This paper gives an example of an auto-regressive (AR) spectral analysis on transient musical sounds. The attack part of many musical sounds is mostly too short to be analysed by a short-time Fourier analysis, whereas this short period of time is long enough for several AR-methods. The AR-spectra obtained from short segments of signals with attack transients have a sufficiently high frequency r...

Given a group G, we can construct a graph relating the elements of G to each other, called the Cayley graph. Using Fourier analysis on a group allows us to apply our knowledge of the group to gain insight into problems about various properties of the graph. Ideas from representation theory are powerful tools for analysis of groups and their Cayley graphs, but we start with some simpler theory t...

In this paper, we focus on Fourier analysis and holographic transforms for signal representation. For instance, in the case of image processing, the holographic representation has the property that an arbitrary portion of the transformed image enables reconstruction of the whole image with details missing. We focus on holographic representation defined through the Fourier Transforms. Thus, We f...

Fourier analysis, a classical topic in analysis, can be applied to many nonanalytic fields. This paper explores Fourier analysis in multiple domains in order to prove several nonanalytic results. Fourier analysis will be explored on a closed interval, the real line, and a finite Abelian group. We use methods from Fourier analysis to prove Weyl’s equidistribution theorem and the Poisson summatio...