Simulating the DFT Algorithm for Audio Processing

Abstract: Since the evolution of digital computers, the storage of data has always been in terms of discrete bits that can store values of either 1 or 0. Hence, all computer programs (such as MATLAB), convert any input continuous signal into a discrete dataset. Applying this to oscillating signals, such as audio, opens a domain for processing as well as editing. The Fourier transform, which is an integral over infinite limits, for the use of signal processing is discrete. The essential feature of the Fourier transform is to decompose any signal into a combination of multiple sinusoidal waves that are easy to deal with. The discrete Fourier transform (DFT) can be represented as a matrix, with each data point acting as an orthogonal point, allowing one to perform complicated transformations on individual frequencies. Due to this formulation, all the concepts of linear algebra and linear transforms prove to be extremely useful here. In this paper, we first explain the theoretical basis of audio processing using linear algebra, and then focus on a simulation coded in MATLAB, to process and edit various audio samples. The code is open ended and easily expandable by just defining newer matrices which can transform over the original audio signal. Finally, this paper attempts to highlight and briefly explain the results that emerge from the simulation