Convolution
Convolution is an operation that appears in many places including Machine Learning. Simply put, its a way to apply a linear combination of weights to a certain interval of a function, and the response is reported as the output of the convolution function.
Fourier Transform
The FT is a way of going from a time domain signal to the frequency domain, and has many applications to sampling theory and signal processing.
Laplace Transform
The Laplace transform is similar to the Fourier transform except that it uses a generic complex exponential. Below is the unilateral/one sided form considering times greater than 0.
The Laplace transform maps a system’s poles and zeros into the S-Plane, which can be used to reason about the stability, causality and other characteristics of a system.
Note that the Laplace transform is a more “generic” version of a mapping function, and the Fourier Transform can be specified as a case of the Laplace transform when .
The Laplace transform then simplifies to
This is equivalent to a contour integral along the x-axis of the Laplace transform.
Pole-Zero plot on the S-Plane for a system function
Visualization of the Fourier transform as a “slice” of the Laplace transform
Properties of S-Plane
Stability
Real pole on the left side of the s-plane indicates negative decaying exponent, so it is stable
Z-Transform
The Z-transform can be described as the discrete time variant of the Laplace transform.