Mathematical background

Boaz Rafaely

doi:10.1007/978-3-662-45664-4_1

Mathematical background

Boaz Rafaely

Department of Electrical & Computer Engineering

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

1 Scopus citations

Abstract

This chapter provides the mathematical background necessary for studying spherical array processing. Spherical arrays typically sample functions on a sphere (e.g. sound pressure); therefore, this chapter begins by presenting the spherical coordinate system as well as some examples of functions on the sphere. Spherical harmonics are a central theme of this book as they form a basis for representing functions on the sphere. Therefore, spherical harmonics are first defined and illustrated, and then an introduction to the spherical Fourier transform and a description of functions on the sphere in Hilbert space follows. The chapter concludes with a presentation of the topics of rotation, convolution, and correlation defined for functions on the sphere.

Original language	English
Title of host publication	Springer Topics in Signal Processing
Publisher	Springer Science and Business Media B.V.
Pages	1-30
Number of pages	30
DOIs	https://doi.org/10.1007/978-3-662-45664-4_1
State	Published - 1 Jan 2015

Publication series

Name	Springer Topics in Signal Processing
Volume	8
ISSN (Print)	1866-2609
ISSN (Electronic)	1866-2617

Keywords

Dirac Delta Function
Legendre Function
Spherical Harmonic
Spherical Harmonic Coefficient
Unit Sphere

ASJC Scopus subject areas

Signal Processing
Computer Networks and Communications
Electrical and Electronic Engineering

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10.1007/978-3-662-45664-4_1

Cite this

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title = "Mathematical background",

abstract = "This chapter provides the mathematical background necessary for studying spherical array processing. Spherical arrays typically sample functions on a sphere (e.g. sound pressure); therefore, this chapter begins by presenting the spherical coordinate system as well as some examples of functions on the sphere. Spherical harmonics are a central theme of this book as they form a basis for representing functions on the sphere. Therefore, spherical harmonics are first defined and illustrated, and then an introduction to the spherical Fourier transform and a description of functions on the sphere in Hilbert space follows. The chapter concludes with a presentation of the topics of rotation, convolution, and correlation defined for functions on the sphere.",

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AB - This chapter provides the mathematical background necessary for studying spherical array processing. Spherical arrays typically sample functions on a sphere (e.g. sound pressure); therefore, this chapter begins by presenting the spherical coordinate system as well as some examples of functions on the sphere. Spherical harmonics are a central theme of this book as they form a basis for representing functions on the sphere. Therefore, spherical harmonics are first defined and illustrated, and then an introduction to the spherical Fourier transform and a description of functions on the sphere in Hilbert space follows. The chapter concludes with a presentation of the topics of rotation, convolution, and correlation defined for functions on the sphere.

KW - Dirac Delta Function

KW - Legendre Function

KW - Spherical Harmonic

KW - Spherical Harmonic Coefficient

KW - Unit Sphere

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Mathematical background

Abstract

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Keywords

ASJC Scopus subject areas

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