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Math Help - Laplace Transform of this signal?

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    Laplace Transform of this signal?

    In my linear systems class, we are doing Laplace transforms using transform tables and the properties. I can usually do the problems that closely resemble the table, but when they involve heavy algebraic manipulations, I don't know what to do.

    Can someone explain the steps of this solution?

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    Why do the +1 and -1 appear? Where did the extra exponential come from? If someone can go line for line, I would appreciate it and how to in general solve these kinds of signals?
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    Re: Laplace Transform of this signal?

    Quote Originally Posted by nottheface View Post
    In my linear systems class, we are doing Laplace transforms using transform tables and the properties. I can usually do the problems that closely resemble the table, but when they involve heavy algebraic manipulations, I don't know what to do.

    Can someone explain the steps of this solution?

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    Why do the +1 and -1 appear? Where did the extra exponential come from? If someone can go line for line, I would appreciate it and how to in general solve these kinds of signals?
    The goal of the algebra is so the can use the t-axis translation theorem. This states that

    L\{f(t-a)u(t-a) \}=e^{-as}F(s)

    The problem is that you have different time shifts in your unit step function and the other function.

    Consider the function

    g(t)=(t-2)^2

    We want to write in with only factors of (t-1) and constants

    (t-2)^2=(t-1-1)^2

    Now for simplicity let u=t-1 this gives

    (u-1)^2=u^2-2u+1=(t-1)^2-2(t-1)+1

    Now we need to do something similar with the exponential term e^{-5t} lets just focus on the exponent

    -5t=-5t+\underbrace{5-5}_{\text{add zero}}=-5(t-1)-5 so

    e^{-5t}=e^{-5(t-1)-5}=e^{-5}e^{-5(t-1)}


    Now lets put everything to gether
    =e^{-5}e^{-5(t-1)}[(t-1)^2-2(t-1)+1]u(t-1)

    Now just distribute and use the t-axis translation theorem to transform each part!
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