Prime-Composition Approach to Ramanujan-Fourier Transform Computation

Ramanujan sums （RS） and their Fourier transforms have attracted more and more attention in signal processing in recent years. Due to their non-periodic and non-uniform spectrum, RS are widely used in low-frequency noise processing, Doppler spectrum estimation and time-frequency analysis. However, th...

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Bibliographic Details
Published inTransactions of Tianjin University Vol. 20; no. 3; pp. 197 - 202
Main Author 周丽娜 王祖林 赵磊
Format Journal Article
LanguageEnglish
Published Heidelberg Tianjin University 01.06.2014
School of Electronic and Information Engineering, Beihang University, Beijing 100083, China
Subjects
Engineering
Humanities and Social Sciences
Mechanical Engineering
multidisciplinary
Science
信号处理
傅立叶变换
傅里叶变换
噪声处理
拉马
组合方法
计算复杂度
黄金
Ramanujan sums
Ramanujan-Fourier transform
prime-composition
Ramanujan
transform
sums
Ramanujan-Fourier
Online AccessGet full text
ISSN1006-4982
1995-8196
DOI10.1007/s12209-014-2201-2

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Summary:Ramanujan sums （RS） and their Fourier transforms have attracted more and more attention in signal processing in recent years. Due to their non-periodic and non-uniform spectrum, RS are widely used in low-frequency noise processing, Doppler spectrum estimation and time-frequency analysis. However, the traditional method for calculating RS values is rather complex since it requires two numbers＇ factorization in two arithmetic functions. For a length-n vector, its Ramanujan-Fourier transform usually involves a series of RS values which will occupy O（n2） memory units. Thus, in this paper an approach based on prime-composition is proposed to reduce the complexity of RS calculation to O（n）. Meanwhile, the complexity of Ramanujan-Fourier transform can be further reduced from O（n2） to O（n In（In（n））） .
Bibliography:Ramanujan sums; Ramanujan-Fourier transform; prime-composition
12-1248/T
Zhou Lina, Wang Zulin , Zhao Lei（School of Electronic and Information Engineering, Beihang University, Beijing 100083, China）
Ramanujan sums （RS） and their Fourier transforms have attracted more and more attention in signal processing in recent years. Due to their non-periodic and non-uniform spectrum, RS are widely used in low-frequency noise processing, Doppler spectrum estimation and time-frequency analysis. However, the traditional method for calculating RS values is rather complex since it requires two numbers＇ factorization in two arithmetic functions. For a length-n vector, its Ramanujan-Fourier transform usually involves a series of RS values which will occupy O（n2） memory units. Thus, in this paper an approach based on prime-composition is proposed to reduce the complexity of RS calculation to O（n）. Meanwhile, the complexity of Ramanujan-Fourier transform can be further reduced from O（n2） to O（n In（In（n））） .
ISSN:1006-4982
1995-8196
DOI:10.1007/s12209-014-2201-2