# Uniform Tube Modeling of Speech Processing – IV

ok so last class we have discussed about that

effect of that different kind of losses nozzle effect of nozzle cavity on sound spectrogram

and we said that with the derived that transfer function of the uniform tube single tube vocal

tune vocal track now think about that ok see vocal track is a single tube model is ok now

can i find out that the radiation loss also you have ah find out the effect of radiation

loss now think about that how the glotal will connected to the vocal track ok if you see

here the air flow from lungs passes through the vocal cords and once it passes through

the vocal cords the since the vocal cord create obstruction in the airflow and due to this

airflow the vocal cord is vibrating and that vibration create the sound and goes through

the vocal track and produce the sound so i can say whole vocal track can be the this

is systems so i can say this is you see that diagram

so this is the vocal track model next this is the vocal track model how we excited that

model using the vocal cord vibration so now vocal cord vibration can a electrically simulate

like this way so this is the airflow that means the air pressure which is coming from

lungs if the vocal cords are close the pressure is increases once it start opening that if

the pressure the this increased pressure force airflow through the vocal cords opening and

that airflow causes a vibration in the vocal cords ok and that vibration create the sound

now if i say if the if i say the if you know that particle ah volume velocity u is nothing

but a analogues to electrical current i so if i say this is the vocal track which is

a system and it is excited by a pressure which is coming from pressure is coming from pg

lungs lungs pressure is excited the vocal track and that pressure sorry that lungs pressure

will connect will will cause an particle velocit[y] or volume velocity u in the vocal track so

that how do you make the what so if i say the impedance of the vocal cords

its nothing but a resistance and inductive impedance then lets it is a lg and it is rg

like the load ok so now this pressure pass through that impedance and create a vocal

cord ah or you can say volume velocity ug which is the input to the vocal track system

ok now if i say this vocal track vocal cords is a time varying vocal cords because vocal

cord is closing and opening closing and opening so once it is completely close so completely

close means ug is equal to zero if it is completely close no volume velocity because air is completely

stopped flow is completely stopped so no volume velocity ug is zero so when this ug will be

zero if the impedance in here is infinite so if i say lets vocal cord opening ag is

a function of t so opening of the vocal cord is a function of t then impedance impedance

is nothing but a one by agt so when the vocal track vocal cords are completely closed then

ag opening agt is equal to zero and that creates infinite impedance so that the although there

is a pressure is there but there is no flow ug is zero ok so thats why that glottal impedance

is infinite volume velocity is equal to zero now this analogy lets try to derive that ok

i know the vocal track transfer function i know the load condition or boundary condition

at the lips which is the radiation load i know the ah input ah input condition ah vocal

vocal vocal cords load into the tube now can i implement that tube in a vocal or within

a circuits or in a digital domains or in a circuits so lets this is my vocal track this

is the vocal track uniform tube is the vocal track so here is an radiation load at the

lip so if this is my zl radiation load in the so this is x equal to zero this is x equal

to length length of the vocal track is l or here is the radiation load this side there

is a some impedance in here and then there is a pressure source pg so this is the complete

electrical model now if it is pg it can be think ok this is pg so if i make it replace

by a current source which is nothing but a ug current source is nothing but a ug then

this impedance come in parallel so this is zg and this is zl this is the complete

model ok now putting these two boundary condition try to simulate this single tube vocal cord

model so if i say this vocal track if i say this is my ah input volume velocity so there

will be output volume velocity this is the there is a this is u equal say that this is

ut ah ut ut plus zero t this is ut minus plus you can say that lt ok this is ut minus and

this is lt and this is ut zero t minus ok so this is a backward wave so there will be

a forward wave there will be a backward wave inside the tube which is u plus and which

is u minus ok so conis considering all those things can i derive this tube mathematical

model of this tube considering two boundary condition ok now so tube can acts nothing

but a delay tube is acting is nothing tube acts nothing but a delay lets forget about

that part lets solve one by one boundary condition so boundary condition at lips so i have a

vocal track this track has a boundary condition at lip so this is nothing but a zl ok so this is nothing but a u ply u minus lt

here and this is u plus lt ok or i can say this current which is coming out it is nothing

but a u lt here it is u ply here it is u minus and here it is u plus ok so this u lt pass

through this zl and create plt plt is nothing but a p at lt is equal to nothing but a zl

into lets complete current u lt or i can say ut lets right that t ah t track ok ut lt so

now find out that lip boundary condition ok so what is p l plt pressure across the load

pressure across the load is nothing but a zl or you can say that zl lets right zl into

ut lt so it is nothing but a zl into what is ut lt ut lt is nothing but a u plus lt

minus ut minus lt ok so it is nothing but this one now if i say what is plt at the output

of the t track plt x equal to l in here and x equal to zero in here so at x equal to l

plt is nothing but the zt into ut plus lt plus ut minus lt ok or not and these two things

are equal because at here this pressure is equivalent to the pressure across the load

so in that case i can say zt ut plus lt plus ut minus lt is equal to zl ut plus lt minus

ut minus lt is ok i can write down now i write down that [vocalized noise] what

is ut lt so zt into u plus ut plus t minus l by c plus ut minus t minus l by c is equal

to zl ut plus t minus l by c this will be plus sorry minus ut minus t plus l by c ok

now from this equation if i want to find out ut minus t plus l by c i want to find out

that ah the the that backward wave so why want if i want to find out backward wave so

lets all the forward wave term i can write one side and on the backward wave term i can

write another side so i can say zt from here i can say zt into ut minus t plus l by c this

term will come plus zl ut minus t plus l by c is equal to zl into ut plus will be there

so zl ut plus t minus l by c minus zt into ut plus t minus l by c ok so i can say ut minus t plus l by c into zt

plus zl equal to ut plus t minus l by c into zl minus zt so i can write ut minus t plus

l by c is equal to see this this is zl minus zt take them minus sign out so minus zt minus

zl divided by zt plus zl into ut plus t minus l by c ok or not lets this whole things zt

minus zl divided by zt plus zl zt minus zl divided by zt plus zl if i say this is nothing

but the rl if i write down this is nothing but a rl so i can write it is nothing but

a minus rl ut plus t minus l by c ok now if i say if it is this minus rl ut plus t minus

l by c is ok so this is like like same that in [vocalized noise] if you see that slides

it is nothing but a same so it is in minus rl into ut minus t minus

l by c is ok now what is the total current then what is ut or you can say the ult what

is ult ult is the current passing through zl current which is passing through zl which

is nothing but a ultimate volume velocity which is passing through this circuit so it

is nothing but a ut lt so i can say utl is nothing but a ut plus t minus l by c minus

ut minus t plus l by c ok so ul you can say ult is nothing but a u at l and t ok so u

at l like this so i can say ok i put that minus value in here so ut plus t minus l by

c minus again minus plus rl into ut plus t minus l by c so i can say it is nothing but

a one plus rl into ut plus ut plus t minus l by c sorry ut plus t minus l by c ok so

this is nothing but a one plus rl now if i want to draw the circuits so what sort of

electrical circuits or transmission line kind of circuits so ult so i can say lets this

is the tube so if i say the glotists producing a lets there is a tube so tube is nothing but a delay so if i say

this is my vocal track so vocal track has a forward wave and as a backward wave so whole

tube is nothing the volume velocity in here and volume velocity here lets the volume velocity

expression only different but lets it is delayed so what will happen lets i say this is a delay

circuits for forward wave so it is nothing but a ut plus zero t ok lets after the delay

i get ut plus lt ok then backward delay either backwards lets this is ut minus lt backward

wave and i can get here also backward wave which is ut minus zero t ok now total output

of this volume output volume velocity is nothing but a ul which is ult which is nothing but

a ut plus ut lt so ut lt what is the ut lt equation ut lt equation is one plus rl ut

plus t minus l by c so it is ut minus l by c so i can say it is multiply by one plus

rl and i get that value what is t minus t minus lt it is nothing but a multiply by the

plus this one multiply by the minus rl as far this equation this equation so i can say it is multiply by the minus rl

i will get this and this whole things is nothing but the delay is ok so it this is the boundary

condition one plus rl multiply by this circuits so this is the signal flow diagram i can say

signal flow diagram is nothing but a one plus rl multiply by ut lt here and here is minus

rl and flow is this side backward wave flow is this side ok now consider if you see the

slide it is like this ok ok now you you you that is a condition if zl is much much greater

than zero much much greater than zero then rl is equal to one if zl what is rl rl is

nothing but a zt minus zl divided by zt plus zl now if say that if zl is much much greater

than zero then i can say rl is equal to ok this condition i will come later this ill

come after discussing the boundary condition i will come this zl zt this ah constant scenario

ok so this is the ah ah lip at the lip condition now if i say the glottis ah glottis site if

discuss about the boundary condition at the glottis side so i can say ok i know the lip side i know

ah lip side is nothing but a zl which is zl ok now this side i have i have said the special

source is replaced by a current source which is nothing but a volume velocity which is

ug zero t or ugt here zero is not here because z equal to zero in here x equal to l in here

now the impedance gotal impedance come in parallel then zg ok so signal flow is like

this this is the forward wave backward wave so i can say that pgt total gotal pressure

is nothing but a pt zero t the pressure across in here will be same in here also gotal pressure

now the current is nothing but a this is a total current so some current will flow this

way and some current will be flow this way some current will be flow this circuits and

some current will be flow this circuits so i can say ut zero t which is the volume velocity

at the input of the vocal track is nothing but a ugt minus so this is the total pressure

is pgt so total pressure is pgt divided by zg so this current is this total pressure is

pgt then the pressure divided by the load is the current so this current will be minus

from the total current is the current which which will be passing through this to the

ah tube so i can say volume velocity at the tube input is nothing but a total ugt total

volume velocity produced by the glottist minus the volume velocity which is passing through

the load glotal load uzg so this total current minus the load current so load current if

the let zg is the load and voltage is pgt then pgt divided zg is the current ok now

what is pgt pgt is nothing but a p zero t or i can say p zero t or pt zero t input track

voltage so pressure at vocal track input so i can write ugt minus one by zg into zt ut

plus zero t plus ut minus zero t ok or not ok so i can write ugt minus zt divided by

zg into ut plus t zero plus ut minus t x equal to zero here x equal to zero ok so from here

what i have to find out again i want to find out the what the volume expression of the

volume velocity which is input to the track forward wave and the backward wave then i

can draw the circuits so from there i want to find out ut plus t from this equation i want to find out ut plus t ok how do i do

it ok so ut zero t here it is ut zero t i write down ut zero t means ut plus t minus

ut minus t ok is equal to ugt minus zt divided by zg into ut plus t plus ut minus t ok ok

so from there i have to find out ut plus t what will be the expression the expression

will be like this ut i am write write deriving it ut plus t will be ugt divided by one plus

zt divided by zg into one minus zt my zg divided by one plus zt by zg into ut minus t ok so

i can simplify it ugt plus ut and here also i can say ugt sorry there will be plus this

is not into plus ok so if i simplify it zg into ugt divided by zg plus zt ok plus zg

minus zt divided by zg plus zt ok into ut minus t ok now now if i say zg minus zt divided by

zg plus zt is equal to rg i say that this terminology this terms is equivalent to rg

ok so what is the value of zg divided by zg plus zt so zg plus zt will be one plus rg lets find out one plus rg by two so what is

one plus rg by two one plus rg divided by two is equal to one plus zg minus zt divided

by zg plus zd whole divided by half will be zg plus zt plus zg minus zt divided by zg

plus zt so zt zt cancel so it will be two zg divided by zg plus zt into half so it is

nothing by but a zg divided by zg plus zt so i can say this term zg by zg plus zt is

nothing but a so i can say ut plus t is nothing but a one plus rg divided by two into ugt

plus rg ut minus t ok now i draw the complex circuit of this side so i can say see this

equation i tried to draw the vocal track circuits ok so lets this is my tube say again this

is my delay in the forward wave this is my delay in the backward wave ok i take this

blue pen so this is the delay circuit ok so this delay circuits this is delay ok so

i can say this is ug is coming from here ugt is coming from here gotal volume velocity

is coming from here so if it has to be ut so this is nothing but the ut plus t ut plus

t ut at that ah ut plus so this is nothing but a one plus rg divided by two multiply

this plus backward wave has to be added here which one is nothing but a rg so ut plus t

is nothing but a one plus rg by two so ugt multiply by one plus rg by two system dialer

signal flow diagram plus delay this is nothing but a ut minus t rg and lip side what i get

if it is [vocalized noise] ut final output lip side we get this kind of this which is

nothing but a one plus rl and this is nothing but a signal flow diagram so this will be

the sign this is nothing but a minus rl has to be fine and this is nothing but a ut minus

lt this is nothing but a ut plus lt ok so if i consider whole tube is nothing but a

delay and put this two boundary condition now if i know rg and rl i can derive the circuit

diagram or transfer function also so in digital circuits how the delay is implemented this

delay can be replaced by z to the power minus n n is the number of sample delay so suppose i have a seventeen point five centimeter

long tube seventeen point five centimeter long tube ok so how much delay if the sound

velocity is three five zero zero zero second ah zero centimeter per second three fifty

meter per second so this how much time sound will take from here from come from here to

here seventeen point five centimeter is the tube length so time is distance divided by

the velocity so you can get it so how much time will required to cover seventeen point

five centimeter if the velocity is v v is velocity c of the sound is equal to three

five zero zero centimeter per seconds i can easily find out that things and once i get

that things so i can say this is three fifty two i can put one so this one t by two zero

zero zero so i can say half zero ah or can say ah point

five half milli second half milli second ten to the power three is half milli seconds zero

point five milli second ok so zero point five milli second time will take from tube come

to here to here now if i say that in digital domain if this circuit i want to implement

if it is sampled at eight kilo hertz then single sample delay means single sample delay

means one by eight kilo hertz is the single sample delay so one by eight kilo hertz i

think one twenty five micro second then how much delay will required point five milli

seconds i can calculate the number of z is required so i can listed out delay i put z

to the power that number of n delay will be simulated and i get the digital circuits ok

if i know the rl and rg ok so next this kind of mathematics you can practice later on next

we d then we discuss about the multi tube modeling so next class will discuss about

the multi tube modeling so single tube modeling we derive the signal

flow diagram you dont have to remember it if you understand that things if you able

to derive these equation then from the equation i can draw the signal flow diagram for a single

tube model considering two boundary condition now instead of the single tube if i consider

that production system is nothing but a multiple tube then whats would be the task and how

do you implement it so next class we will start there ok thank you