Variational Mode Decomposition Part-2: The Maths

This post is the continuation of the part-1 post. Check it out here.

In this part, we will focus on the mathematics of the algorithm. To get a better idea of how VMD works and also I will try my best to clear all the doubts which you might have encountered in the previous part.

Prerequisites

There are few prerequisites before we start the maths. You should be familiar with

1. Fourier Transform
2. Constrained Optimization

If you think you know the above stuff then let’s test it.

There are three important things that VMD does

1. Wiener Filter properties
2. Forming analytical signal
3. Frequency mixing

Wiener Filter

A Wiener filter is used to reconstruct the original image from a blurred image. A similar kind of filtering is done on signals too(Image is nothing but a signal in 2D?). It is a low-narrow band filter whose frequency(w) is close to zero(Keep this point in your mind carefully). Narrowband means the variation across the central frequency will be very less. VMD does almost the same thing. The bandwidth around the central frequency (i.e the frequency which we are interested…