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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 ψ n ( c , t ) can be successfully used for the expansion of a generally complex, bandlimited function f ( t ) :

f ( t ) = ∑ n = 0 ∞ a n ψ n ( c , t ) (1)

the representation is valid for all t , the bandwidth parameter is c = t 0 Ω 0 where Ω 0 represents the finite bandwidth or a cutoff frequency, and t 0 is the time interval. The function f ( t ) is supposed to be Ω 0 -bandlimited. Adopting the criterion of a minimized mean-square error, the expansion coefficients a n in (1) are given by:

a n = ∫ − ∞ ∞ f ( t ) ψ n ( c , t ) d t (2)

There is an alternative way to derive the coefficients a n using only the values of f ( t ) within the finite observation interval [ − t 0 , t 0 ] and introducing the eigenvalues λ n ( c ) . After multiplying (1) by ψ m ( c , t ) , integrating as reported in [

a n = [ λ n ( c ) ] − 1 ∫ − t 0 t 0 f ( t ) ψ n ( c , t ) d t (3)

The latest expression for { a n } together with (1) states that when the bandlimited function f ( t ) is known over a finite interval of extend 2 t 0 then f ( t ) is theoretically known everywhere if one can accurately calculate the coefficients a n for n → ∞ , the functions ψ n ( c , t ) and the eigenvalues λ n ( c ) . λ n ( c ) can be regarded as the index of energy concentration of each function ψ n in [ − t 0 , t 0 ] . Therefore:

f ( t ) ≃ ∑ n = 0 N [ λ n ( c ) ] − 1 ψ n ( c , t ) ∫ − t 0 t 0 f ( t ′ ) ψ n ( c , t ′ ) d t ′ (4)

Accurate computing of ψ n , λ n and a n for n > 2 c / π (where N c r i t = 2 c / π is known as the critical value) turns the orthogonal expansion expression presented in (1) into a signal extrapolation problem. Indeed, for any LPFs set with a fixed c , the energy concentration of the functions within [ − t 0 , t 0 ] decreases as the order n increases and for n = N c r i t , the signal’s maximum concentration reaches the boundary of the observation interval. Hence the summation of { λ n ( c ) } is mostly determined by the first N c r i t terms whose

individual value is very close to 1, and the series ∑ n = 0 ∞ λ n ( c ) converges to a finite value ( 2 c / π ) , as extensively analyzed in [

challenging problem of high-precision numerical integration with an absolute necessity of having ψ n ( c , t ) with a high precision as well [

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

f ( t ) ≃ ∑ n = 0 N [ λ n ( c ) ] − 1 ψ n ( c , t − T ) ∫ T − t 0 T + t 0 f ( t ′ ) ψ n ( c , t ′ − T ) d t ′ (5)

Substituting y = t ′ − T , the following expression for f ( t ) is obtained:

f ( t ) ≃ ∑ n = 0 N [ λ n ( c ) ] − 1 ψ n ( c , t − T ) ∫ − t 0 t 0 f ( y + T ) ψ n ( c , y ) d y (6)

with

a n ( T ) = 1 λ n ( c ) ∫ − t 0 t 0 f ( y + T ) ψ n ( c , y ) d y (7)

Thanks to the significant generalization for the calculation of coefficients { a n } , one can use (6) to perform the signal extrapolation on f ( t ) beyond the observation interval [ − t 0 , t 0 ] using an iterative approach.

Specifically, we start from the assumption that the function f ( t ) is perfectly known in the interval [ − t 0 , t 0 ] . We use (6) with T = 0 to extrapolate the signal by an interval Δ t ( 0 ) using already proposed algorithms to obtain accurate calculations of a n ( 0 ) till N = N ( 0 ) > 2 c / π . Instead of pursuing the more challenging computing for N ≫ 2 c / π , Formula (6) is re-applied for T = Δ t ( 0 ) to extrapolate the signal by an additional interval Δ t ( T ) via accurate calcula- tions of a n ( T ) to N ( T ) > 2 c / π . The procedure is then repeated for the i -th iteration and up to the number of iterations that has been set. Specifically, at each iteration i , we form the function which becomes the input for iteration i + 1 :

f i ( t ) = f ˜ i ( t ) + [ f i − 1 ( t ) − f ˜ i ( t ) ] p Δ t ( t ) = { f i − 1 ( t ) , − t 0 + i Δ t ≤ t ≤ t 0 + ( i − 1 ) Δ t f ˜ i ( t ) , t 0 + ( i − 1 ) Δ t < t ≤ t 0 + i Δ t (8)

with f ˜ i ( t ) being defined in (6). For the sake of simplicity Δ t is chosen to be the same at every iteration. Also,

p Δ t ( t ) = { 1, − t 0 + i Δ t ≤ t ≤ t 0 + ( i − 1 ) Δ t 0, otherwise

A LPFs set with bandwidth parameter c = 20 π and t 0 = 1 is used as the orthogonal basis for the proposed extrapolation method. The functions are discretized in time at a sampling rate of 0.001 for numerical implementation and each discrete sample has a high numerical precision greater than 100 digits. The software Mathematica characterized by high precision computing has been used for the simulations. Extrapolation is carried out on the Ω 0 -bandlimited test function shown in

f ( t ) = c o s ( 2 π t − ( π 11 ) ) + c o s ( 2 π t 7 ) − c o s ( 3 π t 2 )

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 f ( t + T ) in the integral in (6) is known in Mathematica user-defined precision.

tional interval which is up to the 60 % of half of the time range where the function is known. The presence of the truncation error is discussed in Section 5.

We consider a more realistic case when the function f ( t + T ) in the integral in (6) is known in Mathematica user-defined precision only for T = 0 .

obtained by setting i = 1 and hence T = 0 , and reproduces results presented in [

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 T e :

E T = ∫ − t 0 + t 0 + T e | f ( t ) − f N ( t ) | 2 d t = ∫ − t 0 + t 0 | f ( t ) − f N ( t ) | 2 d t + ∫ + t 0 + t 0 + T e | f ( t ) − f N ( t ) | 2 d t (9)

= ∑ n = N + 1 ∞ a n 2 λ n ( c ) + ∫ + t 0 + t 0 + T e | f ( t ) − f N ( t ) | 2 d t (10)

The first term in the summation in (9) represents the error in the fit of f N ( t ) (defined in (6)) to f ( t ) within the interval [ − t 0 , t 0 ] . Specifically, as reported in [

the extrapolation interval and then depends on the quantity T e . It is reasonable to consider the first term not critical for truncation values N above 2 c / π , when the energy factor λ n ( c ) rapidly approaches zero (an example is shown in

sufficiently large n , the integral ( ∫ − t 0 t 0 f ( t ) ψ n ( c , t ) d t ) 2 tends to zero faster than

the corresponding λ n ( c ) at the denominator of the products a n 2 λ n ( c ) . This consideration motivated our work and indirectly highlights again that calculating accurate coefficients a n for large n is critical since both overlap integrals and eigenvalues become very small quantities. This has been a known problem since the 60’s of the last century. The critical aspect is an accurate calculation of ψ n ( c , t ) , which is now possible [

NMSE = ‖ f e − f ‖ 2 ‖ f ‖ 2 = ∫ − t 0 + T e t 0 + T e | f i ( t ) − f ( t ) | 2 d t ∫ − t 0 + T e t 0 + T e | f ( t ) | 2 d t (11)

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 f i ( t ) as concatenation of the known function with segments of optimum estimates. However, we still observe numerical inaccuracies oc- curring at the points of concatenations, which is presently under investigation.

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