# Uniform Tube Modeling of Speech Processing – I

ok so lets ah the in in the first weeks we

are study about that place of particular cells speech acoustics and over view of ah signal

processing lets start the new topics that uniform tube modeling of speech production

what was the purpose of that whole course were a speech processing is that we should

know how the human speech is produced ok after knowing that can i able to mathematically

model that human speech production that is our motto if i know the mathematical modeling

speech production system then studying the system property ah we can know what kind of

speech property you should look for in the acoustic signal ok so our motto is to developed

a mathematical model of human speech production systems now if you see that we have said that this

human speech production system has a two part one is the glottal vibration that glottist

the source of vibration and that sound or vibration passed through the vocal cavity

or ah you can that oral cavity and nasal cavity and produce the different kind of sound depending

on the tube structure or the cavity structure ok now we want to developed if the glottis

is produced sound how can i model this tube model so that i can say i can if i able to

design the mathematical model of the human vocal cord system then i changing the parameters

i can produce the speech or what i know that if i know the exact model or approximate model

of the human speech production system i know the model property so i deriving the model

property i can understand what kind of signal processing i should apply on human speech

so that i can know exactly what is happening in here ok now if i want to model it the complex is

that human speech production model which is not a simple single tube if i have a if you

see in high secondary also you have studied that there is a single tube if this end is

closed and a sound is ah generated here you know what kind of frequency you accept and

what kind of sound we expect from the sound that depends on the l of the length of the

tube but in human vocal tract the this area of the tube in here it is fixed for a rigid

tube but human this area is not same throughout the tube so that area may be look like this

sorry area maybe this is the upper cavity lets lips then started this is the upper cavity

which is almost fixed and if you see the lower cavity the tongue of the tip and then back

tongue of the tip and like this way now if you see depending on the different sound production

system the structure the crosssectional area of the different portion of the cavity for

different sound is different so you can say there are two kind of variation

where i produce the speech speech signal is changing along the time that means that constriction

of the tube or structure of the tube is changing along the time and also for a particular time

lets the produce for this time t one that structure that the cross sectional area of

the tube in all section is not same ok so two complexity one is that cross sectional

area of the tube is not equal in every length so if i say this is the glottis or this is

a vocal cords if this is the x equal to zero and this is at the lip ah if you see this

is the if this is the lip or either if the dental and then there is a lip if i say that

is a lip so and at lip lets x equal to l so length of the tube is l ok but throughout

the length here crosssectional area is same if its a and here it is also a now in here

it is different here it is different here it is different lets here it is different

here it is different here it is different so different kind of crosssectional area is

there now once i produce ka maybe the back of the

tongue touches the upper cavity when i produce po lip lip is touches close together main

cavity structure will be different so i can say for different sound or when its continuously

produce the different kind of sound the cavity structures is changing with respect to time

also ok now we not consider that lets we ah we considering that at if the particular sound

with fix cavity structure is there ok that is ok another simplification will be done

lets think the whole human vocal vocal system is nothing but a single cavity single tube

or single tube which crosssectional area is fixed a so i say throughout the what is glottist

or vocal cords to mouth throughout the whole whole system the cavity is uniform or even

the tube is uniform and it has a crosssectional area a so this is called uniform tube modeling ok now ok lets i consider it is uniform then

we say cross sectional area is a even it is uniform then also the walls of the cavity

is not rigid this this walls are not rigid if it is not rigid then if the air pressure

increases the area crosssectional area may change so this is less vocal cord x equal

to zero and this is lip site x equal to l so if i if the if the walls are not rigid

if this tube is not rigid wall then when the air pressure is passing through the tube walls

will a the cross sectional area of the tube will be change because a wall will be deform

ok now neglect that part also they dont consider that part so i am not considering we said

ok neglect neglect that part lets the wall consider the walls are rigid next one is that there may be a thermal loss

there maybe loss there may be a friction loss lets consider that loss also not there so

we simplify that human vocal tract is nothing but a single tube whos crosssectional area

is constant a there is no frictional loss there is no viscous loss there is no thermal

conduction then try to derive the equation of this human vocal contract ok so uniform

tube model so how do we derive that vocal tract equation so if it is a system if you

if you studied that electrical system signals and system if i have a system if i apply a

input let input is x n let us i consider x n and output is y n then transfer function

if its h n h n then we say h n is nothing but a y n divided by x n ok so we know how

the sound is propagated input sound in here which is input is provided by the glottis

or vocal cords after vibrating it inject the partcular velocity and that particular particules

will be propagate along here and radiated from the mouth radiated from the mouth so i have to know how the wave is propagated

sound wave is propagated along this tube ok along this tube now you have studied already

studied that wave equation you know the wave propagation equation so what is still that

if i produce a sound can i mathematically express the so if i produce a sounds sound

is propagated in the medium who is a condensation and relaxation condensation and relaxation

way can i develop a mathematical equation how that tracer wave is propagated from one

point to another point that is the wave equation ok now considering that amplitude of the pressure

if i say the wave is propagated linearly then you see a linear wave equation here im saying

the amplitude of the sound is not bad enough that propagation become nonlinear so i said

amplitude is less with the limit within the limit and sound is propagated linearly in

the medium and medium is also homogeneous then we can derive the linear wave equation ok similarly here also we will try to derive

the linear wave equation how the wave is propagated along the tube lets try to do that so how

do you do it let consider a single cube in here ok so pressure is so i can say this is

a piston kind of things which produce the pressure in the air and that pressure is propagated

along this cube so if i consider a single cubicle lets i draw the picture of neatly

lets i consider a single cubicle inside that tube whose dimension let us consider the dimension

r this is less del x del y lets tell it del x del y del z so let us volume is v is equal

to del x del y del z ok so this is the volume and this is the cube this small cube inside

the cube i am considering ok now if i see see there is a there is a

pressure wall in here so the pressure is applied in here which is p lets i am saying so if

it is if if the t if the no pressure is applied by the piston then inside the tube it is atmospheric

pressure atmospheric pressure means atmospheric pressure is what lets the what is the average

pressure or atmospheric pressure thats p zero once the piston apply a force a pressure then

this that is p so change pressure is a function of the piston may apply a force in here how

the pressure will propagate it that depends on the position where where i am taking the

position that x equal to zero to l so this depend on the position and time so change of pressure the pressure change

is a function of position and time now if you see if i change if i apply a pressure

ok then what will happen the particle will start moving inside the tube but average velocity

of the particle is zero because medium is homogeneous when sound wave is propagated

medium does not change so that means the particle velocity is exist but the average particle

velocity is zero ok so let the particle velocity is b small b and that also function of x and

t ok now if it is a pressure wave condensation and relaxation condensation and relaxation

so density these also change so density is also function of x and t so in equilibrium

i say medium is homogeneous ok so i have already said some assumption i have

taken that amplitude of the pressure wave is within the limit where the sound is transmitted

linearly thats why you call this linear wave equation ok now if it is linear wave equation

then how do i derive that wave equation so what do you understand my wave equation that

means i want to know how the pressure form x equal to zero is propagated along the tube

lets at x equal to l i want to know what is the pressure so i want to know the pressure

movement equation mathematical equation of the change of pressure with respect to position

and time and also mathematical equation for particle velocity with respect to position

and time and also the relation between the pressure wave pressure with respect to position

time with the particle velocity and position and time so that i want to know thats i want

to know the derive those equation ok so for deriving those equation you already studied

it ah may be in ah acoustic or in wave equation derivation in first year physics we will consider

three principles of physics one is called newtons second law of motion second is called

gas law or thermodynamics and another one is converse conservation of mass think about

it if i apply a pressure in this this plane ok

so change of pressure and the pressure is exist in this plane is lets p two this is

p one then i can say the pressure difference so pressure if p one is the pressure then

i know the pressure into area lets a one is the area and this is also a one so pressure

into a one give me the force lets this is f one which is acting on this plane and force

which is acting from this n is nothing but p two a one so difference of force is the

cause of the motion of this particular in this block ok so i can say newtons second

law of motion the force equal to mass into acceleration so the force resultant force

cause an acceleration of that cube which is nothing but a force equal to mass into acceleration

ok mass of the cube into acceleration ok now second law said the gas law hydroxylated

pressure volume and temperature under the adiabatic condition so what is the meaning

if i apply a pressure on this tube this is there this block suppose a block i apply one

plane one pressure is applied so every day i apply a pressure the volume will be changed

if the volume will change mass would be change but mass will not change its increase the

density due to the conservation of mass mass will remain same density will be change volume

will change because that cube will be deform due to the applied pressure ok so although

the if the pressure is increases the volume is decreased but mass of the cube will be

remain same and this is an adiabatic condition so i say the region when the sound om is propagated

there is no conduction of heat ok now if that is the theory so using this

three law we will have to derive the wave equation so again i can if you see the slides

the assumption i have said media means homogeneous pressure exchange across the small distance

there is no friction in air particle air particle velocity is small sound is adiabatic ok now

if i want to derive that equation what i will do lets this is p one so i can say p one is

nothing but a p one acting on the pressure so it is p of x t not capital p small p ok

what is p two is nothing but a since del x is the length along the x direction so it

is nothing but a small p x plus del x t ok what is p of x plus del x t nothing but

a p of x t minus p of x t and plus change of pressure to for the d x distance so it

is nothing but a del p by del x into del x i said change of pressure is linear i am not

considering the nonlinear term ok this is ah p one is p a p two x so if i say p two

minus p one which is nothing but a p of x or i can say p one minus p two acting of pressure

in p one here it is p two so p one minus p two is nothing but it p of x t minus p of

x plus del x t which is nothing but a p of x t minus p of x t minus del p by del x into

del x ok now if i say what is the force acting on

this plane multiplied by the area so it is a one multiplied by a one so i can say the

one is multiplied so it is a one is multiplied so it is nothing but a one del p by del x

into del x so what is a one is a crosssectional area of this area so this is nothing but the

del z into del y so i can say it is nothing but a minus del p by del x into del x del

y del z because a is a one is equal to del y into del z ok so i get the force so it is

nothing but a force acting on the cube so force is equivalent to mass into excellence

newtons second law of motion so force is equal to mass into acceleration so what is force

minus del p by del x into del x del y del z is equal to mass into acceleration what

is the acceleration del v by d v by d t rate of change of particle velocity is the acceleration

so i can say it is nothing but a into d v d t this v is the particular velocity ok now if i say that then what is d v d v

d t is a total derivative that and v is a function of x and t v is a function of x and

t so i can say d v d t is nothing but a del v by del t plus v into del v by del x ok or

not del v by del x now if i consider this term this is a nonlinear term if i drop this

term that ok this term is common to be the particle velocity is very small particle velocity

is less negligible in that case i can say that del d v by d t is nothing but a del v

by del t ok so in that case i can write minus del p

by del x minus del p by del x is equal to m into del v by del t ok what is m m is nothing

but a density into volume which is sorry this that will be del x del y del z rho into volume

so volume means del x del y del z into del v by d t this will cancel so i can write minus

del p by del x is equal to rho del v by del t this is the equation number wave equation

number one minus del p by del x is equal to rho into del v by del t ok the equation number

one so newtons second law of motion now i come to the gas law what is gas law said p

v is equal to n r t you know n is moles of air r is gas constant t is the temperature

in kelvin that is a gas law adiabatic gas law i can say or i can say p is equal to rho

v divided by m into r t rho is the density v is the volume sorry the p v v is the volume

m is the molecular weight of the air and again r and t is there so if you see here it is

there so p b is equal to n naught t and p v is equal to rho v by m into r t now if i say then p is equal to i can say

rho v v cancel rho r t divided by m ok rho r t into m now you know the c square or c

is equal to bulk modulus divided by that is you know so bulk modulus divided by density

so instead of that i can write c is equal to root over of gamma r t by m where i derived

the im not deriving that one c is equal to so that that you can derive from the pressure

the adiabatic condition that if you require you can see the book and that is available

and i can next day i can also that do that also if we required ok you send me the mail

if it is required then i can derive that things also or you can see that my another lecture

in audio system engineering derivation of linear wave equation ok now if c is equal to then i can say c square

is equal to rho r t by gamma r t by m what is gamma gamma is nothing but specific heat

ratio ok so i can say from here i can say that p into gamma is equal to rho into c square

p into gamma is equal to rho into c square another way i can say p v to the power gamma

is equal to sorry so another another way i can say p v to the power gamma is equal to

constant if i take the derivative with respect to t d d t of p b to the power gamma is nothing

but zero so what is this i can write v to the power gamma into del p by del t plus p

gamma b to the power gamma minus one del v by del t this v is volume this v is capital

v which is volume ok minus del v into del t is equal to zero i can do that that is equal

to zero now if i say what is v volume is nothing but

a a into del x cross sectional area into del x so cube if you see the cube volume of the

cube is nothing but a y z line cross sectional area del y into del z into del x ok now if

i say d v or i if i say del v by del t is nothing but a a into del x by del t which

is del v ok if it is that then i can write a into del

v by del x into del x i can write now i put this one in here so what i get v to the power

gamma del p by del t plus p gamma v to the power v to the power gamma minus one into

a into del x what is a into del x into del v by del x so it is nothing but a v into del

v by del x so i can write v into del v by del x so it is nothing but a v to the equal

to zero so v to the power gamma minus gamma into del p by del t plus rho gamma into p

into v minus one into v v ah v to the power gamma del v by del x is equal to zero ok so this v is volume velocity and this capital

v is the volume because here i have said del x by del t is nothing but a particular velocity

so so particular velocity sorry particular velocity so this v is the particular velocity

ok so i can say v to the power gamma is cancel so i can say del p by del t is nothing but

a rho into p del v by del x is equal to zero ok i can say del p by del t is equal to minus

rho p gamma p sorry gamma p into del v by del x which is nothing but a i said pressure

into gamma is equal to rho c square so i can say it is nothing but a minus rho c square

del v by del x ok so i can write another wave equation so

first wave equation was del p by del p by del x is equal to minus rho into del v by

del t and here i get del p by del t is equal to minus rho c square del v by del x ok or

another way i can write this one same way that i can write minus or minus del p by del

x is equal to rho del v by del t and del p minus del p by del t is equal to rho c square

del v by del x ok now if i want to know the pressure equation so what i will do lets this

equation derive respect to del t x so del square p by del x square lets minus is equal

to rho into del v by del t into one by del x if i again do it in the respect to t minus

del del square p by del t square is equal to rho c square del v by del x into one by

del t so if you see this side and this side what is the difference one by c square only

one by c square if i multiply so i can write minus del square p or minus this side minus

this side minus so i can write del square p by del x square is nothing but a one by

c square into del p square by del t square ok or not similarly i can derive the same

equation which is nothing but a del square v volume particular velocity by del x square

is nothing but a one by c square del square v by del t square so i can say this is the

wave equation so this pressure wave equation this is the particular velocity wave equation ok so i can say this is the wave equation

with a along when the wave is propagated along the x axis i am not considered that wave would

the any particular velocity with the x y and ah y and z direction so lets i say the wave

is propagated along the x axis and there is a not other directional all particular velocity

so friction force is not there viscous viscosity loss is not there so nothing is there ok then

this is my velocity pressure equation this is my velocity equation ok now i what i know

i know in the tube how the pressure wave is propagated how the pressure wave is propagated

i know in the tube now i have to derive so pressure a mathematical equation i know but

i dont know what is the solution of the p so i to a second order differential equation

i can solve that second order differential equation and find out the equation for p ok

solution of that degree second order differential equation and then try to find out how whats

do the transfer function of this vocal tract tube ok so next class we try to derive the

transfer function of single tube ok thank you

## One Reply to “Uniform Tube Modeling of Speech Processing – I”

thank you sir ….superb teaching