ok so now i consider a sequence which is right
sided 
that is it starts at any n one ok and any
n one greater than equal to zero and goes
up to infinity x of n ok to start with n one
greater than equal to zero then we will consider
the case when we starts at a finite negative
point then goes to the infinity but that later
so first is as before x z we will talk take
from n one to infinity only because values
to the left of n one the sample values of
x n as zero so those products are zero no
point in considering then as you before you
take the mod and sum mod of a summation less
than equal to sum of mod it continues mod
of x n and mod of z to the power of minus
n will give raise to r to the power minus
n alright
now suppose suppose this r h s r h s means
right hand side this quantity is is finite
for some r r is nothing but mod of z magnitude
part ok then for some r equal to say suppose
r naught ok if you put r naught here in the
summation this summation is finite then for
any r greater than r naught what happens to
the summation it is r to the power minus n
ok in this summation you see r to the power
r is greater than r naught which means r to
the power minus n is less than r naught to
the power minus n ok r is greater than r naught
so r r to the power minus n means what one
by r to the power n that will be less than
one by r zero to the power n is isn't very
simple four is greater than three so one by
four less than one by three any positive power
of four ok because the powers are positive
powers small n is starting from capital n
one to infinity this all positive numbers
so the positive power of one by r that is
one by four here will be less than positive
power of one by three that is one by r zero
ok so this means actually one by r to the
power n n is positive because this summation
considers n in the positive zone right that
means this summation is for sure any small
nth entry mod x n was present here also here
also but that time i assume that this summation
is finite for r equal to r naught that is
if you have r naught r naught to the power
minus n 
this is finite that what i have assumed that
it exist
now in this summation consider any sample
nth mod x n present here mod x n present here
but that time i was multiplying by r zero
to the power minus n and i am now multiplying
by a smaller number r to the power minus n
so in the previous summation this positive
number this has mod x n multiplied by one
positive number r zero to the power minus
n here again the same particular number mod
x n multiplied by a smaller positive number
r to the power minus n you see r to the power
minus n less so obviously this is lesser than
this so this whole sum is less than this summation
if this summation is finite this is also finite
this summation is less than this summation
because for mod x n mod x n common r to the
power minus n is a positive number less than
r to r zero to the power minus n as you have
seen here because r is greater than r naught
i am taking r greater than n naught so r to
the power minus n is less than r to the their
minus n
which meabs this will be finite therefore
it gives what if this convergence for some
z with magnitude r naught then there is suppose
this much is your n naught and this is z and
this much is this much is your r naught you
draw a circle so these are the radius r naught
therefore z outside the circle magnitude will
be r mod z r will be greater than r naught
right for any z here say z here now ok this
magnitude is r is greater than r naught so
for that r this summation is finite if r is
greater than r naught this summation is finite
ok therefore if z is here then for that z
z transform will exist it means that entire
region outside the circle up to infinity entire
z outside the circle will be your region of
convergence
now i arbitrarily started saying that suppose
say this is finite that is say z transform
exist here for some r equal to r naught now
you start exploring if i go below r naught
still does it converge if yes still i make
it lesser finally a point it will come when
you will be in the bordering point if i make
r naught lesser than these then summation
will not converge so i will stop here and
come to this discussion the entire region
outside the circle will remain region of convergence
so if it is purely right sided sequence starting
from a index n one which is either zero or
positive r o c is region of convergence that
is r o c is outside the circle up to infinity
no restriction it can go up to z equal to
infinity
now i have taken more general that suppose
the sequence is right sided but it starts
at a finite negative value here n one n one
is a negative number dot dot dot dot and then
this that is 
and then to the left all zeros then z transform
as before n will start at n one negative value
up to infinity and this summation i can write
as a summation of two part one will go from
n one the negative value up to minus one here
and other will go from n equal to zero up
to infinity from here to here up to infinity
then mod x z is mod for mod of a summation
is less than equal to summation of the mod
so this is less than equal to summation of
mod of this 
this part is a finite duration sequence you
have to only consider this from some negative
index n one to another negative minus one
ok so this will always exist hm this will
always exist except for z equal to infinity
ok because minus n and n is taking negative
value so this will be plus n ok so when you
write z equal to you know i mean basically
if you make it further it becomes mod of summation
less then equal to summation of mod as purely
as before and summation of mod means product
of the mod ah mod of the product is same as
product of two mod mod x n and mod of z to
the power minus one will be raise to this
and on this side again mod of a summation
less than equal to summation of mod the summation
of the product mod on the product mod on the
product is product of the mods where mod of
z to the power minus n is r to the product
minus n this alright
this summation is always finite we have seen
ok is finite duration it's like a finite duration
sequence from n one to minus one only those
finite number or points in this summation
otherwise no point so this is will always
exist for any r except for r equal to infinity
because n is taking negative values it here
so here it is positive r infinity means it
will diverge so this is converging everywhere
except at r infinity and this summation is
for a sequence that starts starts at n equal
to zero up to infinity so its a purely right
sided sequence so its convergence will be
outside a circle up to infinity ok if i now
for z transform to exist this should be positive
this should be finite this should be finite
because this is a positive number this is
a positive number so they cannot cancel each
other both have to be finite is not that this
is infinity this is infinity but one is plus
one is minus infinity minus infinity is zero
that doesn't happen this is positive this
is positive and this non negative non negative
so this must be finite this must be finite
but this summation is finite means this is
for a right hand sided sequence because n
is from zero to infinity that is this part
this sequence where front side is zeros so
we have also considered this case for this
kind of ah z transform that is right sided
sequences z transform exist the region of
convergence include i means is what what outside
a circle ok which we have like this outside
the circle up to infinity and if this part
is present infinity is not allowed so overall
yes outside a circle but excluding z equal
to infinity r equal to infinity or z equal
to infinity
in this case it starts at n one to the right
of origin or at origin not at negative index
then it converges outside circle outside some
circle goes up to infinity ok this for the
right sided sequences it is similar manner
we can consider a left sided sequence 
ok that is suppose to start with assume that
i am starting at some index say n one and
then i am going it's like this and up to infinity
and then all zeros and suppose n one is less
than zero in this case z transform n will
be from n one which is a negative number up
to minus infinity so you take mod x z mod
of a summation less than equal to all those
things 
suppose r h s that is this summation right
hand side is finite for sum r equal to r naught
remember here n has a minus sign but n is
taking negative indices capital n one is negative
and further you go to the negative so negative
negative positive so basically positive powers
of r are coming ok now if this is true that
means i am assuming n equal to infi[nity]
this is to minus infinity mod x n r naught
to the power minus n if i replace r equal
to i r naught this is finite
now if i take any r less than r naught then
how about this summation everything else is
same mod x n mod x n n one two capital n one
two minus infinity but here it was r zero
to the power n and r to the power minus n
you see minus n is positive but n is taking
n number so r zero to the power some positive
power r to the for the same positive power
therefore if r is less than r naught then
this is also less than this r to the power
minus n for negative n this is also less than
because if n is taken it means it is positive
now minus minus plus minus minus plus if r
is less than r naught means r to the power
of minus n for negative effect will be less
than r r naught to the power minus n
so mod x n here multiplied by r naught to
the power minus n and here multiplied by a
number r to the power minus n but this is
lesser than that so that means summation is
lesser than this summation so if this finite
this is also finite ok so this means if this
converges for any r naught any r naught you
draw a circle same radius r naught for any
z on this circle you have got magnitude r
naught and this is satisfied means z transform
is existing on the circle in that case you
consider any point z inside say here the corresponding
r is a magnitude that is less than r naught
if z is here for that z obviously we we z
transform will exist because this will be
finite because this is finite r is less than
r naught this much is r this much is r naught
now we will start i will start two r naught
arbitrarily now i will start expanding r naught
if i expand the circle make it bigger bigger
finally a point will come that i can't expand
anymore if i expand any more it will not converge
so suppose there is a limiting case ok so
r o c will be full region inside the circle
outside it will diverge ok or it is because
i am considering negative power in this case
so positive powers of r so r is zero doesn't
matter zero to the power positive r no problem
now i consider a sequence which will be left
sided ok but will start at a positive index
that is as a zero zero zero zero in this case
this summation you write as summation of two
component n equal to n one to zero this is
n one n one minus one up to zero you can either
up to zero or may be one or maybe zero ok
zero will be better n one to zero and then
n from to minus infinity therefore mod of
x z will be this which is less than equal
to mod of this plus mod of this mod of this
again less than equal to mod of a summation
less than equal to summation of mod of this
and r to the power minus n and again you get
the same way
if this is finite this is finite this finite
one of this should be infinity and therefore
positive so it cannot happen that this is
infinity this is infinity but infinity minus
infinity is zero ok that cannot happen positive
positive positive thing so there is a positive
sign here alright it cannot be that ok this
is positive plus infinity and this is minus
infinity and they cancel it is not possible
now this is a finite sequence finite sum n
one to zero it's like a z transform ah it's
like a case of z transform with sequence that's
from zero to n one zero to n one so we have
seen z transform that is this summation will
be always existing always finite but since
small n is taking positive values here zero
or positive zero one to also positive will
be negative powers of n ok nega[tive] nega[tive]
negative powers of r therefore r equal to
zero will be ruled out so this is everywhere
converges everywhere except at r equal to
zero and this one is a purely left side sequence
from n equal to minus one to minus infinity
that is this part and from this region of
convergence is inside some circle so together
this means it will be inside some circle r
o c but this is excluded otherwise this whole
area this is excluded
and lastly the more important case of a general
sequence that is both sided for a both sided
sequence it's from minus infinity to infinity
here x z 
ok and this summation you break into two part
n equal to minus one to minus infinity so
mod x z is this which is less than equal to
mod of this plus mod of this and mod of this
is further less than equal to 
r to the power minus n n again this must be
finite this must be finite because both are
positive and there is a plus sign if both
are positive then only right hand side is
less than finite and therefore left hand side
is guaranteed to be finite
now this is corresponding to a purely right
sided sequence from zero to infinity so its
region of convergence is outside a circle
up to infinity and this is a purely left sided
sequence from minus one to minus infinity
its region of convergence is inside the circle
up to origin now suppose the two circles are
such that there is a overlapping region that
is in one case outside this this circle that
is from these up to infinity and in another
case inside a circle that is from this ok
then we have to see the overlap this part
this will be the r o c because here both will
exist but if the two circles are such that
there is no overlap one is i mean from here
it is inside the circle suppose here and from
here it is outside the circle then there is
no overlap between the two shaded region in
that case there is no r o c the sequence is
then so z transform doesn't exist ok
see region of convergence is very important
because you see i can give you the same sequence
i mean you can have 
same z transform expression suppose i give
you a to the power n u n ok what will be the
z transform and r o c ok we mean this is x
n will give raise to what x z summation this
u n is present which is from zero to infinity
and therefore a to the power n z to the power
minus n ok that means a to the power zero
that is zero one if a by z a square by z square
dot dot dot dot this infinite sum will converge
if this is less than one we all know and it
will then converge to one by one minus a by
z this is some basically they have done we
write just one minus a z inverse this are
right hand side sequence its region of convergence
should be outside a circle and what does this
give mod a by z is less than one you know
that mod of two complex numbers z one by z
two is same as mod z one by mod z two if you
don't now remember you don't understand remember
z one if you write as r one e to the power
j theta one this you write as r t two e the
power j theta two so division based r one
by r two into e to the power j theta one minus
theta two mod of that will be r one by r two
into mod of e to the power j theta one minus
theta two but e to the power j theta one minus
theta two has mod one so it will be r one
by r two and r one is this r two is this ok
now mod of a by z less than one means mod
of a less than mod of z which means r o c
is what there is a circle of radius a and
mod z greater than mod a because radius is
mod a a the whole region outside this ok
because mod z greater than mod a so any z
here means mod will be greater than mod a
that is the r o c i consider and into the
r o c the z transform is this i consider a
defined sequence now a to the power minus
n u minus n minus one what is this sequence
u minus n minus one so at n is equal to minus
one into the minus minus plus so plus one
minus one zero ok so if the sequence is like
this at n equal to zero it is u minus one
value is zero so zero at n equal to two n
equal to one minus one minus one u minus two
sorry at n equal to yeah n equal to one u
minus two u minus two is zero n equal to two
u minus three zero this at one equal to minus
one minus minus plus one minus one zero u
zero is one a to the power minus one n is
minus one we have got a that then at there
is minus one then at minus two a to the power
two a square ok because minus two means plus
two minus one is plus one u plus one is one
this way this side will be a the power three
dot dot dot dot
if i take the z transfer of that this sequence
summation will be from minus infinity or minus
one to minus infinity a to the power minus
n that is n minus one means a to the power
plus one n minus two means a to the power
plus two ok z to the power minus n alright
and what will be the summation it is like
you know one plus 
one by a z plus one by a z square and dot
dot dot dot dot
if mod of this is less than one then you have
to converge this but mod of this less than
one means mod of one divided by mod of z a
z and mod of a z is mod of a into mod of z
so if you simplify it will be this ok hm i
change the problem a little to make it more
convenient i just make a little change here
this part is fine this is also fine but and
then it will give raise to if this happens
one by one minus one by a z ok this fine but
i will just change the problem a little bit
instead of a to the power minus n u minus
n minus one suppose i consider a to the power
n in that case the sequence will be form here
only but n is minus one so it will be a to
the power minus one instead of a then a to
the power minus two and dot dot dot dot in
this case z transform will be the same way
a to the power n ok that means a to the power
minus one so z by a then z square by here
there is a mistake i was making sorry here
n was negative right so it cannot be one by
a z a to the power plus it was a to the power
plus one because n is minus one so a is equal
to plus one z to the plus one so it will be
one plus a z then a z square a z cube dot
dot dot dot so this is nothing mod a z will
be less than one it means mod z should be
greater than one by mod a ok mod a z should
be less than one then only it converges and
this will be one by one minus this one i am
sorry this because because n is taking negative
value so a to the power minus minus one so
a to the power one it is minus one z to the
power minus minus one so that a z plus a z
square a z cube dot dot dot dot so you will
take a z common so there will be a z here
also ok alright because a small mistake
here it is z by a plus z square by a square
dot dot dot dot if mod z by a less than one
so it means mod z less than mod a then this
converge this so if you take z by a common
it will be z by a divide by one minus z by
a it is common so it is one plus z by a plus
z square by a square dot dot dot and under
this condit[ion]condition will be this if
you divide numerator denominator by this quantity
it will be one by one minus sorry one by a
by z a by z that is a z inverse minus one
ok if i further make a change if i put a minus
here minus there is sequence is minus a to
the power n u minus a minus one to the minus
sign there is a minus sign here equal to there
will be a minus sign here and if we put the
minus sign inside it will be one by same expression
as this so same one i mean two different sequences
see that was our right sided sequence ok that
was the right sided sequence a to the power
n u n has got these expression and now this
has got a left sided sequence this one that
was these these these both of them have this
form same z transform but what does it differ
in one case it was r o c was mod z greater
than mod a that is you draw a circle of radius
mod a is greater than that and in this case
it is less than mod a that is inside this
ok so that is the difference you see you cannot
for two different sequences you can still
have the same z transfer expression but what
they will differ is from r o c's so that is
all for today we will continue further in
the next class
thank you very much
