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Rajinder K. Jagpal

Publications and source records attributed to Rajinder K. Jagpal.

12 recordsLinked to original sources
arXiv2025

Efficient application of the Voigt functions in the Fourier transform

In this work, we develop a method for rational approximation of the Fourier transform (FT) based on the real and imaginary parts of the complex error function \[ w(z) = e^{-z^2}(1 - {\rm{erf}}(-iz)) = K(x,y) + iL(x,y), \qquad z = x + iy, \] where $K(x,y)$ and $L(x,y)$ are known as the Voigt and imaginary Voigt functions, respectively. In contrast to our previous rational approximation of the FT, the expansion coefficients in this method are not dependent on the values of a sampled function. As the values of the Voigt functions remain the same, this approach can be used for rapid computation with help of look-up tables. Mathematica codes with some examples are presented.

math.GM↗
arXiv2023

A generalized series expansion of the arctangent function based on the enhanced midpoint integration

In this work we derive a generalized series expansion of the acrtangent function by using the enhanced midpoint integration (EMI). Algorithmic implementation of the generalized series expansion utilizes two-step iteration without surd and complex numbers. The computational test we performed reveals that such a generalization improves accuracy in computation of the arctangent function by many orders of the magnitude with increasing integer $M$, associated with subintervals in the EMI formula. The generalized series expansion may be promising for practical applications. It may be particularly useful in practical tasks, where extensive computations with arbitrary precision floating points are needed. The algorithmic implementation of the generalized series expansion of the arctangent function shows a rapid convergence rate in the computation of digits of $π$ in the Machin-like formulas.

math.GM↗
arXiv2022

A rational approximation of the Fourier transform by integration with exponential decay multiplier

Recently we have reported a new method of rational approximation of the sinc function obtained by sampling and the Fourier transforms. However, this method requires a trigonometric multiplier that originates from shifting property of the Fourier transform. In this work we show how to represent the Fourier transform of a function $f(t)$ in form of a ratio of two polynomials without any trigonometric multiplier. A MATLAB code showing algorithmic implementation of the proposed method for rational approximation of the Fourier transform is presented.

math.GM↗
arXiv2022

Effect of the instrument slit function on upwelling radiance from a wavelength dependent surface reflectance

The Radiance Enhancement (RE) method was introduced for efficient detection of clouds from the space. Recently, we have also reported that due to high reflectance of combustion-originated smokes, this approach can also be generalized for detection of the forest fires by retrieving and analyzing datasets collected from a space orbiting micro-spectrometer operating in the near infrared spectral range. In our previous publication, we have performed a comparison of observed and synthetic radiance spectra by developing a method for computation of surface reflectance consisting of different canopies by weighted sum based on their areal coverage. However, this approach should be justified by a method based on corresponding proportions of the upwelling radiance. The results of computations we performed in this study reveal a good match between areal coverage of canopies and the corresponding proportions of the upwelling radiance due to effect of the instrument slit function.

physics.ao-ph↗
arXiv2022

Efficient application of the Radiance Enhancement method for detection of the forest fires due to combustion-originated reflectance

The existing methods for detection of the cloud scenes are applied at relatively small spectral range within shortwave upwelling radiative wavelength flux. We have reported a new method for detection of the cloud scenes based on the Radiance Enhancement (RE). This method can be used to cover a significantly wider spectral range from 1100 nm to 1700 nm by using datasets from the space-orbiting micro-spectrometer Argus 1000. Due to high sunlight reflection of the smoke originated from the forest or field fires the proposed RE method can also be implemented for detection of combustion aerosols. This approach can be a promising technique for efficient detection and continuous monitor of the seasonal forest and field fires. To the best of our knowledge this is the first report showing how a cloud method can be generalized for efficient detection of the forest fires due to combustion-originated reflectance.

physics.ao-ph↗
arXiv2022

A two-domain MATLAB implementation for efficient computation of the Voigt/complex error function

In this work we develop a new algorithm for the efficient computation of the Voigt/complex error function. In particular, in this approach we propose a two-domain scheme where the number of the interpolation grid-points is dependent on the input parameter $y$. The error analysis we performed shows that the MATLAB implementation meets the requirements for radiative transfer applications involving the HITRAN molecular spectroscopic database. The run-time test shows that this MATLAB implementation provides rapid computation, especially at smaller ranges of the parameter $x$.

math.GM↗
arXiv2022

A new form of the Machin-like formula for pi by iteration with increasing integers

We present a new form of the Machin-like formula for $π$ that can be generated by using iteration. This form of the Machin-like formula may be promising for computation of the constant $π$ due to rapidly increasing integers at each step of the iteration. The computational test we performed shows that, with an integer $k \ge 17$, the Lehmer measure remains small and practically does not increase after $18$ steps of iteration.

math.GM↗
arXiv2021

Algorithmic determination of a large integer in the two-term Machin-like formula for pi

In our earlier publication we have shown how to compute by iteration a rational number ${u_{2,k}}$ in the two-term Machin-like formula for pi of kind $$\fracπ{4}=2^{k-1}\arctan\left(\frac{1}{u_{1,k}}\right)+\arctan\left(\frac{1}{u_{2,k}}\right),\qquad k\in \mathbb{Z},\quad k\ge 1,$$ where ${u_{1,k}}$ can be chosen as an integer ${u_{1,k}} = \left\lfloor{{a_k}/\sqrt{2-a_{k-1}}}\right\rfloor$ with nested radicals defined as ${a_k}=\sqrt{2+a_{k-1}}$ and $a_0 = 0$. In this work we report an alternative method for determination of the integer $u_{1,k}$. This approach is based on a simple iteration and does not require any irrational (surd) numbers from the set $\left\{a_k\right\}$ in computation of the integer $u_{1,k}$. Mathematica programs validating these results are presented.

math.GM↗
arXiv2021

Unconditional applicability of Lehmer's measure to the two-term Machin-like formula for $π$

Lehmer defined a measure $$ μ=\sum\limits_{j=1}^J\frac{1}{\log_{10}\left(\left|β_j\right|\right)}, $$ where the $β_j$ may be either integers or rational numbers in a Machin-like formula for $π$. When the $β_j$ are integers, Lehmer's measure can be used to determine the computational efficiency of the given Machin-like formula for $π$. However, because the computations are complicated, it is unclear if Lehmer's measure applies when one or more of the $β_j$ are rational. In this article, we develop a new algorithm for a two-term Machin-like formula for $π$ as an example of the unconditional applicability of Lehmer's measure. This approach does not involve any irrational numbers and may allow calculating $π$ rapidly by the Newton$-$Raphson iteration method for the tangent function.

math.GM↗
arXiv2021

Rapid computation of the total band radiance by using the Spectrally Integrated Voigt Function

In our earlier publication we introduced the Spectrally Integrated Voigt Function (SIVF) as an alternative to the traditional Voigt function for the HITRAN-based applications [Quine & Abrarov, JQSRT 2013]. It was shown that application of the SIVF enables us to reduce spectral resolution without loss of accuracy in computation of the spectral radiance. As a further development, in this study we present more efficient SIVF approximations derived by using a new sampling method based on incomplete cosine expansion of the sinc function [Abrarov & Quine, Appl. Math. Comput. 2015]. Since the SIVF mathematically represents the mean value integral of the Voigt function, this method accounts for area under the curve of the Voigt function. Consequently, the total band radiance, defined as the integrated spectral radiance within a given spectral region, can also retain its accuracy even at low spectral resolution. Our numerical results demonstrate that application of the SIVF may be promising for more rapid line-by-line computation in atmospheric models utilizing the HITRAN molecular spectroscopic database. Such an approach may be particularly efficient to implement a retrieval algorithm for the greenhouse gases from the NIR space data collected by Earth-orbiting micro-spectrometers like Argus 1000 for their operation in a real-time mode. The real-time mode operation of the micro-spectrometers can be advantageous for instant decision making during flight for more efficient data collection from space.

physics.gen-ph↗
arXiv2020

Radiance Enhancement and Shortwave upwelling Radiative Flux methods for efficient detection of cloud scenes

The description, imagery and interpretation of cloud scenes by remote sensing datasets from Earth-orbiting satellites have become a great debate for several decades. Presently, there are many models for cloud detection and its classifications have been reported. However, none of the existing models can efficiently detect the clouds within the small band of shortwave upwelling radiative wavelength flux (SWupRF) in the spectral range from 1100 to 1700 nm. Therefore, in order to detect the clouds more effectively, a method known as the radiance enhancement (RE) can be implemented (Siddiqui et al., 2015). This article proposes new approaches how with RE and SWupRF methods to distinguish cloud scenes by space orbiting Argus 1000 micro-spectrometer utilizing the GENSPECT line-by-line radiative transfer model (Quine and Drummond, 2002; Siddiqui, 2017). This RE approach can also be used within the selected wavelength band for the detection of combustion-originated aerosols due to seasonal forest fires.

physics.space-ph↗