A Visual Way to Understand Fourier Transform
Dot Product as Similarity Measure/Matching Algorithm
The two plots above visually demonstrate the concept of the dot product as a measure of similarity or "closeness" between two vectors in a 2D space.
This illustrates how the dot product can be used to measure the degree of similarity between two vectors: the larger the dot product, the more similar the vectors are in terms of their direction and magnitude. Conversely, a smaller dot product indicates less similarity, and a dot product of zero indicates no similarity at all (i.e., the vectors are orthogonal).
Use Basis Vectors for Any Vector Representation
The length of each basis vector is 1 unit, and the scaling factors for each scaled vector are labeled in the plot. This demonstrates how the original vector can be expressed as a linear combination (i.e. vector sum of scaled basis vectors aka a dot product) of its components along the basis vectors.
Matrix-Vector Dot Product Operation and Its Relation to Eigenvalue Problem
The figure above illustrates the concepts of scaling a vector by a factor λ and transforming a vector by a basis matrix side by side. On the right the basis vector A is transformation is an identify matrix which means that it does not change the vector. On the left, the blue array represents a 3D vector of (1,1,1). and red arrow represents which is the original vector v scaled by a factor λ=2 This shows that multipling a vector by a scalar only changes it's magnitude but the direction is preserved.
Understanding the Discrete Fourier Transform: Treating it as a Matrix-Vector Dot Product Operation with Fourier Basis Vectors
Practical Implementation of Discrete Fourier Transform
Let's illustrate this with Visual example:
Here's visual of how this matrix vector product works in practice:
A Close Examination of DFT Matrix
As we move from Row 1 to Row 6, we can see that the frequency of the sinusoids increases. This reflects the fact that the DFT matrix includes basis functions that correspond to a range of frequencies, allowing us to decompose a signal into its frequency components. Even though we only plotted the first 6 rows, the DFT matrix actually contains 64 rows, each corresponding to a different frequency. The higher frequency components are not shown in this plot.