The Experts below are selected from a list of 18498 Experts worldwide ranked by ideXlab platform
P. P. Vaidyanathan - One of the best experts on this subject based on the ideXlab platform.
-
On the Zeros of Ramanujan Filters
IEEE Signal Processing Letters, 2020Co-Authors: Pranav Kulkarni, P. P. VaidyanathanAbstract:Ramanujan filter banks have been used for identifying periodicity structure in streaming data. This letter studies the locations of zeros of Ramanujan filters. All the zeros of Ramanujan filters are shown to lie on or inside the unit circle in the z-plane. A convenient factorization appears as a corollary of this result, which is useful to identify common factors between different Ramanujan filters in a filter bank. For certain families of Ramanujan filters, further structure is identified in the locations of zeros of those filters. It is shown that increasing the number of periods of Ramanujan sums in the filter definition only increases zeros on the unit circle in z-plane. A potential application of these results is that by identifying common factors between Ramanujan filters, one can obtain efficient implementations of Ramanujan filter banks (RFB) as demonstrated here.
-
Efficient multiplier-less structures for Ramanujan filter banks
2017 IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP), 2017Co-Authors: P. P. Vaidyanathan, S. TennetiAbstract:Ramanujan filter banks (RFB) are useful to generate time-period plane plots which allow one to localize multiple periodic components in the time domain. For such applications, the RFB produces more satisfactory results compared to short time Fourier transforms and other conventional methods, as demonstrated in recent years. This paper introduces a novel multiplier-less, hence computationally very efficient, structure to implement Ramanujan filter banks, based on a new result connecting Ramanujan sums and natural periodic bases.
-
Multidimensional Ramanujan-sum expansions on nonseparable lattices
2015 IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP), 2015Co-Authors: P. P. VaidyanathanAbstract:It is well-known that the Ramanujan-sum cq(n) has applications in the analysis of periodicity in sequences. Recently the author developed a new type of Ramanujan-sum representation especially suited for finite duration sequences x(n): This is based on decomposing x(n) into a sum of signals belonging to so-called Ramanujan subspaces Sqi. This offers an efficient way to identify periodic components using integer computations and projections, since cq(n) is integer valued. This paper revisits multidimensional signals with periodicity on possibly nonseparable integer lattices. Multidimensional Ramanujan-sum and Ramanujan-subspaces are developed for this case. A Ramanujan-sum based expansion for multidimensional signals is then proposed, which is useful to identify periodic components on nonseparable lattices.
-
Properties of Ramanujan filter banks
2015 23rd European Signal Processing Conference (EUSIPCO), 2015Co-Authors: P. P. Vaidyanathan, S. TennetiAbstract:This paper studies a class of filter banks called the Ramanujan filter banks which are based on Ramanujan-sums. It is shown that these filter banks have some important mathematical properties which allow them to reveal localized hidden periodicities in real-time data. These are also compared with traditional comb filters which are sometimes used to identify periodicities. It is shown that non-adaptive comb filters cannot in general reveal periodic components in signals unless they are restricted to be Ramanujan filters. The paper also shows how Ramanujan filter banks can be used to generate time-period plane plots which track the presence of time varying, localized, periodic components.
-
Ramanujan sums in the context of signal processing part ii fir representations and applications
Sport Psychologist, 2014Co-Authors: P. P. VaidyanathanAbstract:The mathematician Ramanujan introduced a summation in 1918, now known as the Ramanujan sum c_q(n). In a companion paper (Part I), properties of Ramanujan sums were reviewed, and Ramanujan subspaces S_q introduced, of which the Ramanujan sum is a member. In this paper, the problem of representing finite duration (FIR) signals based on Ramanujan sums and spaces is considered. First, it is shown that the traditional way to solve for the expansion coefficients in the Ramanujan-sum expansion does not work in the FIR case. Two solutions are then developed. The first one is based on a linear combination of the first N Ramanujan-sums (with N being the length of the signal). The second solution is based on Ramanujan subspaces. With q_1, q_2,..., q_K denoting the divisors of N; it is shown that x(n) can be written as a sum of K signals x_(qi) (n) ∈ S_(qi). Furthermore, the i_(th) signal x_(qi) (n) has period q_i, and any pair of these periodic components is orthogonal. The components x_(qi) (n) can be calculated as orthogonal projections of x(n) onto Ramanujan spaces S_(qi). Then, the Ramanujan Periodic Transform (RPT) is defined based on this, and is useful to identify hidden periodicities. It is shown that the projection matrices (which compute x_(qi) (n) from x(n)) are integer matrices except for an overall scale factor. The calculation of projections is therefore rendered easy. To estimate internal periods N_∞ <; N of x(n), one only needs to know which projection energies are nonzero.
S. Tenneti - One of the best experts on this subject based on the ideXlab platform.
-
Efficient multiplier-less structures for Ramanujan filter banks
2017 IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP), 2017Co-Authors: P. P. Vaidyanathan, S. TennetiAbstract:Ramanujan filter banks (RFB) are useful to generate time-period plane plots which allow one to localize multiple periodic components in the time domain. For such applications, the RFB produces more satisfactory results compared to short time Fourier transforms and other conventional methods, as demonstrated in recent years. This paper introduces a novel multiplier-less, hence computationally very efficient, structure to implement Ramanujan filter banks, based on a new result connecting Ramanujan sums and natural periodic bases.
-
Properties of Ramanujan filter banks
2015 23rd European Signal Processing Conference (EUSIPCO), 2015Co-Authors: P. P. Vaidyanathan, S. TennetiAbstract:This paper studies a class of filter banks called the Ramanujan filter banks which are based on Ramanujan-sums. It is shown that these filter banks have some important mathematical properties which allow them to reveal localized hidden periodicities in real-time data. These are also compared with traditional comb filters which are sometimes used to identify periodicities. It is shown that non-adaptive comb filters cannot in general reveal periodic components in signals unless they are restricted to be Ramanujan filters. The paper also shows how Ramanujan filter banks can be used to generate time-period plane plots which track the presence of time varying, localized, periodic components.
-
Ramanujan subspaces and digital signal processing
2014 48th Asilomar Conference on Signals Systems and Computers, 2014Co-Authors: P. P. Vaidyanathan, S. TennetiAbstract:Ramanujan-sums have in the past been used to extract hidden periods. In a recent paper it was shown that for finite duration (FIR) sequences, the traditional representation is not suitable. Two new types of Ramanujan-sum expansions were proposed for the FIR case, each offering an integer basis, and applications in the extraction of hidden periodicities were developed. Crucial to these developments was the introduction of Ramanujan spaces. The aim of this paper is to develop some properties of these subspaces in the context of signal processing. The design of near orthogonal bases for these spaces is emphasized.
Bruce C Berndt - One of the best experts on this subject based on the ideXlab platform.
-
The Influence of Carr’s Synopsis on Ramanujan
Number Theory and Discrete Mathematics, 2020Co-Authors: Bruce C BerndtAbstract:Almost all biographers of Ramanujan (e.g., P.V. Seshu Aiyar and R. Ramachandra Rao [13, p. xii]) point to G.S. Carr’s A Synopsis of Elementary Results in Pure Mathematics [10] as the book which kindled the fire of Ramanujan’s devotion to mathematics. How much did Carr’s Synopsis influence Ramanujan? Which published papers and which entries in his notebooks [14] have their seeds in the Synopsis? We cannot provide definitive answers to these questions, because we know very little about the other books Ramanujan might have studied in his formative years. However, upon a close examination of Carr’s book, we can suggest some topics which Ramanujan might have learned from Carr.
-
Eisenstein Series in Ramanujan's Lost Notebook
Ramanujan Journal, 2020Co-Authors: Bruce C Berndt, Heng Huat Chan, Jaebum SohnAbstract:In his lost notebook, Ramanujan stated without proofs several beautifulidentities for the three classsical Eisenstein series (in Ramanujan's notation) P(q), Q(q), and R(q). The identities are given in terms of certain quotients of Dedekind eta-functions called Hauptmoduls. These identities were first proved by S. Raghavan and S.S. Rangachari, but their proofs used the theory of modular forms, with which Ramanujan was likely unfamiliar. In this paper we prove all these identities by using classical methods which would have been well known to Ramanujan. In fact, all our proofs use only results from Ramanujan's notebooks.
-
Ramanujan s series for 1 π a survey
American Mathematical Monthly, 2009Co-Authors: Nayandeep Deka Baruah, Bruce C Berndt, Heng Huat ChanAbstract:When we pause to reflect on Ramanujan’s life, we see that there were certain events that seemingly were necessary in order that Ramanujan and his mathematics be brought to posterity. One of these was V. Ramaswamy Aiyer’s founding of the Indian Mathematical Society on 4 April 1907, for had he not launched the Indian Mathematical Society, then the next necessary episode, namely, Ramanujan’s meeting with Ramaswamy Aiyer at his office in Tirtukkoilur in 1910, would also have not taken place. Ramanujan had carried with him one of his notebooks, and Ramaswamy Aiyer not only recognized the creative spirit that produced its contents, but he also had the wisdom to contact others, such as R. Ramachandra Rao, in order to bring Ramanujan’s mathematics to others for appreciation and support. The large mathematical community that has thrived on Ramanujan’s discoveries for nearly a century owes a huge debt to V. Ramaswamy Aiyer. 1. THE BEGINNING. Toward the end of the first paper [57], [58 ,p . 36] that Ramanujan published in England, at the beginning of Section 13, he writes, “I shall conclude this paper by giving a few series for 1/π.” (In fact, Ramanujan concluded his paper a couple of pages later with another topic: formulas and approximations for the perimeter of an ellipse.) After sketching his ideas, which we examine in detail in Sections 3 and 9, Ramanujan records three series representations for 1/π .A s is customary, set
-
Ramanujan s forty identities for the rogers Ramanujan functions
2007Co-Authors: George E Andrews, Bruce C BerndtAbstract:The Rogers-Ramanujan identities are perhaps the most important identities in the theory of partitions. They were first proved by L.J. Rogers in 1894 and rediscovered by Ramanujan prior to his departure for England. Since that time, they have inspired a huge amount of research, including many analogues and generalizations. Published with the lost notebook is a manuscript providing 40 identities satisfied by these functions. In contrast to the Rogers-Ramanujan identities, the identities in this manuscript are identities between the two Rogers-Ramanujan functions at different powers of the argument. In other words, they are modular equations satisfied by the functions. The theory of modular forms can be invoked to provide proofs, but such proofs provide us with little insight, in particular, with no insight on how Ramanujan might have discovered them. Thus, for nearly a century, mathematicians have attempted to find “elementary” proofs of the identities. In this chapter, “elementary” proofs are given for each identity, with the proofs of the most difficult identities found only recently by Hamza Yesilyurt.
-
Ramanujan s lost notebook part i
2005Co-Authors: George E Andrews, Bruce C BerndtAbstract:In the spring of 1976, George Andrews of Pennsylvania State University visited the library at Trinity College, Cambridge, to examine the papers of the late G.N. Watson. Among these papers, Andrews discovered a sheaf of 138 pages in the handwriting of Srinivasa Ramanujan. This manuscript was soon designated, "Ramanujan's lost notebook." Its discovery has frequently been deemed the mathematical equivalent of finding Beethoven's tenth symphony.This volume is thefourthof fivevolumes thatthe authors plan to write on Ramanujans lost notebook.In contrast to thefirst three books on Ramanujan's Lost Notebook, the fourth book does not focus on q-series. Most of the entries examined in this volume fall under the purviews of number theory and classical analysis. Several incomplete manuscripts of Ramanujan published by Narosa with the lost notebook are discussed. Three of the partial manuscripts are on diophantine approximation, and others are in classical Fourier analysis and prime number theory. Most of the entries in number theory fall under the umbrella of classical analytic number theory. Perhaps the most intriguing entries are connected with the classical, unsolved circle and divisor problems.Reviewfrom the second volume:"Fans of Ramanujan's mathematics are sure to be delighted by this book. While some of the content is taken directly from published papers, most chapters contain new material and some previously published proofs have been improved. Many entries are just begging for further study and will undoubtedly be inspiring research for decades to come. The next installment in this series is eagerly awaited."- MathSciNetReview from the first volume:"Andrews and Berndt are to be congratulated on the job they are doing. This is the first step...on the way to an understanding of the work of the genius Ramanujan. It should act as an inspiration to future generations of mathematicians to tackle a job that will never be complete."- Gazette of the Australian Mathematical Society
Krishnaswami Alladi - One of the best experts on this subject based on the ideXlab platform.
-
P.A. MacMahon: Ramanujan’s Distinguished Contemporary
Ramanujan's Place in the World of Mathematics, 2020Co-Authors: Krishnaswami AlladiAbstract:P.A. MacMahon, a noted combinatorialist, wrote a famous treatise on Combinatory Analysis in which he included a discussion of the combinatorial significance of the Rogers–Ramanujan identities that neither Rogers nor Ramanujan emphasized. This article describes MacMahon’s life as a Major in the British Army stationed in India and later as an assistant to G.H. Hardy of Cambridge University. MacMahon’s mathematical connections with Ramanujan are not just due to the Rogers–Ramanujan identities, but also due the table of partitions he compiled in order to verify the famous Hardy–Ramanujan asymptotic formula for the partition function. The story of how Ramanujan discovered his famous partition congruences upon seeing this table is also described.
-
L.J. Rogers: A Contemporary of Ramanujan
Ramanujan's Place in the World of Mathematics, 2020Co-Authors: Krishnaswami AlladiAbstract:The British mathematician L.J. Rogers had talents similar to Ramanujan in the theory of q-hypergeometric series and had discovered and proved the Rogers–Ramanujan identities about two decades before Ramanujan discovered them. In this article the life and mathematical contributions of Rogers are described and the story told of how Ramanujan in England accidentally came across certain papers of Rogers, and the recognition Rogers received after Ramanujan’s rediscovery of his work.
-
Ramanujan and Partitions
Ramanujan's Place in the World of Mathematics, 2020Co-Authors: Krishnaswami AlladiAbstract:The theory of partitions, founded by Euler in the mid-eighteenth century, underwent a glorious transformation under the magic touch of Ramanujan in the early twentieth century. The discussion includes the revolutionary work of Ramanujan on partition congruences, the Hardy–Ramanujan asymptotic formula for the partition function, and the Rogers–Ramanujan identities, and story of their discoveries. Also included is a discussion of the current state of research on partitions and the continuing influence of Ramanujan’s ideas in this area.
-
A Review of “Ramanujan: Letters and Commentary”
Ramanujan's Place in the World of Mathematics, 2020Co-Authors: Krishnaswami AlladiAbstract:This is a review of the book “Ramanujan: Letters and Commentary” by Bruce Berndt and Robert Rankin, which analyzes in detail from a mathematical, historical, and social point of view, letters written to, from and about Ramanujan.
-
Fermat and Ramanujan: A Comparison
Ramanujan's Place in the World of Mathematics, 2020Co-Authors: Krishnaswami AlladiAbstract:The famous French mathematician Pierre Fermat, like Ramanujan, made notes of his observations without proofs and communicated his findings in letters to contemporaries. In this article a comparison is made of certain mathematical contributions of Fermat and Ramanujan, and of their mathematical tastes. The famous Ramanujan taxi-cab equation is discussed as a Diophantine equation in four variables having solutions, but this equation becomes the cubic version of Fermat’s Last Theorem when one of the variables is set equal to zero, in which case there are no non-trivial solutions.
Michael D. Hirschhorn - One of the best experts on this subject based on the ideXlab platform.
-
A “Difficult and Deep” Identity of Ramanujan
The Power of q, 2017Co-Authors: Michael D. HirschhornAbstract:An identity, sent by Ramanujan in his first letter to Hardy, one of a handful that convinced Hardy that Ramanujan was a “mathematician of the highest class”, is presented. The proof is begun here, and finished after the Rogers–Ramanujan continued fraction is given in Chapter 15.
-
The Rogers–Ramanujan Identities and the Rogers–Ramanujan Continued Fraction
The Power of q, 2017Co-Authors: Michael D. HirschhornAbstract:The Rogers–Ramanujan continued fraction is introduced, and two proofs of the Rogers–Ramanujan identifies are given.
-
Some Modular Equations of Ramanujan
The Power of q, 2017Co-Authors: Michael D. HirschhornAbstract:Various identities given by Ramanujan involving \(R(q), R(q^2)\) and \(R(q^3)\), where R(q) is the Rogers–Ramanujan continued fraction, are stated and proved.
-
Factorizations that involve Ramanujan’s function k(q) = r(q)r ^2(q ^2)
Acta Mathematica Sinica English Series, 2011Co-Authors: Shaun Cooper, Michael D. HirschhornAbstract:In the “lost notebook”, Ramanujan recorded infinite product expansions for $$\frac{1} {{\sqrt r }} - \left( {\frac{{1 - \sqrt 5 }} {2}} \right)\sqrt r and \frac{1} {{\sqrt r }} - \left( {\frac{{1 + \sqrt 5 }} {2}} \right)\sqrt r ,$$ , where r = r ( q ) is the Rogers-Ramanujan continued fraction. We shall give analogues of these results that involve Ramanujan’s function k = k ( q ) = r ( q ) r ^2( q ^2).
