The Laplace Transform is a more general form of the Fourier transform. The equations corresponding to Analysis and Synthesis are given below.
The Laplace transform equation is identical to the Fourier transform equation, when s=jw. The synthesis equation for the Laplace transform is not always easy to use as it involves a line integral, region of convergence and so on; therefore we shall not use it here. Instead we’ll use a look-up table of common Laplace transforms to obtain the inverse transform.
For convenience we’ll consider the same rectangular pulse used in the previous lecture to illustrate the Fourier Transform.
Applying the Laplace Transform formula we can calculate the transform, X(s)
In the result above if we substitute s=jw we’ll get the Fourier transform as obtained in the example in the previous lecture.
A table of common Laplace Transforms is given below.
The Laplace transform and the Fourier transform have some important properties, and the following table lists them. Their use will become apparent in some examples.
Of particulat importance among these properties is the convolution property. The convolution of two time signals becomes simple multiplication of the Laplace transforms of those signals. This is much easier to do than convolution.
Find the inverse Laplace transform of the result obtained in the last example:
There are two terms in the function and both contain an exponential in ‘s’. The "time-shift" property of the Laplace transform can be first used. Then noting that the other factor is 1/s, we can look-up the inverse in the first table and obtain the inverse transform (the time function) as u(t), i.e., the step function. Now, combining the two steps, we get the inverse transform as two time shifted step functions as elaborated below.
This function plotted against time, is a rectangular pulse as shown earlier.
© Suresh Devasahayam