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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| Published in | Transactions of Tianjin University Vol. 20; no. 3; pp. 197 - 202 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Heidelberg
Tianjin University
01.06.2014
School of Electronic and Information Engineering, Beihang University, Beijing 100083, China |
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| Online Access | Get full text |
| ISSN | 1006-4982 1995-8196 |
| DOI | 10.1007/s12209-014-2201-2 |
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| Abstract | 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））） . |
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| AbstractList | 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））） . 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 ( n 2 ) memory units. Thus, in this paper an approach based on prime-composition is proposed to reduce the complexity of RS calculation to O ( n 2 ). Meanwhile, the complexity of Ramanujan-Fourier transform can be further reduced from O ( n 2 ) to O ( n ln(ln( n ))). |
| Author | 周丽娜 王祖林 赵磊 |
| AuthorAffiliation | School of Electronic and Information Engineering, Beihang University, Beijing 100083, China |
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| Cites_doi | 10.1109/LSP.2005.843775 10.1112/plms/s2-34.1.1 10.1016/j.dsp.2009.03.007 10.1109/LSP.2006.881527 10.1109/LSP.2012.2194142 |
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| Keywords | Ramanujan sums Ramanujan-Fourier transform prime-composition Ramanujan transform sums Ramanujan-Fourier |
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| Notes | 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））） . |
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| References | Noblom (CR13) 2008; 41 Eric, Jeffrey (CR12) 1996 Samadi, Ahmad, Swamy (CR8) 2005; 12 Planat, Rosu, Perrine (CR4) 2002 Carmichael (CR2) 1932; s2–34 Ramanujan (CR1) 1918; 22 Sugavaneswaren, Xie, Umapathy (CR7) 2012; 19 Planat, Minarovjech, Saniga (CR10) 2009; 85 Hardy, Ramanujan (CR11) 1917; 48 Lagha, Bensebti (CR5) 2009; 19 Gadiyar, Padma (CR3) 2006 Pei, Chang (CR9) 2007; 14 Mainardi, Bertinelli, Sassi (CR6) 2008 G H Hardy (2201_CR11) 1917; 48 S Samadi (2201_CR8) 2005; 12 M A Noblom (2201_CR13) 2008; 41 H G Gadiyar (2201_CR3) 2006 S Ramanujan (2201_CR1) 1918; 22 R D Carmichael (2201_CR2) 1932; s2–34 S C Pei (2201_CR9) 2007; 14 L Sugavaneswaren (2201_CR7) 2012; 19 M Planat (2201_CR4) 2002 M Planat (2201_CR10) 2009; 85 L T Mainardi (2201_CR6) 2008 B Eric (2201_CR12) 1996 M Lagha (2201_CR5) 2009; 19 |
| References_xml | – start-page: 605 year: 2008 end-page: 608 ident: CR6 article-title: Analysis of T-wave alternans using the Ramanujan transform[C] publication-title: . Bologna, Italy – year: 2006 ident: CR3 publication-title: Linking the circle and the sieve: Ramanujan-Fourier series [EB/OL] – volume: 12 start-page: 293 issue: 4 year: 2005 end-page: 296 ident: CR8 article-title: Ramanujan sums and discrete Fourier transforms[J] publication-title: Signal Processing Letters, IEEE doi: 10.1109/LSP.2005.843775 – volume: 41 start-page: 27 issue: 1 year: 2008 end-page: 31 ident: CR13 article-title: Counting the perfect powers[J] publication-title: Applied Probability Trust – volume: 48 start-page: 76 year: 1917 end-page: 92 ident: CR11 article-title: The normal number of prime factors of a number[J] publication-title: Quarterly Journal of Mathematics – volume: 85 start-page: 1 issue: 4 year: 2009 end-page: 5 ident: CR10 article-title: Ramanujan sums analysis of long-period sequences and 1/ noise[J] publication-title: Europhysics Letters(EPL) – volume: s2–34 start-page: 1 issue: 1 year: 1932 end-page: 26 ident: CR2 article-title: Expansions of arithmetical functions in infinite series[J] publication-title: Proceedings of the London Mathematical Society doi: 10.1112/plms/s2-34.1.1 – year: 1996 ident: CR12 publication-title: [M] – start-page: 715 year: 2002 end-page: 720 ident: CR4 article-title: Ramanujan sums for signal processing of low-frequency noise[C] publication-title: . Louisiana, USA – volume: 19 start-page: 843 issue: 5 year: 2009 end-page: 851 ident: CR5 article-title: Doppler spectrum estimation by Ramanujan-Fourier transform(rft)[J] publication-title: Digital Signal Processing doi: 10.1016/j.dsp.2009.03.007 – volume: 14 start-page: 20 issue: 1 year: 2007 end-page: 23 ident: CR9 article-title: Odd Ramanujan sums of complex roots of unity[J] publication-title: Signal Processing Letters, IEEE doi: 10.1109/LSP.2006.881527 – volume: 22 start-page: 259 year: 1918 end-page: 276 ident: CR1 article-title: On certain trigonometric sums and their applications in the theory of numbers[J] publication-title: Trans. Camb. Phil Soc. – volume: 19 start-page: 352 issue: 6 year: 2012 end-page: 355 ident: CR7 article-title: Timefrequency analysis via Ramanujan sums[J] publication-title: Signal Processing Letters, IEEE doi: 10.1109/LSP.2012.2194142 – start-page: 605 volume-title: Computer in Cardiology. Bologna, Italy year: 2008 ident: 2201_CR6 – volume-title: Algorithmic Number Theory[M] year: 1996 ident: 2201_CR12 – volume: s2–34 start-page: 1 issue: 1 year: 1932 ident: 2201_CR2 publication-title: Proceedings of the London Mathematical Society doi: 10.1112/plms/s2-34.1.1 – volume: 19 start-page: 352 issue: 6 year: 2012 ident: 2201_CR7 publication-title: Signal Processing Letters, IEEE doi: 10.1109/LSP.2012.2194142 – start-page: 715 volume-title: IEEE International Frequency Control Symposium and PDA Exhibition. Louisiana, USA year: 2002 ident: 2201_CR4 – volume: 14 start-page: 20 issue: 1 year: 2007 ident: 2201_CR9 publication-title: Signal Processing Letters, IEEE doi: 10.1109/LSP.2006.881527 – volume: 48 start-page: 76 year: 1917 ident: 2201_CR11 publication-title: Quarterly Journal of Mathematics – volume: 41 start-page: 27 issue: 1 year: 2008 ident: 2201_CR13 publication-title: Applied Probability Trust – volume: 22 start-page: 259 year: 1918 ident: 2201_CR1 publication-title: Trans. Camb. Phil Soc. – volume-title: Linking the circle and the sieve: Ramanujan-Fourier series [EB/OL] year: 2006 ident: 2201_CR3 – volume: 85 start-page: 1 issue: 4 year: 2009 ident: 2201_CR10 publication-title: Europhysics Letters(EPL) – volume: 19 start-page: 843 issue: 5 year: 2009 ident: 2201_CR5 publication-title: Digital Signal Processing doi: 10.1016/j.dsp.2009.03.007 – volume: 12 start-page: 293 issue: 4 year: 2005 ident: 2201_CR8 publication-title: Signal Processing Letters, IEEE doi: 10.1109/LSP.2005.843775 |
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| Snippet | 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... 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... |
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| Title | Prime-Composition Approach to Ramanujan-Fourier Transform Computation |
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