What are they?
In system engineering, there are two important transforms, Fast Fourier and Laplace.
Fast Fourier transform is a tool for signal processing and Laplace is mainly applied to
circuit analysis controller design.
What do they do?
Many of the parameters in our universe interact through differential equations. For
example, the voltage across an inductor is proportional to the derivative of the current
through the device. Likewise, the force applied to a mass is proportional to the
derivative of its velocity.
Fast Fourier and Laplace transforms analyze differential equations.
How do they work?
A system's differential equation is solved by mapping it onto a linear algebraic
equation with Laplace Transform. The solution is then mapped back onto the system's
Fast Fourier transform converts a signal from its original domain (often time and
space) to a representation in the frequency domain and vice versa.
Fast Fourier transform is used:
to detect seizure problems in EEG electrical signals,
to convert 3D images into electrical signals,
to convert electrical signals into 3D images for analysis.
Laplace transform is used to solve analysis problems in electrical circuits (often filters).