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Find the convolution of the following two functions: (15 points) f() * 9(t) Kro)gc T)dt for f(0) = e" and g(t) = t...

Question

Find the convolution of the following two functions: (15 points) f() * 9(t) Kro)gc T)dt for f(0) = e" and g(t) = t

Find the convolution of the following two functions: (15 points) f() * 9(t) Kro)gc T)dt for f(0) = e" and g(t) = t



Answers

Find $f(t)$ using convolution if $\mathbf{F}(s)$ is $\mathbf{F}(s)=\frac{1}{(s+1)(s+2)}$

Let's use convolution tedium to find the universal applause transform of SBS two divided by as a square into S plus one. So this is the capital fFS given. But we know that laplace transform of convolution of hefty. A star GOP star means contribution is the product of the individual oblast transformed. So that is HFS into gFS. Now let's uh factor this as a form of head office times the office so I can take something like this. One by S plus one has my head office SBS two divided by the square as my job office. So that means head office is one by it's best one. And what is the inverse laplace transform this? Hfts ipod negativity. You are? So this is the inverse laplace transform and what were the inverse laplace transform? The office? The office is basically Esposto divided by the square. So that implies uh GFS is equal to one bias plus two by square. So what were the inverse laplace transform, inverse laplace transform, one bias is simply one, whereas inverse laplace transform one by square is T. So it's duty. So this is GFT. So we have got head of the N G O T of course understood into your body. So this is the off the end, this is hft. Now let's find the convolution. It's convolution. Head start G of T is given by integration of minus infinity to infinity. I like something like this. So it is G of tao into E power sorry into head off t minus style detail. So this is the definition of convolution because head start G S M S G star hatch convolutions always communicating. So this integration of minus infinity infinity G of tower which is to tao that's one into your of tao whereas hedge of t minus targets deep negative t minus style into europe t minus style detail. Now, since there are two units stuff functions you have to win, you have t minus stop these boats will be one only when tower lies between zero and T because when tories Pastor this is one and mental weighs less than the this is one. All right. And this is applicable when T is positive and if the host is negative then obviously uh what should be zero. All right. So when uh when T is negative so there is no question of uh zero has done deal has done uh town is that T? So we'll split this into two cases. So in case one So case one is basically 20 is positive and zero less than tao less than T. So we get the transform of uh I mean the contribution of integration of minus infinity to infinity. Now both the unit steps will be one. So it will be simply 2000 plus one into uh the par minus soft t minus tao detailed. But since to between zero and t I can change the limits to zero and t Zero and t. To tell plus one E. Part tao into the par minus t detail. Since we are integrating with respect to towel I can take it, take it F R minus T. Outside that will be par minus t. Integration of zero to T. To tell plus one E. Power town detail. So let's integrate by parts so we get 2000 plus one into a potato minus two into a pop top. Alright now substitute upper and lower limits 20. So we get the final answer s minus T. Two to t plus one. A party minus to a party. And the lower limit for zero it will be minus one plus two. So it is a par minus T. You do two d plus one, everybody minus three party plus one. So that will be a par minus t into two t uh minus one party plus one. So take the par minus the insights which will be par minus t plus two t minus one. So what is Okay? It's one case 1 20 is positive zero, less than Towel is Dante. My head start G of tea is basically what you got. It is a people ar minus T plus two t minus one. And what if if these negative 20 is negative? Obviously your head start GFT will be zero so I can clap both as a unit step function. So let's start. GFT is nothing but epa minus d. Keep ar minus D plus two D minus one into uh U. F. T. So this is the final answer. So in words laplace transform of SBS two divided by the square into SPS one. Is this particular function, time domain function


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