On the Applications of Ramanujan Sums in Signal Processing

Abstract

In 1918, Sri. Srinivasa Ramanujan, one of the world s great mathematical geniuses newlineproposed a summation, known today as Ramanujan sums (RS). Ramanujan used these newlinesums to represent many arithmetic functions. These sums have a significant role in newlinenumber theory. A decade later, after the proposal of these sums, RS has been introduced into the signal processing domain, specifically in period extraction. The main motivation for this thesis originates from the recent development in the area of RS transforms and newlinesubspaces. This thesis aims at a better understanding of the capabilities of RS from the newlineview point of signal processing. newlineTransmultiplexer (TMUX) is a key component in multicarrier communication newlinesystems. TMUX is a multi-input multi-output system, that allows users to share a newlinecommon channel. In this thesis, we propose a perfect reconstruction transmultiplexer newlinefilter bank design based on RS. A key feature of this filter bank is that the filter coefficients newlineare integer-valued, which enables them to be computationally efficient in hardware newlineimplementation. The proposed filterbank has been also used to transmultiplex audio, newlineimage, and electrocardiogram (ECG) signals. newlinePeriod extraction of signals is a classic problem in digital signal processing. Accurate identification of periods in a composite signal is important in different fields like audio and speech signal processing, DNA sequence analysis, and biosignal processing. A signal newlineperiodicity detection method based on RS is proposed in this thesis. This periodicity newlinedetection method employs a two-tier approach based on autocorrelation and RS subspace newlineprojection. The potential periods are chosen based on the autocorrelation values of newlinethe signal. The signal is projected into RS subspaces corresponding to these potential newlineperiods for the selection of final periods. The selection of candidate periods based on the autocorrelation function reduces the number of RS subspace projections which saves the computational time. newline