WEBVTT
Kind: captions
Language: en
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hello friends welcome to my lecture on applications
of z transforms we have seen that the z
00:00:25.690 --> 00:00:31.320
transform of the sequence u n equal to one
is z over z minus one when mod of z is greater
00:00:31.320 --> 00:00:36.860
than one and the z transform of the sequence
un equal to n is z over z minus one whole
00:00:36.860 --> 00:00:44.190
square when mod of z is greater than one
and z of z transform of the sequence un equal
00:00:44.190 --> 00:00:49.329
to n square is z square plus z over z minus
one whole cube and mod of z is greater than
00:00:49.329 --> 00:00:56.000
one now we can also find the z transform of
the sequence un equal to n cube by using the
00:00:56.000 --> 00:01:05.070
recurrence relation z transform of n
to the power p z transform of this equals
00:01:05.070 --> 00:01:14.909
n to the power p equal to minus z d over dz
of z transform of n to the power p minus one
00:01:14.909 --> 00:01:22.270
where p is a positive integer so here z transform
of n cube where we have taken n equal to three
00:01:22.270 --> 00:01:28.140
here so z transform of a n cube is minus z
d over dz dz of z transform of n square
00:01:28.140 --> 00:01:34.229
so we write the z transform of a n square
that is z square plus z over z minus one cube
00:01:34.229 --> 00:01:40.159
here and then differentiate with respect to
z and multiply by z we arrive at z cube
00:01:40.159 --> 00:01:45.120
plus four z square plus z over z minus one
to the power four when mod of z is greater
00:01:45.120 --> 00:01:52.479
than one this z transform we will need
when we solve difference equations so we
00:01:52.479 --> 00:01:59.690
have to drive this z transform of n cube now
let us what we i have done is that we are
00:01:59.690 --> 00:02:04.820
taking z transforms inverse z transform of
some expression functions where we shall see
00:02:04.820 --> 00:02:11.250
that the when we write the partial fractions
of uz over z we have to write uz over z the
00:02:11.250 --> 00:02:20.460
partial fractions of uz over z according
to the the inverse z formulas that is
00:02:20.460 --> 00:02:26.209
we we will have to use while inverting will
have to use the z transforms of special sequences
00:02:26.209 --> 00:02:30.669
so we will have to write in in the uz over
z also in that form
00:02:30.669 --> 00:02:36.529
so for example here when we say z the inverse
of z cube over twenty z over z minus two whole
00:02:36.529 --> 00:02:44.239
cube equal to z minus four let us see how
we write the uz by z into partial fractions
00:02:44.239 --> 00:02:54.620
so what we see do is let us say that uz
equal to z cube minus twenty z divided by
00:02:54.620 --> 00:03:08.309
z minus two whole cube into z minus four
then uz by z by in the partial fractions method
00:03:08.309 --> 00:03:17.230
we have to write uz by z into partial fractions
so we write a over z minus four b over z minus
00:03:17.230 --> 00:03:26.150
two c over z minus two whole square see we
have z minus four power one so we will ave
00:03:26.150 --> 00:03:31.250
one fract partial fraction corresponding
to the factor z minus four but correspond
00:03:31.250 --> 00:03:36.780
to z minus two whole cube we will have three
fractions one with z minus two with the
00:03:36.780 --> 00:03:42.239
other one z minus two whole square and the
third one with z minus two whole cube
00:03:42.239 --> 00:03:49.221
now let us see how we will have to write these
partial fractions while inverting we will
00:03:49.221 --> 00:03:56.049
write uz uz will be a times z over z minus
four so while inverting the inverse z transform
00:03:56.049 --> 00:04:01.650
of z over z minus four will be four to the
power n so we will be able to invert and here
00:04:01.650 --> 00:04:06.430
we will get z over z minus two inverse z transform
of z over z minus two will be two to the power
00:04:06.430 --> 00:04:12.459
n so that is also possible to invert and here
we will have z over z minus two whole square
00:04:12.459 --> 00:04:23.090
let us recall that z transform of n into
a to the power n is az upon z minus a whole
00:04:23.090 --> 00:04:31.860
square when mod of z is greater than mod of
a so here if you take because here if z minus
00:04:31.860 --> 00:04:37.530
two whole square so you take a equal to two
so z transform of n into two to the power
00:04:37.530 --> 00:04:45.870
n will be equal to two z upon z minus two
whole square and so we can say that z transform
00:04:45.870 --> 00:04:53.620
of n two to the power n divided by two will
be z upon z minus two whole square
00:04:53.620 --> 00:04:59.460
so inverse z transform of z over z minus two
whole square will be n times two to the power
00:04:59.460 --> 00:05:04.900
n by two so we will have to we will be able
to invert this term also z over z minus two
00:05:04.900 --> 00:05:11.070
whole square but here when we come to z minus
two whole cube let us recall the formula
00:05:11.070 --> 00:05:21.350
for n square a to the power n we have z
transform of n square a to the power n this
00:05:21.350 --> 00:05:33.410
is equal to a z square plus a square z divided
by z minus a whole cube so this is actually
00:05:33.410 --> 00:05:48.180
z times az plus a square divided by z minus
a whole cube now if i take a equal to two
00:05:48.180 --> 00:06:02.300
taking a equal to two we get z transform of
n square into two to the power n equal
00:06:02.300 --> 00:06:15.030
to z into two z plus four divided by
z minus two whole cube so z while while writing
00:06:15.030 --> 00:06:21.430
uz this z we will will get come here so here
we have z z here and then two z plus four
00:06:21.430 --> 00:06:32.130
is also needed so what we do is we write here
d times two z plus four now now when we
00:06:32.130 --> 00:06:38.930
will be able when we will when we will find
the values of abc and d this term we can also
00:06:38.930 --> 00:06:44.780
invert because this term will be z into two
z plus four divided by z minus two whole cube
00:06:44.780 --> 00:06:50.780
so its in in its inverse z transform will
be n square into two to the power n now the
00:06:50.780 --> 00:06:56.750
only thing that remains is to do here is
that we have to find the values of abcd so
00:06:56.750 --> 00:07:06.160
a can be found easily a is corresp a corresponds
to z square minus twenty because uz by
00:07:06.160 --> 00:07:16.750
z is z square minus twenty divided by z minus
two whole cube z minus two whole cube so this
00:07:16.750 --> 00:07:23.830
is when you put z equal to four so you
put z equal to four so we get sixteen minus
00:07:23.830 --> 00:07:30.460
twenty divided by four minus two that is two
whole cube so that is eight and we get here
00:07:30.460 --> 00:07:39.150
minus four by eight so we get minus half
we can then write this is let me say that
00:07:39.150 --> 00:07:48.120
we write it as z square minus twenty divided
by z minus two whole cube into z minus four
00:07:48.120 --> 00:07:54.520
so equating both sides we can also find
the value of d directly what you do
00:07:54.520 --> 00:08:03.270
is let us write from this i equality we
write z square minus twenty equal to a times
00:08:03.270 --> 00:08:18.230
z minus two whole cube plus b times z minus
two whole square into z minus four plus c
00:08:18.230 --> 00:08:34.320
times z minus two into z minus four and d
times two z plus four into z minus four
00:08:34.320 --> 00:08:38.320
now you can see that when you take z equal
to two this term is zero this term is zero
00:08:38.320 --> 00:08:44.560
this term is zero and this term is non zero
so we will get the value of d so putting z
00:08:44.560 --> 00:08:57.399
equal to two we have two square means four
minus twenty so we get minus sixteen equal
00:08:57.399 --> 00:09:05.879
to d times two into two that is four four
plus four is eight and then we get two minus
00:09:05.879 --> 00:09:12.709
four that is minus two so minus sixteen d
equal to minus sixteen this gives you d equal
00:09:12.709 --> 00:09:19.120
to one and the next thing that we can do is
let us equate the coefficients of various
00:09:19.120 --> 00:09:25.310
powers of z so equating the coefficient of
z cube on both sides here we have term
00:09:25.310 --> 00:09:26.950
in z cube is zero
00:09:26.950 --> 00:09:32.230
so the coefficient of z cube is zero there
here the coefficient of z cube is a plus b
00:09:32.230 --> 00:09:54.579
so equating the coefficient of z cube both
sides we get a plus b equal to zero now a
00:09:54.579 --> 00:10:05.790
we have already found a is equal to minus
half so b is equal to half then we can
00:10:05.790 --> 00:10:10.809
equate the constants on both sides which means
that we can put z equal to zero in this equation
00:10:10.809 --> 00:10:22.600
so putting z equal to zero we get
let me call this equation as equation number
00:10:22.600 --> 00:10:29.730
one so putting z equal to zero in one what
we get is i will write here so minus twenty
00:10:29.730 --> 00:10:37.889
equal to a times minus two whole cube so
minus eight into a and then we get minus two
00:10:37.889 --> 00:10:42.600
whole square which is four four into minus
four that is minus sixteen
00:10:42.600 --> 00:10:50.279
so minus sixteen b and then we will get c
times minus two and minus four so that is
00:10:50.279 --> 00:10:59.230
eight times c and here we shall have z equal
to zero means d times four into minus four
00:10:59.230 --> 00:11:10.369
so minus sixteen d let us use the values of
abc a a d b and d so a is equal to minus half
00:11:10.369 --> 00:11:20.430
so minus eight into minus half and then we
will have the value of the d b equal to half
00:11:20.430 --> 00:11:28.509
and then we have to find the value of c minus
sixteen d is equal to one so how much is this
00:11:28.509 --> 00:11:42.059
or minus twenty is equal to now this is four
eight by two is four here we get minus eight
00:11:42.059 --> 00:11:48.769
here we get eight c minus sixteen
00:11:48.769 --> 00:11:56.209
so this is four minus twenty four that is
minus twenty so c is equal to zero now so
00:11:56.209 --> 00:12:10.120
we get c equal to zero and thus we have the
following fractions so hence uz by z a is
00:12:10.120 --> 00:12:19.790
equal to a is equal to minus half so minus
one by two z one over z minus four then we
00:12:19.790 --> 00:12:27.929
have b equal to half so we get one by two
z minus two and c is zero d is one so we get
00:12:27.929 --> 00:12:38.290
two z plus four divided by z minus two whole
cube now let us multiply by z so r uz equal
00:12:38.290 --> 00:12:52.860
to minus half z over z minus four plus half
z over z minus two and then two z square plus
00:12:52.860 --> 00:13:01.649
four z divided by z minus two whole cube now
lets take inverse z transform both sides so
00:13:01.649 --> 00:13:09.110
taking inverse z transform we get un equal
to minus one by two four to the power n then
00:13:09.110 --> 00:13:20.910
one by two two to the power n and this is
n square into two to the power n and for these
00:13:20.910 --> 00:13:25.999
transforms ok mod of z is greater than two
but here mod of z is greater than four so
00:13:25.999 --> 00:13:32.940
we will take mod of z greater than for the
common region so this is the inverse z transform
00:13:32.940 --> 00:13:40.300
of z cube minus twenty z over z minus two
to the power three over z minus four
00:13:40.300 --> 00:13:46.179
so while writing the partial fraction of uz
by z we have to write them in such a way that
00:13:46.179 --> 00:13:55.279
while inverting we get the standard
sequences so standard results we have we will
00:13:55.279 --> 00:14:02.249
have to use now the again next in the next
example we have taken this difference equation
00:14:02.249 --> 00:14:06.579
and we have to find the response of the system
that is we have to find the sequence yn
00:14:06.579 --> 00:14:13.489
such that y naught equal to zero y one
equal to one and here we are given that un
00:14:13.489 --> 00:14:18.540
is equal to unit step sequence un equal to
one for n equal to zero one two three and
00:14:18.540 --> 00:14:24.670
so on and which is nothing but the unit
step sequence and for the unit step sequence
00:14:24.670 --> 00:14:30.850
if you remember we have seen that the z transform
is z over z minus one where mod of z is greater
00:14:30.850 --> 00:14:38.189
than one so here we shall see that the sequence
yn is equal to one by two minus two
00:14:38.189 --> 00:14:41.569
times two to the power n plus three by two
three to the power n
00:14:41.569 --> 00:14:48.449
let us see how we get the solution of this
it is very simple difference equation so
00:14:48.449 --> 00:14:54.350
as usual let us take the z transform of the
given difference equation yn plus z transform
00:14:54.350 --> 00:15:04.459
of yn plus two and then we have minus five
times z transform of yn plus one then we have
00:15:04.459 --> 00:15:14.449
six times z transform of the sequence yn equal
to z six times z transform of this sequence
00:15:14.449 --> 00:15:21.170
yn and then we have the z transform of the
unit step sequence un z transform yn plus
00:15:21.170 --> 00:15:31.759
two we can find by the shifting property
so this is z square times yz where yz is the
00:15:31.759 --> 00:15:40.851
transform of the sequence yn minus y naught
minus y one by z minus five times z of yn
00:15:40.851 --> 00:15:50.339
plus one is z times yz minus y naught plus
six times z transform yn we have taken as
00:15:50.339 --> 00:15:57.559
yz equal to unit z transform of the unit step
sequences z over z minus one
00:15:57.559 --> 00:16:03.199
so here we have taken mod of z greater than
mod of one now let us use the values of y
00:16:03.199 --> 00:16:14.779
naught and y one y naught is zero y one is
one so z square times yz minus one by z because
00:16:14.779 --> 00:16:26.740
y naught is zero y one is one minus five times
z into yz because y naught is zero plus
00:16:26.740 --> 00:16:35.350
six yz equal to z over z minus one this
is how much let us collect the coefficient
00:16:35.350 --> 00:16:48.730
of yz here so z square minus five z plus six
times yz equal to z over z minus one plus
00:16:48.730 --> 00:16:58.850
z which is z square divided by z minus
one now the factors here are z minus two and
00:16:58.850 --> 00:17:08.930
z minus three so we can write yz equal to
z square divided by z minus one z minus two
00:17:08.930 --> 00:17:13.380
z minus three ok
00:17:13.380 --> 00:17:25.810
so as usual let us write yz by z so then yz
by z will be z over z minus one z minus two
00:17:25.810 --> 00:17:33.780
z minus three which we shall write as a over
z minus one because z minus one z minus two
00:17:33.780 --> 00:17:39.750
z minus three all occur in power one so for
each one we will have one fraction so b over
00:17:39.750 --> 00:17:48.370
z minus two c over z minus three and we can
see that while writing yz each term here
00:17:48.370 --> 00:17:52.410
will be multiplied by z so and we will be
able to determine the inverse z transform
00:17:52.410 --> 00:17:57.700
of z over z minus one z over z minus two z
over z minus three and so we just have to
00:17:57.700 --> 00:18:06.440
find values of abc so a you can see a is equal
to z over z minus two z minus three at
00:18:06.440 --> 00:18:17.000
z equal to one which comes out to be one over
one minus two one minus three so this is minus
00:18:17.000 --> 00:18:24.150
two this s minus one so we have one over two
and similarly b will be z over z minus one
00:18:24.150 --> 00:18:33.740
z minus three at z equal to two so we get
two over two minus one is one two minus
00:18:33.740 --> 00:18:41.680
three is minus one so we get minus two and
then c similarly will be z over z minus one
00:18:41.680 --> 00:18:48.960
z minus two evaluated at z equal to three
so we shall have three over three minus one
00:18:48.960 --> 00:19:00.900
is two three minus two is one so we get three
by two and thus thus yz will be equal to let
00:19:00.900 --> 00:19:10.140
us multiply by z the value of a we have found
as half so half z over z minus one the value
00:19:10.140 --> 00:19:15.380
of b we have found to be minus two so minus
two times z over z minus two and then the
00:19:15.380 --> 00:19:22.980
value of c we have found three by two so three
by two z over z minus three so inverse z transform
00:19:22.980 --> 00:19:31.920
of yz which we will be equal to the sequence
yn this is equal to half one to the power
00:19:31.920 --> 00:19:42.050
n minus two times two to the power n and then
we get three by two three to the power n for
00:19:42.050 --> 00:19:46.040
this mod of z greater than three for this
mod of z greater than two and for this mod
00:19:46.040 --> 00:19:47.040
of z greater than one
00:19:47.040 --> 00:19:55.740
so we have mod of z greater than three
which is the common portion so he inverse
00:19:55.740 --> 00:20:01.420
z transform is half minus two into two to
the power n plus three by two into three to
00:20:01.420 --> 00:20:09.950
the power n now let us go to this problem
where we shall see that we have to use
00:20:09.950 --> 00:20:16.560
z transform of n cube which we have found
earlier ok so lets consider the difference
00:20:16.560 --> 00:20:35.670
equation un plus two minus two un plus one
two un plus one plus un equal to three
00:20:35.670 --> 00:20:41.180
n plus five you can see that here we are not
given any conditions on the sequence un so
00:20:41.180 --> 00:20:47.200
the u naught and u one u two which we will
occur which will occur while taking the z
00:20:47.200 --> 00:20:52.370
transform of this will be taken as some arbitrary
constants we will see that there will occur
00:20:52.370 --> 00:20:57.440
u naught and u one so u naught and un will
be taken as some constants and writing the
00:20:57.440 --> 00:21:03.570
solutions solution of this problem we shall
see that we have to choose c naught as u naught
00:21:03.570 --> 00:21:10.240
and c one as u one minus u naught to arrive
at this sequence the solution of the problem
00:21:10.240 --> 00:21:28.460
so let us take z transform of this so taking
z transform on both sides we get
00:21:28.460 --> 00:21:40.250
ok so z transform will be equal to z square
uz minus u naught minus u one by z and then
00:21:40.250 --> 00:21:49.430
minus two times z into uz minus u naught the
z transform of un plus one plus z transform
00:21:49.430 --> 00:21:57.540
of un will be uz z trans three times z transform
of n let us recall that z transform of n is
00:21:57.540 --> 00:22:04.740
equal to z over z minus one whole square where
mod of z is greater than one so this is
00:22:04.740 --> 00:22:15.520
z over z minus one whole square and z transform
of five five is a constant so we need to know
00:22:15.520 --> 00:22:22.130
the z transform of one z transform of a to
the power n was equal to z over z minus a
00:22:22.130 --> 00:22:34.401
provided mod of z is greater than a so taking
a equal to one here you have z transform
00:22:34.401 --> 00:22:41.460
of one is equal to z over z minus one so provided
mod of z is greater than one so this is
00:22:41.460 --> 00:22:55.610
z over z minus one provided mod of z is greater
than one for these to be true ok so now
00:22:55.610 --> 00:23:06.260
let us take the coefficient of uz so z square
minus two z plus one into uz and here we shall
00:23:06.260 --> 00:23:17.140
have minus u naught into z square here we
shall have u one z and here we will get two
00:23:17.140 --> 00:23:28.290
u naught z equal to three times z over z minus
one whole square plus five times z over z
00:23:28.290 --> 00:23:37.000
minus one ok so let us collect let us find
the value of uz so uz will be equal to
00:23:37.000 --> 00:23:41.010
ok so uz will be equal to this is z minus
one whole square
00:23:41.010 --> 00:23:47.380
so let me write it as z minus one whole square
into uz and let us take these terms with the
00:23:47.380 --> 00:23:57.080
other side we have three z over z minus one
whole square plus five times z over z minus
00:23:57.080 --> 00:24:08.880
one and then we have here u naught times z
square minus two z u naught times these
00:24:08.880 --> 00:24:12.840
terms when go to the other side will give
you u naught times z square by two z plus
00:24:12.840 --> 00:24:23.380
u one z plus divide by z minus one whole square
to get the uz so uz is equal to three z
00:24:23.380 --> 00:24:34.030
over z minus one to the power four plus five
z over z minus one to the power three then
00:24:34.030 --> 00:24:44.870
u naught times z minus square minus two z
divided by z minus one whole square and then
00:24:44.870 --> 00:24:54.390
u one z over z minus one whole square now
let us write uz by z from here and then we
00:24:54.390 --> 00:25:02.000
shall write the fractions of uz by z in such
a way that we can easily get the inverse z
00:25:02.000 --> 00:25:14.920
transform so uz by z is equal to three by
z minus one to the power four five upon z
00:25:14.920 --> 00:25:24.690
minus one whole cube and then we have here
u naught into z minus two divided by z minus
00:25:24.690 --> 00:25:34.960
one square then we have u one upon z minus
one whole square
00:25:34.960 --> 00:25:40.570
now we can see that in the denominator we
have z minus one power of z minus one as two
00:25:40.570 --> 00:25:46.970
three four so when we write the partial fractions
corresponding to this uz by z we have to
00:25:46.970 --> 00:25:54.130
write four fractions corresponding to z
minus one to the power four so this we shall
00:25:54.130 --> 00:26:07.660
write as a over z minus one b over z minus
two sorry a over z minus two one b over z
00:26:07.660 --> 00:26:16.470
minus one square now so far when you will
write uz you will multiply this y a and b
00:26:16.470 --> 00:26:22.690
y z so z over z minus one can be inverted
z over z minus one whole square can also be
00:26:22.690 --> 00:26:29.000
inverted but here when you will multiply by
z the term corresponding to z minus one
00:26:29.000 --> 00:26:38.930
to the power three so let us recall that
z transform of n square a to the power n this
00:26:38.930 --> 00:26:47.640
was equal to a z square plus a square z
divided by z minus a to the power three
00:26:47.640 --> 00:26:54.890
so corresponding to z minus one cube we will
have to have in the numerator term like this
00:26:54.890 --> 00:27:03.590
so what we will do let us take a equal to
one so z of n square will by taking a equal
00:27:03.590 --> 00:27:18.450
to one is how much z square plus z divided
by z minus one whole cube z minus one whole
00:27:18.450 --> 00:27:26.920
cube so we will have so this z is to be multiplied
to z plus one in order to get z square plus
00:27:26.920 --> 00:27:36.090
one so we get here c times z plus one so that
when we multiply by z we get c times z square
00:27:36.090 --> 00:27:42.800
plus z divided by z minus one whole cube whose
inverse z transform will be n square into
00:27:42.800 --> 00:27:47.180
a to the power n but a is equal to one so
n square so this is how much we this is how
00:27:47.180 --> 00:27:51.930
we write the corresp the term corresponding
to z minus one whole cube then the term corresponding
00:27:51.930 --> 00:27:55.320
to z minus one to the power four we have to
write
00:27:55.320 --> 00:28:01.680
now let us recall the z transform of n
cube which we had found earlier so this
00:28:01.680 --> 00:28:06.340
is what we have so corresponding to z minus
one to the power four in the numerator we
00:28:06.340 --> 00:28:12.880
must have z cube plus four z square plus z
one z will come from uz by z so we need to
00:28:12.880 --> 00:28:23.600
have z square plus four z plus one so we write
here d times z square plus four z plus one
00:28:23.600 --> 00:28:29.680
so this how we write uz by z and now later
our aim will be to find out the values of
00:28:29.680 --> 00:28:39.260
abcd so which we can do easily so we shall
write so this is uz by z uz by z come is equal
00:28:39.260 --> 00:28:47.910
to let me i think i have omitted that ok
so this is equal to a times z minus one
00:28:47.910 --> 00:28:57.940
to the power three plus b times z minus one
to the power two plus c times z plus one into
00:28:57.940 --> 00:29:08.010
z minus one and then we have d times z square
plus four z plus one
00:29:08.010 --> 00:29:17.850
now uz was equal to now uz by z we have seen
uz by z is equal to three over z minus one
00:29:17.850 --> 00:29:29.280
to the power four and then we have five over
z minus one cube and then we have u naught
00:29:29.280 --> 00:29:43.270
times z minus two upon z minus one to the
power two plus u one upon z minus one whole
00:29:43.270 --> 00:29:53.840
square
so this is uz by z is equal to this so
00:29:53.840 --> 00:30:05.350
let us equate the both equate the both their
side this is this is divided by z
00:30:05.350 --> 00:30:09.700
minus one to the power four here also we can
write z minus one to the power four so this
00:30:09.700 --> 00:30:19.740
is z minus one to the power four so three
plus five times z minus one plus u naught
00:30:19.740 --> 00:30:29.980
times z minus two into z minus one whole square
plus u one times z minus one whole square
00:30:29.980 --> 00:30:36.110
so this way this is equal to this so we have
a times z minus one whole cube plus b times
00:30:36.110 --> 00:30:41.200
z minus one whole square plus c times z plus
one into z minus one plus d times this equal
00:30:41.200 --> 00:31:30.280
to the numerator here so then plus this
will be equal to u one times z minus one whole
00:31:30.280 --> 00:31:36.990
square now from here if when we solve this
we can get the value we get the from the equations
00:31:36.990 --> 00:31:48.290
that we get we shall be able to determine
the values of abcd let us see if we
00:31:48.290 --> 00:31:58.340
put ok we can easily see that the a is
equal to u naught how we get a is equal to
00:31:58.340 --> 00:32:11.680
u naught here by taking z equal to if
we ok z q the term z q ok so here we see that
00:32:11.680 --> 00:32:31.460
ok let us go equate the coefficient of z q
both sides so equating
00:32:31.460 --> 00:32:36.140
here the coefficient of z q is a here the
co efficient of z q s u naught so we get a
00:32:36.140 --> 00:32:44.160
equal to u naught and we will similarly
we can equate the coefficient of z square
00:32:44.160 --> 00:32:49.850
the coefficient of z the constants we will
get the questions and then we can see that
00:32:49.850 --> 00:32:59.350
a comes out to be u naught b comes out to
be u one minus u naught minus three by two
00:32:59.350 --> 00:33:07.890
and c comes out be equal to one d is equal
to half so a is equal to u naught b is equal
00:33:07.890 --> 00:33:12.720
to u one minus u naught minus three by two
c is equal to one d is equal to half so we
00:33:12.720 --> 00:33:20.620
put this values here and then take the inverse
z transform so we shall have so using this
00:33:20.620 --> 00:33:31.890
values of a b c d we then get u z equal to
a times z over z minus one so u naught
00:33:31.890 --> 00:33:41.570
times z over z minus one and then b is u one
minus u naught minus three by two in to
00:33:41.570 --> 00:33:50.330
z over z minus one whole square and we
will get c is equal to one so z plus one so
00:33:50.330 --> 00:33:58.900
z square plus z divided by z minus one whole
cube and then we will get d times d is equal
00:33:58.900 --> 00:34:06.830
to half so d equal to half will give us a
one by two times z cube plus four z square
00:34:06.830 --> 00:34:12.389
plus z divided by z minus one to the power
four
00:34:12.389 --> 00:34:20.940
now take the inverse z transform so z inverse
u z which is equal to u one sequence will
00:34:20.940 --> 00:34:30.530
be equal to u naught in to one to the power
n u one minus u naught minus three by two
00:34:30.530 --> 00:34:41.649
and then we get here z transform of
n into a to the power n is a z over z minus
00:34:41.649 --> 00:34:47.730
a whole square so taking inverse transform
take n is a equal to one and then take the
00:34:47.730 --> 00:34:52.450
inverse transform the inverse transform of
z over z minus one whole square will be n
00:34:52.450 --> 00:34:59.560
so we get n here then we get here z transform
of n square so the z transform of n square
00:34:59.560 --> 00:35:09.950
so we get n square here and then we get half
of n cube so if we simplify this we will get
00:35:09.950 --> 00:35:16.970
u n equal to c naught plus c one and plus
half of n in to n minus one n plus three we
00:35:16.970 --> 00:35:21.610
can easily check that where we are chosen
c naught equal to u naught and c one equal
00:35:21.610 --> 00:35:29.190
to u one minus u naught so this is how
we will find the solution of the difference
00:35:29.190 --> 00:35:36.900
equation given in an example three i think
with those examples the how to deal with the
00:35:36.900 --> 00:35:40.450
solution of a difference equation must be
very clearly to you
00:35:40.450 --> 00:35:42.940
thank you very much for your kind attention