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Laplace Transform

Can someone help explain to me the concept of convolution and how it is used?

 

It is used when you want to take inverse Laplace transforms.

 

If you recognize that the function of s is the product of 2 Laplace transforms for example

\(H(s)=\frac{1}{s-3}\frac{s}{s^2+4}={\cal L}\{e^{3t}\}{\cal L}\{\cos 2t\}\)

 

You could do a partial fraction decomposition or you could use convolution.

 

The product of 2 Laplace transforms is the Laplace transforms of the convolution of the 2 functions.

 

In this case \(H(s)={\cal L}\{e^{3t}∗\cos 2t\}\) where * is the convolution sign

 

Using the definition of the convolution \({\cal L}^{−1}\{H\}=e^{3t}∗\cos 2t=∫_0^t e^{3v}\cos(2(t−v))dv=∫_0^te^{3(t−v)}\cos(2v)dv\)