Auto-Tuning Using Fourier Coefficients

The Fourier transform is an integral part of signal processing of any kind. To be able to analyze an input signal as a superposition of infinitely many signals of different frequency allows for manipulation of the coefficients and, after completing the inverse Fourier transform, interesting effects on the initial signal. Thus, in my project I focused on creating pitch correction effects as seen in software Auto-Tune, and in this paper I will talk about the process of creating a Matlab code that performs this effect. The Fast Fourier Transform algorithm is effective at breaking down mathematical functions into a Fourier series of different frequency complex exponentials and the corresponding coefficients. However, when analyzing audio signals it is impractical and inefficient to take one Fouier Transform of an entire audio clip; for a band-limited signal, most of the 44100t Fourier coefficients will contain no imformation about the original signal and there is no way to tell which frequencies are dominant at different times in the signal. Thus, the way that audio signals are most often processed is using something called the Short-Time Fourier Transform (STFT). The basic idea of the STFT is that one can window the actual signal multiple times and take the FFT of the resulting modified input. The windowing process is done simply by multiplying the signal by another function that is mostly zero and has a maximum of 1. This will kill off the signal at other times that are not of interest and allow for getting the frequencies at a certain point in time via the FFT. There are several types of windowing functions that can be used in the STFT process. The most commonly used set of windows are of the form