Exploring Ramanujan Sums in Digital Signal Processing.

Date of Submission

December 2017

Date of Award

Winter 12-12-2018

Institute Name (Publisher)

Indian Statistical Institute

Document Type

Master's Dissertation

Degree Name

Master of Technology

Subject Name

Computer Science


Computer Vision and Pattern Recognition Unit (CVPR-Kolkata)


Palit, Sarbani (CVPR-Kolkata; ISI)

Abstract (Summary of the Work)

The problem of estimation of period lengths from a “mixture” of periodic signals is a well-studied topic in the field of digital signal processing. Various methods have been proposed in the past, to extract signal periods. The extraction of periods of sinosoidal components in particular has attracted much attention. The more general problem of period determination when the mixture is of periodic but non-sinosoidal signals was first addressed in 1997 to separate foetal ECG from the maternal ECG. The algorithm used the Singular Value Decomposition method, but it had the stringent requirement of a large difference in the strengths of the individual signals. That is, if we have two signals having similar amplitude levels added together, the SVD method cannot be used. (Since the foetal ECG is generally much weaker than the maternal one, this method was sufficient for the problem.) Recently, using a summation proposed by the great Indian mathematician S. Ramanujan in 1918, certain methods have been developed. These algorithms also have good noise performance, which was usually absent in the older methods.We examine these algorithms in the first part of the thesis. We also point out some of the drawbacks of a few of the older methods. While one of the main problems of signal processing is the identification of periods from a signal mixture, another equally important problem is their separation, that is, reconstructing the original signals, the “mixture” of which produced the signal under consideration. We will consider two such “mixing” operations, addition and multiplication in this thesis. There are methods such as filtering which have been used in the past to reconstruct the signals. Even Singular Value Decomposition has been used for this purpose. We propose an alternative method to “almost” separate the signals given a few conditions. The conditions are the mutual co-primality amongst the signal periods and a signal length of the LCM of the periods. We prove that “exact” reconstruction is never possible, for any given added or point by point multiplied signals. We then go on to show that using very little computation time and hardware support, we can very simply get back the original signals, with nominal changes. For the additive case, we get the initial signal with an offset and for the multiplicative case, we get a scaled version. We also show that this method gives better noise performance for white noise, under some conditions. Finally we show how, under few more strict conditions, even exact reconstruction of the initial signals in the multiplicative case is possible.


ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843092

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Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.



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