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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.

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. Moreover, unlike other functions, they are also complete and orthogonal over the infinite interval. An additional property is that the finite Fourier transform (FT) of a linear prolate function is proportional to the same prolate function. Although there are other functions which are their own infinite Fourier transform, only the prolate functions enjoy the property for the finite transform: this property uniquely defines the prolate functions [

Specifically, in the research area of bandlimited signal extrapolation, there have been contributions with iterative and non-iterative algorithms for extrapolation of signals in the LCT (linear canonical transform) domain that is a generalization of the Fourier transform. The challenges of convergence of algorithms based on the Gerchberg-Papoulis (GP) algorithm [

In this paper, we benefit from a proprietary algorithm developed theoretically and implemented numerically by Cada [

As sets of bandlimited functions, orthogonal on the finite interval and orthonormal on the infinite interval, the linear prolate functions

the representation is valid for all

There is an alternative way to derive the coefficients

The latest expression for

Accurate computing of

individual value is very close to 1, and the series

challenging problem of high-precision numerical integration with an absolute necessity of having

Our main objective in signal extrapolation using linear prolate functions aims to take advantage of a generalized expression stated in [

Substituting

with

Thanks to the significant generalization for the calculation of coefficients

Specifically, we start from the assumption that the function

with

A LPFs set with bandwidth parameter

In order to test the proposed approach for signal extrapolation as described in Section 3, we consider the ideal case first. This assumption means that at each iteration of the extrapolation, the function

tional interval which is up to the

We consider a more realistic case when the function

obtained by setting

In

The difference quotients in

exact | 36 iterations ( | 16 iterations ( | 1 iteration (no shift) | |
---|---|---|---|---|

6.83248 | 6.83238 | 6.36935 | ||

6.49618 | 6.49601 | 5.82323 | ||

6.11953 | 6.11922 | 5.18117 | ||

5.71649 | 5.71596 | 4.45234 | ||

5.30341 | 5.30248 | 3.651 |

exact | 36 iterations ( | 16 iterations ( | |
---|---|---|---|

3.8751 | 3.61211 | ||

3.8203 | 3.45436 | ||

3.67135 | 3.16975 | ||

3.41957 | 2.74234 | ||

0.532228 | −2.40967 |

The proposed method is subject to an inherent series truncation error. Its mean squared error expression is the following, after an extrapolation interval

The first term in the summation in (9) represents the error in the fit of

the extrapolation interval and then depends on the quantity

sufficiently large

the corresponding

It is clear from

NMSE-36 iterations ( | NMSE-16 iterations ( | NMSE-1 iteration (no shift) | |
---|---|---|---|

notation in (8). The extrapolation can be optimized by reconstructing the extra- polated function

In this paper, we have proposed and implemented a low complexity iterative algorithm for bandlimited signal extrapolation based on orthogonal projections over real-valued eigenvectors: the linear prolate functions. The method is valid for an arbitrary large range of frequencies with immediate applications in signal processing. The main contribution of our work is a theoretical derivation such that given a truncated series (up to a selectable value) of prolate functions, it is possible to extrapolate the bandlimited function (initially known in a limited time interval) elsewhere if each extrapolated portion of the function is subject only to moderate series truncation errors. These errors are controllable by the depth of extrapolation at each iteration. By doing so and with the aim of finding an alternative solution to the initial problem of implementing an accurate summation of infinite terms, we have investigated a property of the signal de- composition formula based on LPFs according to which the integration interval does not need to be symmetric with respect to the origin while time-shifted prolate functions are used in the summation. Also, we have investigated the effects of different sources of errors by implementing and analyzing the iterative algorithm as a generalization of the special case presented in [

Valente, D., Cada, M. and Ilow, J. (2017) Linear Prolate Functions for Signal Extrapolation with Time Shift. Applied Mathematics, 8, 417-427. https://doi.org/10.4236/am.2017.84034