On the Reduced Noise Sensitivity of a New Fourier Transformation Algorithm
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In this study, a new inversion method is presented for performing one-dimensional Fourier transform, which shows highly robust behavior against noises. As the Fourier transformation is linear, the data noise is also transformed to the frequency domain making the operation noise sensitive especially in case of non-Gaussian noise distribution. In the field of inverse problem theory it is well known that there are numerous procedures for noise rejection, so if the Fourier transformation is formulated as an inverse problem these tools can be used to reduce the noise sensitivity. It was demonstrated in many case studies that the method of most frequent value provides useful weights to increase the noise rejection capability of geophysical inversion methods. Following the basis of the latter method the Fourier transform is formulated as an iteratively reweighted least squares problem using Steiner’s weights. Series expansion was applied to the discretization of the continuous functions of the complex spectrum. It is shown that the Jacobian matrix of the inverse problem can be calculated as the inverse Fourier transform of the basis functions used in the series expansion. To avoid the calculation of the complex integral a set of basis functions being eigenfunctions of the inverse Fourier transform is produced. This procedure leads to the modified Hermite functions and results in quick and robust inversion-based Fourier transformation method. The numerical tests of the procedure show that the noise sensitivity can be reduced around an order of magnitude compared to the traditional discrete Fourier transform.
KeywordsNoise sensitivity Noise rejection Fourier transformation Series expansion-based inversion Hermite functions
The investigations were based on the previous research work of the TÁMOP (Project No. 4.2.1.B-10/2/KONV-2010-0001) Center of Excellence. The research was partly supported by the Hungarian Research Fund OTKA (Project No. K109441). The second author was supported from the TÁMOP 4.2.4. A/2-11-1-2012-0001 National Excellence Program—Elaborating and operating an inland student and researcher personal support system convergence program. The project was subsidized by the European Union and co-financed by the European Social Fund. The authors thank the editor and two anonymous reviewers for their constructive comments, which helped us to improve the manuscript.
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