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Find the convolution of the following two functions. f(t) = { (top) 1 -1 < t < 1 , { (bottom) 0 otherwiseAnd the function. g(t) = u(t)e^-t...

Question

Find the convolution of the following two functions. f(t) = { (top) 1 -1 < t < 1 , { (bottom) 0 otherwiseAnd the function. g(t) = u(t)e^-t

Find the convolution of the following two functions. f(t) = { (top) 1 -1 < t < 1 , { (bottom) 0 otherwise And the function. g(t) = u(t)e^-t



Answers

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

We need to use the convolution here. Um to find the inwards laplace transform of one by SPS one into supposed to fall. Let's call F one office pass one by X plus one F. to office as one x 1. No what is F. one of T In front of these? In words like last transform one by Express one. This is very well known at the Par -150. And what is F. Two of the It is immerse the blast transformer one by express to that is a part negative duty new york. Now, if you recall the convolution team FN Star F two of T. These integration of negative infinity to negative or positive infinity tower into two of T minus tower detail. And Loblaw stands for more conclusion is the product of individual la la strengths. So that means our aim is to find the convolution of these two punches a round of the NFL let's find it. So F. F. T. Is a par minus T. Your teeth after D. This E Power -2 D. Your feet. All right now let's find it. If it's star, I have two of T. There's nothing but mitigation of negative infinity to negative infinity if enough, doubt that if r minus dow you of uh into E power minus two times of t minus tower new york t minus tao. Now view of tower is one, mentor is positive, but this is one when T is less than also how is less than deep. So that means the limits of integration can be converted to zero to T. Because apart from this through the value of U. Of town and U. Of t minus zero because that is the definition of unit step function. So integration zero T. E par minus tao -2 T Plus two Towers Down. So you can daily power minus to the outside deviation of zero T. The par -2 and two dogs detailed. So finally, what do we want to be in the arteries? Evan Starr F two of T is equal to the par minus duty. The integration of the battle with the bottle and substitute the limits from zero T. So we get the par minus two T into a party minus minus two T into zero. So it's finally we get the par minus two T into a party minus the par minus duty, which is nativity part minus t minus the par minus duty. So, hence in Bruce laplace, transform one by SPS one into SPS too. Nothing about the convolution. The par minus t minus F. R minus duty. And since we are in the unit step function domain, so, I have to multiply it. Unit step function. All right. This is the final answer. Because this integration is rallying 20 is positive. I'm in here because if he is negative, what will happen Both of zero, right? And he is native because when these negative stories less than negative numbers, tab is negative. So that means this is zero. So when he is positive, only this zero to make sense. So this is actually applicable amenities faster. If he is negative, the whole integrity zero. And that is why we are multiplying with you. Okay? So this is the final answer.

In the problem. Find Nablus waas competition. So you just even ask lab lesson. What are if it's GS? This is all is one into lab Last worse off. One of our is last three spot. This is cough tea into treaty into Theo de the three. And you don did, uh so this is written as didn't India. You are three into the oh God, The job is indeed a d You see glass into down into cause t minus down thing is the answer.


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