<p>In this manuscript we have proposed a modified Newton-Raphson algorithm to obtain efficient estimators of the elementary chirp signal parameters in presence of stationary noise. The asymptotic distribution of the least squares estimators which minimizes the residual sum of squares has been provided. The least squares optimization function is a highly non-linear function in its chirp parameters and an iterative procedure is needed to optimize the criterion function. The Newton-Raphson algorithm does not work well for this kind of models. We have used Newton-Raphson algorithm by a step factor modification. The proposed algorithm starts with an initial estimator of order <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="34_2025_3165_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$O_p(n^{-2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>O</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. This modified Newton-Raphson method provides estimators with the same rate of convergence <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="34_2025_3165_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$O_p(n^{-5/2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>O</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow> <mo>-</mo> <mn>5</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> as the least squares estimators. Further, the asymptotic variances of the proposed estimators are <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="34_2025_3165_Article_IEq3.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="8" /> </InlineMediaObject> <EquationSource Format="TEX">$\frac{4}{9}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>4</mn> <mn>9</mn> </mfrac> </math></EquationSource> </InlineEquation> times smaller than those of the least squares estimators. Extensive numerical experiments are conducted to study the performances of the proposed estimators and compared with the least squares estimators and fixed iteration efficient estimators using mean squared errors. The analyses of two sonar data, one sonar rocks data and one sonar mines data are demonstrated using the elementary chirp model and estimating the parameters using the proposed modified Newton-Raphson algorithm.</p>

Estimating Parameters of Elementary Chirp Model Using Modified Newton-Raphson Algorithm

期刊论文
发表日期：

2025年5月20日

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Swagata Nandi,
Debasis Kundu

摘要

In this manuscript we have proposed a modified Newton-Raphson algorithm to obtain efficient estimators of the elementary chirp signal parameters in presence of stationary noise. The asymptotic distribution of the least squares estimators which minimizes the residual sum of squares has been provided. The least squares optimization function is a highly non-linear function in its chirp parameters and an iterative procedure is needed to optimize the criterion function. The Newton-Raphson algorithm does not work well for this kind of models. We have used Newton-Raphson algorithm by a step factor modification. The proposed algorithm starts with an initial estimator of order $O_p(n^{-2})$ $O_{p} (n^{- 2})$ . This modified Newton-Raphson method provides estimators with the same rate of convergence $O_p(n^{-5/2})$ $O_{p} (n^{- 5 / 2})$ as the least squares estimators. Further, the asymptotic variances of the proposed estimators are $\frac{4}{9}$ $\frac{4}{9}$ times smaller than those of the least squares estimators. Extensive numerical experiments are conducted to study the performances of the proposed estimators and compared with the least squares estimators and fixed iteration efficient estimators using mean squared errors. The analyses of two sonar data, one sonar rocks data and one sonar mines data are demonstrated using the elementary chirp model and estimating the parameters using the proposed modified Newton-Raphson algorithm.