Small Argument Approximations Research Articles

Spherical expansions of sound radiation from a finite line source with reduced error.

The sound pressure due to a finite length line source is calculated using a spherical harmonic expansion. An improved expansion is then derived which uses a small argument approximation for the product of a spherical Bessel and Hankel function and exploits the fact that the approximation also works when the order is large compared to the argument. The higher order terms can then be approximated using the equivalent potential integral. This method can be applied to any calculation of radiated fields using spherical expansions. The improved expansion is most useful for the calculation of the sound pressure in the near field.

A General Theory to Determine the Exact Radiated Power, Directivity, and Radiation Resistance of a Line-Source Radiator

Instead of integrating the power pattern function directly and discounting the effect of the element factor to find the total radiated power of a line source, we show herein that an exact analytical expression for the radiated power can be obtained by singular knowledge of the source’s current distribution. The expression is found by using the Fourier principles associated with convolution integrals, autocorrelation integrals, Parseval’s Identity, and the Helmholtz operator in one dimension. Knowledge of the exact radiated power is used to find the expressions for directivity and radiation resistance. Furthermore, we show that the efficiency of the line source is explicitly a function of the relative proportion of two autocorrelation integrals. This newly developed theory is validated by considering large and small argument approximations in which closed-form results are readily known. Additional validation is obtained by comparing the exact result with the data obtained from numerical integration of the power pattern function. Examples associated with the half-wave dipole, cosine distribution, cosine-squared distribution, generalized dipole, triangular distribution, and the uniform distribution are provided. Other than the half-wave and generalized dipole solutions, we believe that the expressions obtained in this investigation have never been reported in the open literature. This is particularly true for the uniform line source in which the classical asymptotic result for the directivity (i.e., $2L/\lambda$ ) is shown to be deficient.

Asymptotic Error Rate Analysis of Selection Combining on Generalized Correlated Nakagami-m Channels

From the canonical representation of the Marcum Q-function, a simple and highly accurate small argument approximation for the Marcum Q-function is first obtained. Utilizing this approximation, we derive the asymptotic error rate and outage probability expressions for multi-branch selection combining over generalized correlated Nakagami-m fading channels. These closed-form solutions can be used to provide rapid and accurate estimation of the error rates and outage probabilities in large signal-to-noise ratio regimes. It is shown that asymptotic error rates and outage probability over correlated Nakagami-m branches can be obtained by scaling the asymptotic error rates and outage probability over independent branches with a factor detm(M), where det(M) is the determinant of matrix M whose elements are the square root of the corresponding elements in the branch power covariance coefficient matrix.

Accurate simple closed-form approximations to rayleigh sum distributions and densities

Sums of Rayleigh random variables occur extensively in wireless communications. A closed-form expression does not exist for the sum distribution and consequently, it is often evaluated numerically or approximated. A widely used small argument approximation for the density is shown to be inaccurate for medium and large values of the argument. Highly accurate, simple closed-form expressions for the sum distributions and densities are presented. These approximations are valid for a wide range of argument values and number of summands.

New approximations to J/sub 0/ and J/sub 1/ Bessel functions

New approximate solutions to the 0th- and 1st order Bessel functions of the first kind are derived. The formulations are based upon using a new integral with no previously known solution. The new integral in the limiting case is identical to the 0th-order Bessel function integral. It is solved in closed form, and the solution is expressed as a simple even order polynomial with integer coefficients. The polynomial coefficients are all of integer value. The 1st-order Bessel function approximation can then be found through a simple derivative. Comparisons are made between the exact solution, classic solutions, and the new approximation. The new approximation proves to be much more accurate than the classic small argument approximation. It is also sufficiently accurate to bridge the gap between the classic large and small argument approximations and has potential applications in allowing one to analytically evaluate integrals containing Bessel functions. >

Technical memorandum: approximate cross-polar characteristics for microstrip patch antennas

Simple, approximate formulas describing the peak cross-polar characteristics of circular and rectangular microstrip patches are derived from the governing equations using small argument approximations. Results from the formulas compare favourably with those obtained elsewhere using more stringent numerical methods. In the case of the circular patch, a novel approximation is employed to circumvent the use of Bessel functions.

The aperture admittance of a circumferential slot in a circular cylinder

A circumferential slot in an infinitely long cylinder when fed with a tangential electric field is considered. Parseval's theorem, in which Fourier integrals appear, is used to express the aperture admittance as the integral from zero to infinity of an infinite series. Recognizing that the first two terms of this infinite series have a nonintegrable singularity when the variable of integration w is equal to the wavenumber k, the authors distort slightly the contour of integration to avoid k. The terms of the infinite series are then approximated when w is close to k. A combination of numerical and analytical schemes employing both large and small argument approximations of Hankel functions is used to perform the integration. It has been found that these integration schemes work very well, and a simple and fast algorithm has resulted. >

Point inertance of an infinite plate with respect to a force acting in its plane

A corrected expression is given for the point inertance of an infinite plate with respect to a force acting in its plane, as presented by Nikiforov [1]. It is noted that the small argument approximation is no longer valid. However, the large argument approximation remains the same.