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Signal Processing

Volume 120, March 2016, Pages 8-12
Signal Processing

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Pole position problem for Meixner filters

https://doi.org/10.1016/j.sigpro.2015.08.009Get rights and content

Highlights

  • We solved the two-parameter pole position problem for the Meixner filters with an extra parameter.

  • We extended the theorem provided for the discrete Laguerre filters to the Meixner filters.

  • We proposed an approach to optimizing the extra parameter using connection coefficients method.

Abstract

This paper is motivated by previous research that demonstrates the importance of the explicit solution to the pole position problem. The main purpose of this paper is to solve the two-parameter pole position problem for the Meixner filters with an extra parameter. To attain the objective, we extended the theoretical results provided for the discrete Laguerre filters and proposed the approach to optimizing the extra parameter using the connection coefficients method. The present research yields a series of computational experiments to test this approach, to verify the theoretical results, and to point up the positive outcomes of using the Meixner filters.

Introduction

Seydnejad and Ebrahimi [1] stress the importance of an explicit solution to the pole position problem in broadband beamforming. This study indicates that finding the optimal pole of the discrete Laguerre filters enables to obtain a better beamformer response and, by that, to ensure the output without any distortion. According to an extensive body of knowledge on using the Meixner-like functions in signal processing applications [2], [3], [4], [5], [6], the Meixner filters can be an effective alternative to the discrete Laguerre filters in attempt to improve the results to a greater degree. This fact is attributed to an extra parameter that permits providing better series expansions [2]. As a consequence, a number of studies have set out the solutions to the extra parameter optimization problem [7], [8]. But, there seemed little attention to be drawn to the explicit solution to this problem. Thus, the present study raises the issue of solving the two-parameter pole position problem drawing primarily on the work [9] that provides the theoretical outcomes in solving the pole position problem for the discrete Laguerre filters that needs to be extended. In addition, this research casts doubt on Brinker׳s statement [2] that “Meixner filters are not realizable since Meixner functions do not have a rational z-transform” putting forward rational Meixner filters as rational Laplace transforms of the generalized Laguerre functions [9] converted into z-transforms.

Section snippets

Problem statement

For each fixed pole γΓ, where Γ={γR:γ>0}, and α{αR:α>1}, kN0 in the Hilbert space L2(R+), the generalized Laguerre functions Lk(τ,γ,α) given by exp(γτ/2)Lk(γτ,α)k=0 are orthogonal with respect to the nonnegative weight function ω(τ,α)=τα over the interval τR+ and the norm Lk(γ,α)2=Γ(k+α+1)/(k!γα+1). Therefore, the Laplace transform Λk(s,γ,α) of Lk(τ,γ,α) can be defined as [9]Λk(s,γ,α)=(γs+γ/2)α+1(sγ/2s+γ/2)k.The filters Λk(s,γ,α) can be mapped onto a rational z-transform Gk(z,ξ,α)

Computational experiments

To test the proposed approach and to bear out the theoretical outcomes, we now provide a couple of examples chosen from the literature. The first example was previously considered in [9] to illustrate the pole position problem solving results for the discrete Laguerre filters. So, for the sake of comparison, we provided the same example to prove the validity of the proposed extension to the Meixner filters. The choice of the second example is motivated by the literature review. As noted in [2],

Conclusion

In this study, we posed the two-parameter pole position problem for the Meixner filters. To solve this problem, we first extended the theoretical findings in solving the pole position problem for the discrete Laguerre filters to the case of the proposed Meixner filters. Then, we proposed an approach to solving (ξ,α)=argmin|ξ|<1α>1Δm(ξ,α) based on the connection coefficients method. To test this approach, to verify the theoretical results, and to point up the positive outcomes of using the

Acknowledgment

The authors would like to thank the reviewers for the valuable comments and suggestions. This work was supported by the Ministry of Education and Science of the Russian Federation grant 074-U01.

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