hello friends welcome to my lecture on applications of z transforms we have seen that the z
transform of the sequence u n equal to one is z over z minus one when mod of z is greater
than one and the z transform of the sequence un equal to n is z over z minus one whole
square when mod of z is greater than one and z of z transform of the sequence un equal
to n square is z square plus z over z minus one whole cube and mod of z is greater than
one now we can also find the z transform of the sequence un equal to n cube by using the
recurrence relation z transform of n to the power p z transform of this equals
n to the power p equal to minus z d over dz of z transform of n to the power p minus one
where p is a positive integer so here z transform of n cube where we have taken n equal to three
here so z transform of a n cube is minus z d over dz dz of z transform of n square
so we write the z transform of a n square that is z square plus z over z minus one cube
here and then differentiate with respect to z and multiply by z we arrive at z cube
plus four z square plus z over z minus one to the power four when mod of z is greater
than one this z transform we will need when we solve difference equations so we
have to drive this z transform of n cube now let us what we i have done is that we are
taking z transforms inverse z transform of some expression functions where we shall see
that the when we write the partial fractions of uz over z we have to write uz over z the
partial fractions of uz over z according to the the inverse z formulas that is
we we will have to use while inverting will have to use the z transforms of special sequences
so we will have to write in in the uz over z also in that form
so for example here when we say z the inverse of z cube over twenty z over z minus two whole
cube equal to z minus four let us see how we write the uz by z into partial fractions
so what we see do is let us say that uz equal to z cube minus twenty z divided by
z minus two whole cube into z minus four then uz by z by in the partial fractions method
we have to write uz by z into partial fractions so we write a over z minus four b over z minus
two c over z minus two whole square see we have z minus four power one so we will ave
one fract partial fraction corresponding to the factor z minus four but correspond
to z minus two whole cube we will have three fractions one with z minus two with the
other one z minus two whole square and the third one with z minus two whole cube
now let us see how we will have to write these partial fractions while inverting we will
write uz uz will be a times z over z minus four so while inverting the inverse z transform
of z over z minus four will be four to the power n so we will be able to invert and here
we will get z over z minus two inverse z transform of z over z minus two will be two to the power
n so that is also possible to invert and here we will have z over z minus two whole square
let us recall that z transform of n into a to the power n is az upon z minus a whole
square when mod of z is greater than mod of a so here if you take because here if z minus
two whole square so you take a equal to two so z transform of n into two to the power
n will be equal to two z upon z minus two whole square and so we can say that z transform
of n two to the power n divided by two will be z upon z minus two whole square
so inverse z transform of z over z minus two whole square will be n times two to the power
n by two so we will have to we will be able to invert this term also z over z minus two
whole square but here when we come to z minus two whole cube let us recall the formula
for n square a to the power n we have z transform of n square a to the power n this
is equal to a z square plus a square z divided by z minus a whole cube so this is actually
z times az plus a square divided by z minus a whole cube now if i take a equal to two
taking a equal to two we get z transform of n square into two to the power n equal
to z into two z plus four divided by z minus two whole cube so z while while writing
uz this z we will will get come here so here we have z z here and then two z plus four
is also needed so what we do is we write here d times two z plus four now now when we
will be able when we will when we will find the values of abc and d this term we can also
invert because this term will be z into two z plus four divided by z minus two whole cube
so its in in its inverse z transform will be n square into two to the power n now the
only thing that remains is to do here is that we have to find the values of abcd so
a can be found easily a is corresp a corresponds to z square minus twenty because uz by
z is z square minus twenty divided by z minus two whole cube z minus two whole cube so this
is when you put z equal to four so you put z equal to four so we get sixteen minus
twenty divided by four minus two that is two whole cube so that is eight and we get here
minus four by eight so we get minus half we can then write this is let me say that
we write it as z square minus twenty divided by z minus two whole cube into z minus four
so equating both sides we can also find the value of d directly what you do
is let us write from this i equality we write z square minus twenty equal to a times
z minus two whole cube plus b times z minus two whole square into z minus four plus c
times z minus two into z minus four and d times two z plus four into z minus four
now you can see that when you take z equal to two this term is zero this term is zero
this term is zero and this term is non zero so we will get the value of d so putting z
equal to two we have two square means four minus twenty so we get minus sixteen equal
to d times two into two that is four four plus four is eight and then we get two minus
four that is minus two so minus sixteen d equal to minus sixteen this gives you d equal
to one and the next thing that we can do is let us equate the coefficients of various
powers of z so equating the coefficient of z cube on both sides here we have term
in z cube is zero
so the coefficient of z cube is zero there here the coefficient of z cube is a plus b
so equating the coefficient of z cube both sides we get a plus b equal to zero now a
we have already found a is equal to minus half so b is equal to half then we can
equate the constants on both sides which means that we can put z equal to zero in this equation
so putting z equal to zero we get let me call this equation as equation number
one so putting z equal to zero in one what we get is i will write here so minus twenty
equal to a times minus two whole cube so minus eight into a and then we get minus two
whole square which is four four into minus four that is minus sixteen
so minus sixteen b and then we will get c times minus two and minus four so that is
eight times c and here we shall have z equal to zero means d times four into minus four
so minus sixteen d let us use the values of abc a a d b and d so a is equal to minus half
so minus eight into minus half and then we will have the value of the d b equal to half
and then we have to find the value of c minus sixteen d is equal to one so how much is this
or minus twenty is equal to now this is four eight by two is four here we get minus eight
here we get eight c minus sixteen
so this is four minus twenty four that is minus twenty so c is equal to zero now so
we get c equal to zero and thus we have the following fractions so hence uz by z a is
equal to a is equal to minus half so minus one by two z one over z minus four then we
have b equal to half so we get one by two z minus two and c is zero d is one so we get
two z plus four divided by z minus two whole cube now let us multiply by z so r uz equal
to minus half z over z minus four plus half z over z minus two and then two z square plus
four z divided by z minus two whole cube now lets take inverse z transform both sides so
taking inverse z transform we get un equal to minus one by two four to the power n then
one by two two to the power n and this is n square into two to the power n and for these
transforms ok mod of z is greater than two but here mod of z is greater than four so
we will take mod of z greater than for the common region so this is the inverse z transform
of z cube minus twenty z over z minus two to the power three over z minus four
so while writing the partial fraction of uz by z we have to write them in such a way that
while inverting we get the standard sequences so standard results we have we will
have to use now the again next in the next example we have taken this difference equation
and we have to find the response of the system that is we have to find the sequence yn
such that y naught equal to zero y one equal to one and here we are given that un
is equal to unit step sequence un equal to one for n equal to zero one two three and
so on and which is nothing but the unit step sequence and for the unit step sequence
if you remember we have seen that the z transform is z over z minus one where mod of z is greater
than one so here we shall see that the sequence yn is equal to one by two minus two
times two to the power n plus three by two three to the power n
let us see how we get the solution of this it is very simple difference equation so
as usual let us take the z transform of the given difference equation yn plus z transform
of yn plus two and then we have minus five times z transform of yn plus one then we have
six times z transform of the sequence yn equal to z six times z transform of this sequence
yn and then we have the z transform of the unit step sequence un z transform yn plus
two we can find by the shifting property so this is z square times yz where yz is the
transform of the sequence yn minus y naught minus y one by z minus five times z of yn
plus one is z times yz minus y naught plus six times z transform yn we have taken as
yz equal to unit z transform of the unit step sequences z over z minus one
so here we have taken mod of z greater than mod of one now let us use the values of y
naught and y one y naught is zero y one is one so z square times yz minus one by z because
y naught is zero y one is one minus five times z into yz because y naught is zero plus
six yz equal to z over z minus one this is how much let us collect the coefficient
of yz here so z square minus five z plus six times yz equal to z over z minus one plus
z which is z square divided by z minus one now the factors here are z minus two and
z minus three so we can write yz equal to z square divided by z minus one z minus two
z minus three ok
so as usual let us write yz by z so then yz by z will be z over z minus one z minus two
z minus three which we shall write as a over z minus one because z minus one z minus two
z minus three all occur in power one so for each one we will have one fraction so b over
z minus two c over z minus three and we can see that while writing yz each term here
will be multiplied by z so and we will be able to determine the inverse z transform
of z over z minus one z over z minus two z over z minus three and so we just have to
find values of abc so a you can see a is equal to z over z minus two z minus three at
z equal to one which comes out to be one over one minus two one minus three so this is minus
two this s minus one so we have one over two and similarly b will be z over z minus one
z minus three at z equal to two so we get two over two minus one is one two minus
three is minus one so we get minus two and then c similarly will be z over z minus one
z minus two evaluated at z equal to three so we shall have three over three minus one
is two three minus two is one so we get three by two and thus thus yz will be equal to let
us multiply by z the value of a we have found as half so half z over z minus one the value
of b we have found to be minus two so minus two times z over z minus two and then the
value of c we have found three by two so three by two z over z minus three so inverse z transform
of yz which we will be equal to the sequence yn this is equal to half one to the power
n minus two times two to the power n and then we get three by two three to the power n for
this mod of z greater than three for this mod of z greater than two and for this mod
of z greater than one
so we have mod of z greater than three which is the common portion so he inverse
z transform is half minus two into two to the power n plus three by two into three to
the power n now let us go to this problem where we shall see that we have to use
z transform of n cube which we have found earlier ok so lets consider the difference
equation un plus two minus two un plus one two un plus one plus un equal to three
n plus five you can see that here we are not given any conditions on the sequence un so
the u naught and u one u two which we will occur which will occur while taking the z
transform of this will be taken as some arbitrary constants we will see that there will occur
u naught and u one so u naught and un will be taken as some constants and writing the
solutions solution of this problem we shall see that we have to choose c naught as u naught
and c one as u one minus u naught to arrive at this sequence the solution of the problem
so let us take z transform of this so taking z transform on both sides we get
ok so z transform will be equal to z square uz minus u naught minus u one by z and then
minus two times z into uz minus u naught the z transform of un plus one plus z transform
of un will be uz z trans three times z transform of n let us recall that z transform of n is
equal to z over z minus one whole square where mod of z is greater than one so this is
z over z minus one whole square and z transform of five five is a constant so we need to know
the z transform of one z transform of a to the power n was equal to z over z minus a
provided mod of z is greater than a so taking a equal to one here you have z transform
of one is equal to z over z minus one so provided mod of z is greater than one so this is
z over z minus one provided mod of z is greater than one for these to be true ok so now
let us take the coefficient of uz so z square minus two z plus one into uz and here we shall
have minus u naught into z square here we shall have u one z and here we will get two
u naught z equal to three times z over z minus one whole square plus five times z over z
minus one ok so let us collect let us find the value of uz so uz will be equal to
ok so uz will be equal to this is z minus one whole square
so let me write it as z minus one whole square into uz and let us take these terms with the
other side we have three z over z minus one whole square plus five times z over z minus
one and then we have here u naught times z square minus two z u naught times these
terms when go to the other side will give you u naught times z square by two z plus
u one z plus divide by z minus one whole square to get the uz so uz is equal to three z
over z minus one to the power four plus five z over z minus one to the power three then
u naught times z minus square minus two z divided by z minus one whole square and then
u one z over z minus one whole square now let us write uz by z from here and then we
shall write the fractions of uz by z in such a way that we can easily get the inverse z
transform so uz by z is equal to three by z minus one to the power four five upon z
minus one whole cube and then we have here u naught into z minus two divided by z minus
one square then we have u one upon z minus one whole square
now we can see that in the denominator we have z minus one power of z minus one as two
three four so when we write the partial fractions corresponding to this uz by z we have to
write four fractions corresponding to z minus one to the power four so this we shall
write as a over z minus one b over z minus two sorry a over z minus two one b over z
minus one square now so far when you will write uz you will multiply this y a and b
y z so z over z minus one can be inverted z over z minus one whole square can also be
inverted but here when you will multiply by z the term corresponding to z minus one
to the power three so let us recall that z transform of n square a to the power n this
was equal to a z square plus a square z divided by z minus a to the power three
so corresponding to z minus one cube we will have to have in the numerator term like this
so what we will do let us take a equal to one so z of n square will by taking a equal
to one is how much z square plus z divided by z minus one whole cube z minus one whole
cube so we will have so this z is to be multiplied to z plus one in order to get z square plus
one so we get here c times z plus one so that when we multiply by z we get c times z square
plus z divided by z minus one whole cube whose inverse z transform will be n square into
a to the power n but a is equal to one so n square so this is how much we this is how
we write the corresp the term corresponding to z minus one whole cube then the term corresponding
to z minus one to the power four we have to write
now let us recall the z transform of n cube which we had found earlier so this
is what we have so corresponding to z minus one to the power four in the numerator we
must have z cube plus four z square plus z one z will come from uz by z so we need to
have z square plus four z plus one so we write here d times z square plus four z plus one
so this how we write uz by z and now later our aim will be to find out the values of
abcd so which we can do easily so we shall write so this is uz by z uz by z come is equal
to let me i think i have omitted that ok so this is equal to a times z minus one
to the power three plus b times z minus one to the power two plus c times z plus one into
z minus one and then we have d times z square plus four z plus one
now uz was equal to now uz by z we have seen uz by z is equal to three over z minus one
to the power four and then we have five over z minus one cube and then we have u naught
times z minus two upon z minus one to the power two plus u one upon z minus one whole
square so this is uz by z is equal to this so
let us equate the both equate the both their side this is this is divided by z
minus one to the power four here also we can write z minus one to the power four so this
is z minus one to the power four so three plus five times z minus one plus u naught
times z minus two into z minus one whole square plus u one times z minus one whole square
so this way this is equal to this so we have a times z minus one whole cube plus b times
z minus one whole square plus c times z plus one into z minus one plus d times this equal
to the numerator here so then plus this will be equal to u one times z minus one whole
square now from here if when we solve this we can get the value we get the from the equations
that we get we shall be able to determine the values of abcd let us see if we
put ok we can easily see that the a is equal to u naught how we get a is equal to
u naught here by taking z equal to if we ok z q the term z q ok so here we see that
ok let us go equate the coefficient of z q both sides so equating
here the coefficient of z q is a here the co efficient of z q s u naught so we get a
equal to u naught and we will similarly we can equate the coefficient of z square
the coefficient of z the constants we will get the questions and then we can see that
a comes out to be u naught b comes out to be u one minus u naught minus three by two
and c comes out be equal to one d is equal to half so a is equal to u naught b is equal
to u one minus u naught minus three by two c is equal to one d is equal to half so we
put this values here and then take the inverse z transform so we shall have so using this
values of a b c d we then get u z equal to a times z over z minus one so u naught
times z over z minus one and then b is u one minus u naught minus three by two in to
z over z minus one whole square and we will get c is equal to one so z plus one so
z square plus z divided by z minus one whole cube and then we will get d times d is equal
to half so d equal to half will give us a one by two times z cube plus four z square
plus z divided by z minus one to the power four
now take the inverse z transform so z inverse u z which is equal to u one sequence will
be equal to u naught in to one to the power n u one minus u naught minus three by two
and then we get here z transform of n into a to the power n is a z over z minus
a whole square so taking inverse transform take n is a equal to one and then take the
inverse transform the inverse transform of z over z minus one whole square will be n
so we get n here then we get here z transform of n square so the z transform of n square
so we get n square here and then we get half of n cube so if we simplify this we will get
u n equal to c naught plus c one and plus half of n in to n minus one n plus three we
can easily check that where we are chosen c naught equal to u naught and c one equal
to u one minus u naught so this is how we will find the solution of the difference
equation given in an example three i think with those examples the how to deal with the
solution of a difference equation must be very clearly to you
thank you very much for your kind attention