubes of cross-sectional areas and
as shown in Fig. 2.When the volume velocity enters from the tube into the
tube, it splitsinto an axial component along the vocal tract and a
radial component in a transverse direction. We use local coordinate system in
the divergent duct. Therefore, the origin for the splitting of volume velocity into
the axial and radial directions lays at the junction of the and cylinders
as shown in Fig. 2. When the volume flow is along the direction of the vocal tract,
the acoustic impedance depends on the cross-sectional area of the cylinder. The
cross-sectional area of a cylinder is an area in which the volume flow occur normal
to this area. If we consider the volume flow in a radial direction then the volume
flow occurs normal to the surface area which leads to the assumption that the
impedance of volume flow in a radial direction may depend on the surface area of
the cylinder. For this purpose, we can assume an imaginary cylinder of appropriate
cross-sectional area intruded into the cylinder shown by dotted line in
43. 43
Fig. 2 in which volume velocity along the vocal tract is . A transverse component
may be regarded as the volume velocity coming out of the surface of this
imaginary cylinder in a transverse direction so that it may be considered as
proportional to its surface area. In this way we can control axial and transverse
volume velocity components and by changing the radius of the imaginary
cylinder. It may be noted that the radius of the imaginary cylinder will necessarily be
a fraction of the radius of the cylinder because otherwise there can be
notransverse component in the cylinder, and may be expressed as ,
where . The surface area of this imaginary cylinder whose length is
equal to that of the cylinder, may be written as
(1)
We denote by and respectively, the volume velocity and the acoustic
pressure at position x and time t within the cylindrical tube. Then by solving the
well-known momentum equation and mass continuity equation (Markel and Gray,
1976; Rabiner and Shafer, 1978), we obtain
(2)
(3)
Where c is the velocity of sound in air, ρ is the density of air and the + and − signs
denote the forward and backward traveling components, respectively. Let l be the
length of any cylindrical tube as all tubes have same length. Under the above
assumptions, the acoustic pressure at the junction of the two cylinders forming a
divergent duct is identical in either direction and the total volume velocity is
44. 44
preserved. We, then, have the following boundary conditions at the junction of the
and cylinders.
(4)
(5)
Where represents pressure in the transverse direction and other quantities are as
defined earlier. We have used local coordinate system in which is the
entrance location of the cylinder and is its exit location. Substituting (2)
and (3) into (4) and (5), we get
(6)
(7)
Where , is the time required to travel the cylindrical tube.
From Eq. (6), we have
(8)
(9)
Using Equations (8) and (9) in Eq. (7) we have
If we let , then the above equation becomes
45. 45
(10)
Which can be re-arranged to give a
(11)
In these equations is known as reflection coefficient.
Using Equations (10) and (11) in Eq. (7), then the following matrix form can be
obtained:
(12)
Now, we consider the boundary conditions at the lips and the glottis. For these cases,
we use standard approach of1D digital waveguide model (Kelly and Lochbaum,
1962).
A mathematical relation for the lips radiation is given as (Markel and Gray, 1976;
Rabiner and Shafer, 1978)
(13)
Where is the reflection coefficient at the lips?
Let , then by using Eq. (13), the output volume velocity at the lips can be
written as (Markel and Gray, 1976; Rabiner and Shafer, 1978)
(14)
Similarly, for the glottis section, we have the following mathematical relation
(Markel and Gray, 1976; Rabiner and Shafer, 1978)
46. 46
(15)
Where is the reflection coefficient at the glottis? The time domain representation
of the present vocal tract model consists of Equations (12) to (15). However, this
representation is not computationally convenient for the study of vocal tract formant
frequencies. In the next section, we derive another representation of this model in the
z-domain using z-transformation.
Transfer function of the Model:
In this section we derive the transfer function of the above vocal tract model in pole-
zero type form by transforming the model from time-domain to z-domain using z-
transformation. This representation provides convenient means for studying the
model characteristics.
First of all we assume that we have a vocal tract model with cylindrical tubes of
equal length and delay in each tube is considered as half-sample delay, i.e., we
sample every sample, where is the time required to traverse each tube. We
denote by and as per convention the -transformed representations of
volume velocity components and respectively. We put in Eq.
(12) and apply z-transformation on it which leads to the following matrix form:
(16)
By applying , Eq. (6) takes the form
Where (17)
47. 47
So far we have derived a simple expression for the current model in the form of three
equations represented by Equations (16) and (17). These three equations are not
suitable for the derivation of the transfer function and need to be reduced into two
equations. For this, we define as
(18)
From Equations (17) and (18), we obtain
(19)
By using Eq. (19) into Eq. (16), we have
(20)
Where
,
,
,
Where is defined as earlier.
Now Eq. (20) leads to the desired system of two equations for the derivation of the
transfer function. For the boundary conditions at the lips, we add a fictitious
48. 48
cylindrical tube of infinite length such that there is no negative-going
wave component. We, then, have (Markel and Gray, 1976; Rabiner and Shafer,
1978)
(21)
So Eq. (20) can be written for the lips as
(22)
Similarly, by taking the z-transformation of Eq. (15), the boundary conditions at the
glottis can be written as (Markel and Gray, 1976; Rabiner and Shafer, 1978)
(23)
The transfer function is evaluated by the relation
By combining Equations (20), (22) and (23), the transfer function is thus obtained as
(24)
Eq. (24) gives the transfer function of the current model in z-domain.
49. 49
In our model of the vocal tract, represents the delay in a transverse direction
for divergence duct. For theevaluation of transfer function, we develop an expression
for in terms of z-variable. For this, we assume that the radii of the first and
second cylindrical tubes are and . Then, transverse delay time in the
cylinder tube denoted by can be written as:
Where (25)
Let Where l is the length of the ith cylindrical tube. (26)
Where is a real number? (27)
When (28)
Eq. (27) represents a transverse delay in terms of the delay
Let we introduce be the reflection coefficient at the wall, then on the wall of the
cylindrical tube (as shown in Fig. 2).
(29)
Eq. (29) can be rewritten as
(30)
Taking z-transformation of Eq. (30), we have
50. 50
(31)
Which in view of Equation (18) give the following representation of in
terms of z-variable
(32)
This completes all the requirements for the evaluation of the transfer function. The
block diagram is shown in Fig. 3.
Figure3. Block Diagram of divergent duct.
Numerical Simulation:
Here, we give the numerical solution procedure that was adopted for solving a
waveguide model.
A waveguide model is found to give more accurate formant synthesis, producing
vowels that give a good match to the real-world targets. The current approach has its
advantage of better frequency formants than those of a digital waveguide and
comparable with a waveguide while maintaining its computational efficiency
comparable to that of a digital waveguide.
51. 51
In this work, the length of the vocal tract has been chosen as 17.5 cm. The vocal tract
model has been divided into 10 equal cylindrical segments starting from the glottal
end in order to gain sampling frequency approximately 32 kHz for the speech. In all
simulations, boundary reflection, at lips and is chose as 0.90 respectively.
By using MATLAB 7.0 we derive the graphs and table of Vowels /a/, /e/, /i/, /o/ and
/u/ respectively.
Table:
List of the cross-sectional areas of five vowels given in cm2. The glottal end of each
area of vowel is at section 1 and lip end at section 10
Section /a/, /e/, /i/, /o/, /u/,
1 2.6 2.6 3.2 2.6 2.6
2 1.5855 2.001 2.5871 1.5616 2.6209
3 1.0995 1.4108 1.8044 0.9763 1.0589
4 1.8246 2.1091 2.991 3.4816 8.9055
5 3.8876 7.0104 8.4481 5.257 10.471
8
6 1.9417 6.3825 8.5665 4.0256 9.8715
7 1.2451 6.4014 10.8156 3.0124 8.026
8 0.7165 7.2167 10.4176 1.7726 5.6223
9 0.6466 7.8945 10.5203 1.4555 3.0957
10 0.6331 9.4523 10.4964 1.0436 1.3908
54. 54
Vocal Tract Response:
In this section, we present the accuracy and efficiency of our current waveguide
model in the simulation of vowels and The comparison of its
formant frequencies and efficiency has been made with those of waveguide model.
The parameter k appearing in Eq. (1) determines the size of the imaginary cylinder
relative to the cylinder that carries an axial velocity component within the
cylinder. Its value varies with the variation of the vocal tract length and
the number of segments constituting the vocal tract. We have tested our model for
different values of the parameter k in the range 0 to 1. As we increase the value of
from 0, the formant frequencies of the proposed model start to match with that of the
standard model. It has been found that the best matching of formant frequencies of
the present model with the standard model is achieved at . Therefore, the
55. 55
radius of the imaginary cylinder has been taken to be the same as that of the
cylinder in all the present simulations which corresponds to the choice of k = 1.
Table 1 represents cross-sectional areas of the 10 tubes, which constitute the vocal
tract for each vowel. These cross-sectional areas have been obtained by the spline
interpolation of the cross-sectional areas given in this topic.
References:
1. Beeson, M.J., Murphy, D.T., 2004. Room Weaver: a digital waveguide mesh
based room acoustics research tool. In: Proceedings of the Seventh
International Conference on Digital Audio Effects (DAFX-04), Naples, Italy,
pp. 268–273.
2. Campos, G.R., Howard, D.M., 2005. On the computational efficiency of
different waveguide mesh topologies for room acoustic simulation. IEEE
Trans. Speech Audio Process. 13, 1063–1072.
3. Cooper, C., Murphy, D., Howard, D., Tyrrell, A., 2006. Singing synthesis
with an evolved physical model. IEEE Trans. Audio Speech Lang.
Process.14, 1454–1461.
4. Kang, M.G., Lee, B.G., 1988. A generalized vocal tract model for pole-zero
type linear prediction. In: Proceedings of International Conference on
Acoustics, Speech and Signal Processing (ICASSP), vol. 1, pp. 687–690.
5. Kelly, J.L., Lochbaum, C.C., 1962. Speech synthesis. In: Proceedings of
Fourth International Congress on Acoustics, Copenhagen, Denmark, pp.1–4.
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