The Laplace transform *F*(*s*) of a function *f*(*t*) is defined as follows:

where ‘*s*’ is a complex variable called the Laplace operator and is equal to *jw*(*s* = *σ* + *jω*) · *e*^{−st} is called the kernel of the Laplace transformation.

1. Step input

*f*(*t*) = 0 for t < 0

= *u*_{0} for *t* ≥ 0

For a unit step,

2. Pulse input

*f*(*t*) = 0 for *t* < 0 and *t* > *t*_{0}

= *u*_{0} for 0 < *t* < *t*_{0}

3. Exponentially decaying function

*f*(*t*) = *e ^{−at}*

4. Ramp function

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