Abstract— An efficient and reliable method to extrapolate bandlimited signals up to an arbitrarily high range of frequencies is proposed. The orthogonal properties of linear prolate spheroidal wave functions (PSWFs), also known as Slepian functions, are exploited to form an orthogonal basis set needed for synthesis. Higher order piecewise polynomial approximation is used for the calculation of overlap integral required for obtaining the expansion coefficients accurately with very high precision. A PSWFs set having a fixed Slepian frequency is utilized for performing extrapolation. Numerical results of extrapolation of some standard test signals using our algorithm are presented, compared, discussed, and some striking conclusions are made. Index Terms—Signal extrapolation; Slepian series; Bandlimited signals; Linear prolate spheroidal wave functions; Overlap integral I. INTRODUCTION ANDLIMITED signal extrapolation is a well-known problem in signal analysis for which several iterative and non-iterative solutions have been proposed in the past. Recent advancements in this field are mainly based on using Slepian functions [1-4] (also known as prolate spheroidal wave functions; henceforth abbreviated as PSWFs) as an orthogonal series expansion of the bandlimited signal. Improved numerical techniques and superior computational power have aided in the numerical evaluation of these functions, which otherwise would have been an insurmountable task. Significant research has been going on especially within the past decade on Slepian functions [5-8]. In [9], Senay et al. proposed sampling and reconstruction of bandlimited as well as non-bandlimited signals using Slepian functions. They discussed the idea of modifying the Whittaker-Shannon sampling theory by replacing the sinc basis by Slepian functions for reconstruction of signals. Further to this, in [10,11], they showed signal reconstruction using non-uniform sampling and level-crossing sampling with Slepian functions. Considering signal extrapolation to be the subject of this paper and not just reconstruction or interpolation, we will Manuscript received July 19, 2013; revised August 9, 2013. This work was supported in part by Applied Science in Photonics and Innovative Research in Engineering (ASPIRE), a program under Collaborative Research and Training Experience (CREATE) program, funded by Natural Sciences and Engineering Research Council (NSERC) of Canada. 1 Amal Devasia is a MASc. candidate in the Department of Electrical and Computer Engineering, Dalhousie University, Halifax, NS, B3H 4R2, Canada (e-mail: amal.devasia@dal.ca). 2 Michael Cada is with the Department of Electrical and Computer Engineering, Dalhousie University, Halifax, NS, B3H 4R2, Canada (e- mail: michael.cada@dal.ca). shift our focus to the recent advancements in this context. While we consider the signals to be bandlimited in the Fourier transform domain, much attention has recently been on extrapolation of signals bandlimited in linear canonical transform (LCT) domain, this being a four-parameter family of linear integral transform [12,13] that generalizes Fourier transform as one of its special cases. For extrapolation of LCT bandlimited signals, several iterative and non-iterative algorithms have been proposed [14-17]. Most of the iterative algorithms are centered on modifying the Gerchberg- Papoulis (GP) algorithm [18,19] that relies on successive reduction of error energy. Although theoretical convergence of the result has been shown, there is still some uncertainty associated with the swiftness with which this is achieved. With respect to the non-iterative algorithms proposed in [15], the authors themselves admit that the extrapolation could become unstable with an increase in the number of observations. A comparison of the extrapolation of an LCT bandlimited signal, using an iterative GP algorithm and another algorithm based on signal expansion into a series of generalized PSWFs [17] is presented in [16]. The comparison showed better results for the iterative method (proposed by [16]) over the one described in [17], in terms of the normalized mean square error (NMSE). Gosse, in [20], performed Fourier bandlimited signal extrapolation by handling lower and higher frequencies of the signal separately. He used PSWFs for extrapolating lower frequency components while the higher frequencies were dealt with compressive sampling [21,22] algorithms. The efficiency of the proposed method was highly dependent on the correlation between low and high frequencies in the signal (it should be weak for better results), the existence of a sparse representation of higher frequencies in the Fourier basis, and on a reasonable choice of extrapolation domain. In this paper, we propose a non-iterative and simple method for bandlimited signal extrapolation valid up to an arbitrarily high range of frequencies using Slepian functions. Although we concentrate mainly on Fourier bandlimited signals, it however might also be applied for LCT bandlimited cases as is shown in one of our results below. Several comparisons are made with the results obtained in earlier related publications. They show that, within the prescribed bandwidth, the proposed method is far superior over several other methods referenced in this paper. PSWFs for analysis purposes need to be computed accurately and with rather high precision. Here, we rely on a proprietary algorithm developed theoretically and implemented numerically by Cada [23], for accurately generating the linear prolate functions (one-dimensional PSWFs, henceforth abbreviated as LPFs) set with desired high precision. Once the LPFs set is obtained with the Extrapolation of Bandlimited Signals Using Slepian Functions Amal Devasia 1 , Member, IAENG and Michael Cada 2 B Proceedings of the World Congress on Engineering and Computer Science 2013 Vol I WCECS 2013, 23-25 October, 2013, San Francisco, USA ISBN: 978-988-19252-3-7 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) WCECS 2013