Montgomery Whalen

One of the most important tools for time-frequency analysis is the Fourier transform. The Fourier transform of a function is a complex valued function, where the modulus of the transform represents the amount of certain frequencies in the original function. It is incredibly useful for finding the dominant frequencies of complicated signals. For example, the waveform of a song looks quite noisy, however when taking the Fourier transform of the signal, one can see what frequencies/notes are used the most during the song, and thus find the key. Once we go over the intuition behind the Fourier transform, we will investigate some natural extensions, such as the short-time Fourier transform (STFT), which can describe the prominent frequencies over different sections of the domain (or different moments in time), instead of the entire function at once. The inversion formula for the STFT will inspire a search for a discrete expansion of functions in L² with respect to a countable set of time-frequency shifts of a certain function, which will lead us to Gabor frames. Ideas from the Fourier transform are widely used in various applications, such as engineering, digital sound and music, quantum mechanics, and image processing.