Introduction to Laplace Transform Analysis

The *one-sided Laplace transform* of a signal
is defined
by

where is real and is a complex variable. The one-sided Laplace transform is also called the

When evaluated along the
axis (*i.e.*,
), the
Laplace transform reduces to the unilateral *Fourier transform*:

The Fourier transform is normally defined bilaterally ( above), but for causal signals , there is no difference. We see that the Laplace transform can be viewed as a generalization of the Fourier transform from the real line (a simple frequency axis) to the entire complex plane. We say that the Fourier transform is obtained by evaluating the Laplace transform along the axis in the complex plane.

An advantage of the Laplace transform is the ability to transform signals which have no Fourier transform. To see this, we can write the Laplace transform as

Thus, the Laplace transform can be seen as the Fourier transform of an

- Existence of the Laplace Transform
- Analytic Continuation
- Relation to the
*z*Transform - Laplace Transform Theorems

- Laplace Analysis of Linear Systems

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