Music Through Fourier Space pp 27-49 | Cite as

# Homometry and the Phase Retrieval Problem

## Summary

This chapter studies in depth the notion of homometry, i.e. having identical internal shape, as seen from Fourier space, where homometry can be seen at a glance by the size (or magnitude) of the Fourier coefficients. Finding homometric distributions is then a question of choosing the phases of these coefficients, hence this problem is often called *phase retrieval* in the literature. Such a choice of phases is summed up in the objects called *spectral units*, which connect homometric sets together. I included the original proof of the one difficult theorem of this book (Theorem 2.10), which non-mathematicians are quite welcome to skip. Some generalisations of the hexachord theorem are given, followed by the few easy results on higher-order homometry which deserve some room in this book because they rely heavily on DFT machinery. An original method for phase-retrieval with singular distributions (the difficult case) is also given. Some knowledge of basic linear algebra may help in this chapter.

## Keywords

Galois Group Inverse Fourier Transform Galois Theory Compact Abelian Group Phase Retrieval## Preview

Unable to display preview. Download preview PDF.