Thread: 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?

Originally Posted by nottheface

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?

ImageShack® - Online Photo and Video Hosting

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



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

Consider the function



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



Now for simplicity let this gives



Now we need to do something similar with the exponential term lets just focus on the exponent

so




Now lets put everything to gether


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