Comparison of the DFT and FFT via Matrices
The purpose of this article is to illustrate the differences of the Discrete Fourier Transform (DFT) versus the Fast Fourier Transform (FFT). In addition, I will provide an alternative view of the FFT calculation path as described in Week 2 of Lab 6 in ECE 438. A link can be found here [1]. Please note the following explanation of the FFT will use the "divide and conquer" method.
To start, we will define the DFT as,
$ X_N[k] = \sum_{n=0}^{N-1} x[n] e^{-j2{\pi}kn/N} (Eq. 1) $
It is fairly easy to visualize this 1 point DFT, but how does it look when x[n] has 8 points, 1024 points, etc. That's where matrices come in. For an N point DFT, we will define our input as x[n] where n = 0, 1, 2, ... N-1. Similarly, the output will be defined X[k] where k = 0, 1, 2, ... N-1. Referring to our definition of the Discrete Fourier Transform above, it should be apparent that we are simply repeating Eq. 1 N times. For every value of x[n] in the discrete time domain, there is a corresponding value, X[k].
INPUT
$ \begin{pmatrix} x[0] \\ x[1] \end{pmatrix} $
$ \begin{pmatrix} \alpha& \beta^{*}\\ \gamma^{*}& \delta \end{pmatrix} $
