Using Orthogonal Legendre Polynomials for Filtering Noisy Signals over a Limited Interval in Coordinate Space

Authors V.M. Fitio1, H.A. Petrovska1, Y.V. Bobitski1,2

1Lviv Polytechnic National University, 12, S. Bandera St., 71013 Lviv, Ukraine

2College of natural sciences Institute of Physics University of Rzeszow, St. Pigonia 1, 35-310, Rzeszow, Poland

Issue Volume 14, Year 2022, Number 1
Dates Received 15 November 2021; revised manuscript received 25 February 2022; published online 28 February 2022
Citation V.M. Fitio, H.A. Petrovska, Y.V. Bobitski, J. Nano- Electron. Phys. 14 No 1, 01032 (2022)
PACS Number(s) 42.40.i, 42.40.Kw, 42.30.d, 42.30.d
Keywords Digital holography, Digital holographic interferometry, Noise (3) , Signal filtering, Legendre polynomials, Orthogonality of polynomials.

The paper shows that orthogonal Legendre polynomials in the interval [– 1, 1] can be effectively used to filter noisy signals, including filtering interferograms and phase maps in digital holographic interferometry. They can also be used to effectively approximate harmonic signals, and the approximation accuracy increases with the number of polynomials used. Filtering is based on the use of the optimal number of Legendre polynomials when approximating the signal. It is impractical to filter directly digital holograms and phase maps, since in this case it is necessary to use several hundred polynomials, which significantly increases the time of numerical calculations. Therefore, in digital holographic interferometry, it is necessary to filter directly the field amplitudes calculated from the digital hologram. Interferograms and phase maps can be calculated using filtered field amplitudes for different states of the object under study. If for the real or imaginary part of the signal the minimum distance between adjacent local minima (maxima) is equal to ∆l, then for a satisfactory approximation of such a signal by Legendre polynomials, 6/∆l polynomials are required. The efficiency of filtering by Legendre polynomials is higher if the noise signal contains harmonic components with a frequency greater than the frequency of the useful signal.

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