Abstract
We propose a low complexity iterative algorithm for band limited signal extrapolation. The extrapolation method is based on the decomposition of finite segments of the signal via truncated series of real-valued linear prolate functions. Our theoretical derivation shows that given a truncated series (up to a selectable value) of prolate functions, it is possible to extrapolate the band limited function elsewhere if each extrapolated portion of the function is subject only to moderate truncation errors that we quantify in this paper. The effects of different sources of errors have been analyzed via extensive simulations. We have investigated a property of the signal decomposition formula based on linear prolate functions whereby the integration interval does not need to be symmetric with respect to the origin while time-shifted prolate functions are used in the series.
Highlights
In the early 60’s David Slepian and his colleagues discovered the bandlimited function that is maximally concentrated, in the mean-square sense, within a given time interval; this function is the prolate spheroidal wave function (PSWF) of zero-order.The linear prolate functions (LPFs) are the one-dimensional version of the prolate spheroidal functions and they form sets of bandlimited functions which are orthogonal and complete over a finite interval
Our theoretical derivation shows that given a truncated series of prolate functions, it is possible to extrapolate the band limited function elsewhere if each extrapolated portion of the function is subject only to moderate truncation errors that we quantify in this paper
We have investigated a property of the signal decomposition formula based on linear prolate functions whereby the integration interval does not need to be symmetric with respect to the origin while time-shifted prolate functions are used in the series
Summary
In the early 60’s David Slepian and his colleagues discovered the bandlimited function that is maximally concentrated, in the mean-square sense, within a given time interval; this function is the prolate spheroidal wave function (PSWF) of zero-order. The linear prolate functions (LPFs) are the one-dimensional version of the prolate spheroidal functions and they form sets of bandlimited functions which are orthogonal and complete over a finite interval. Approaches based on the use of the prolate spheroidal wave functions [7] need to provide efficient ways to compute the prolate functions. We benefit from a proprietary algorithm developed theoretically and implemented numerically by Cada [8], for accurate generation of linear prolate functions with desired high precision to use LPFs for signal extrapolation. We introduces the basics of signal expansion using the linear prolate functions in Section 2; Section 3 presents our approach to signal extrapolation based on LPFs. In Section 4 and Section 5 simulation results, error analysis and numerical examples are presented and discussed.
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