00:01
In this question we need to find out the z transform of the following function.
00:05
So the first function is given to be n plus two times u of x.
00:11
Okay, so how we are going to do this? so first of all, we know that x of n is equal to given by n ux plus two times ux.
00:23
Okay, and also we know that x of z, z transform is given by my.
00:31
Minus z d over d z of z over z minus one, correct? plus two times z over z minus one.
00:40
So x of z will become what minus z times? this becomes z minus one minus z divided by z minus one square, applying the cosine rule of derivative.
00:53
And this is 2 z divided by z minus one.
00:56
So from this i get xz is equals to we can split this.
01:00
So z over z minus 1 square plus 2 z over z minus 1.
01:06
So from this i am getting 2 plus 2 z times z minus 1 divided by z minus 1 square, which is equals to 2 z square minus z divided by z minus 1 square.
01:22
So our z transform comes out to be two times, sorry, z times 2z squared, i'm just taking z common and this will be z minus one is square okay so this is the x transform for the given function now let's move to the next part in the next part we have been given x cube times u of x okay x cube times u of sorry here what we are having is n cube plus u of n correct n -cube plus u of n so let me write properly n q times u of n so now what we are going to do so this is nothing but n square n u of n this i can write here then n times of n times of n times of u of okay this we can write in this iterative form so this is nothing but z plus n times of u of n and this value we already found out so we are getting this to be equals to z over z minus one times of z over z minus one square so this is z plus n square u of n is equals to minus z d over d z of z over z minus one square so this is same as minus z apply the cocent rule of derivative z minus 1 square minus 2 z times z minus 1 divided by denominator whole square it becomes whole raise to 4 simplifying we get this to be minus z z minus 1 minus 2 z divided by z minus 1 whole cube this is what we are getting as of now now what is the next step here so we can write this two times 2 t of x n square t of n square u n is equal to this we know two times z times of z plus 1 divided by z minus 1 whole cube.
04:02
So 2 t n square u of n is equal to z square plus z divided by z minus 1 whole cube.
04:13
So 2t of n q, u of n will be equals to.
04:18
Again, we have to apply the same rule, minus z derivative of z plus z squared divided by z minus one whole cube, correct? so now what we are going to get is minus z and solving this, okay, we are going to get z minus 1, 2z plus 1 minus 3 z square minus 3 z.
04:44
Divided by z minus one whole cube square if we do it becomes whole raise to six.
04:55
So this is nothing but minus z, z minus one, 2z plus one, okay, minus 3z square minus 3z.
05:09
Or here only i have written already, you know if you write again.
05:13
So here we are getting whole raise to four.
05:16
So what we get two times of z? this is not two actually, okay? so this is z.
05:27
So this you can write as z times of t, correct? so z times of t of n cube u of n is equals to z, z squared plus 4 z, plus 1 divided by z minus 1, whole raise to 4.
05:44
So this is the required z transform in this case.
05:47
And the next question, we need to find out the inverse z transform for the given function.
05:55
So in this case, the first one is given to be z inverse of z inverse of z mod 2, sine 3 divided by z square minus 2 mod 2, cosine of 3 times, z plus modulus of two square so for this what i am going to do let us see that so it becomes this is equals to z t of sine omega not u of n okay this is what i will get so this is nothing but z sine omega -not z squared minus 2 z cos omega -not plus 1 so from this i get omega -not to be plus to 3 okay then z t of sine 3 and un is equal to z sine 3 divided by z square minus here we are getting 2 z cosine of omega -knot which is 3 plus 1...