Incomplete Lipschitz-Hankel Integrals Research Articles

Analytic Expressions and Bounds for Special Functions and Applications in Communication Theory

This paper is devoted to the derivation of novel analytic expressions and bounds for a family of special functions that are useful in wireless communication theory. These functions are the well-known Nuttall $Q{-}$ function, incomplete Toronto function, Rice $Ie-$ function, and incomplete Lipschitz–Hankel integrals. Capitalizing on the offered results, useful identities are additionally derived between the above functions and Humbert, $\Phi _{1}$ , function as well as for specific cases of the Kampede Feriet function. These functions can be considered as useful mathematical tools that can be employed in applications relating to the analytic performance evaluation of modern wireless communication systems, such as cognitive radio, cooperative, and free-space optical communications as well as radar, diversity, and multiantenna systems. As an example, new closed-form expressions are derived for the outage probability over nonlinear generalized fading channels, namely, $\alpha {-}\eta {-}\mu $ , $\alpha {-}\lambda {-}\mu $ , and $\alpha {-}\kappa {-}\mu $ as well as for specific cases of the $\eta {-}\mu $ and $\lambda {-}\mu $ fading channels. Furthermore, simple expressions are presented for the channel capacity for the truncated channel inversion with fixed rate and corresponding optimum cutoff signal-to-noise ratio for single-antenna and multiantenna communication systems over Rician fading channels. The accuracy and validity of the derived expressions is justified through extensive comparisons with respective numerical results.

Analysis of Adaptive MIMO Transmit Beamforming Under Channel Prediction Errors Based on Incomplete Lipschitz–Hankel Integrals

This paper provides new results for a family of incomplete Lipschitz-Hankel integrals (ILHIs) which also lead to the evaluation of certain integrals involving the generalized Marcum Q function. These mathematical results are then applied to analyze the bit error rate (BER) of adaptive modulation over multiple-input-multiple-output (MIMO) fading channels under imperfect channel state information (CSI). A novel exact closed-form expression for the average BER of adaptive modulation under MIMO transmit beamforming with maximal ratio combining, assuming prediction errors at the receiver for the adaptation CSI required by the transmitter, is obtained. The benefit of this result with respect to previous analysis is threefold. First, the expression is an exact closed form. Second, it is applicable to any antenna configuration, and third, it allows a design improvement of the cutoff SNR thresholds, which leads to better system performance in terms of average spectral efficiency at no extra cost.

Network Oriented Dyadic Time Domain Green's Function for a Sequentially Excited Infinite Planar Array of Dipoles in Free Space

During the past few years we have performed an effective wideband analysis for characterizing the electrodynamic behavior of phased array antennas, infinite periodic structures, frequency selective surfaces and related applications, with emphasis on gaining physical insight into the phenomenology of short-pulse radiation. The present contribution shows the current status of our network-oriented dyadic time domain Green's function (TD-GF) for a planar array of sequentially excited dipoles that constitutes a prototype study of sequentially short-pulsed radiation by infinite periodic arrays. The dispersive effects of the TD-Floquet waves (FW) and the consequences of a TE and TM field decomposition are discussed in details. In this network formulation, the equivalent TD transmission line (TL) voltages and currents excited by equivalent TD current generators are determined. An expression for the TM voltage TL-GF is provided for the first time in terms of incomplete Lipschitz-Hankel integrals and Bessel functions together with some remarks for its evaluation. As shown in previous publications for the scalar potentials, real-TD physical observables are determined by pairing TD Floquet waves with positive and negative indices. It is shown that the vectorial TD radiated field is reconstructed at any time with the superposition of a few TD-FWs.

Exact Closed-Form Expression of a Sommerfeld Integral for the Impedance Plane Problem

A Sommerfeld integral for an impedance half-plane is one of the classical problems in electromagnetic theory. In this paper, the integral is evaluated into two series representations, which are expressed in terms of the exponential integral and the Lommel function, respectively. It is shown that the series expansions absolutely converge for any parameters, such as distance or surface impedance. Then based on the Lommel function expansion, an exact closed-form expression of the integral is formulated. This expression is written in terms of incomplete Weber integrals, which are directly related to incomplete cylindrical functions such as incomplete Lipschitz-Hankel integrals. Additionally, a complete asymptotic series of the integral is obtained based on the exponential integral expansion. With conventional methods such as steepest descent method, it is cumbersome to derive the divergent series. The validity of all the formulations derived in this paper is demonstrated through a comparison with a numerical integration of the integral for various situations

Numerical computation of incomplete Lipschitz‐Hankel integrals of the Hankel type for complex‐valued arguments

Incomplete Lipschitz‐Hankel integrals (ILHIs) form an important class of special functions since they appear in numerous applications in engineering and physics. While ILHIs of the Hankel type can be expressed as linear combinations of ILHIs of the Bessel and Neumann types, this procedure can lead to computational inaccuracies. As shown in this paper, these inaccuracies can be avoided by directly computing ILHIs of the Hankel types (both the first and second kinds). If desired, these results can then be combined to obtain accurate values for the ILHIs of the Bessel and Neumann types. Series representations for complementary incomplete Lipschitz‐Hankel integrals (CILHIs) are also derived in this paper. CILHIs are often needed to avoid numerical inaccuracies caused by finite precision arithmetic. In order to better understand this group of special functions, the characteristics of ILHIs and CILHIs of the Hankel type are also discussed.

Exact, Closed-Form Representations for the Time-Domain Surface Impedances of a Homogeneous, Lossy Half-Space

In this paper, we derive exact, closed-form representations for the transverse electric and transverse magnetic time-domain, surface impedances associated with a plane wave that is obliquely incident on a homogeneous, lossy half-space. The time-domain surface impedances are expressed in terms of modified Bessel functions and incomplete Lipschitz-Hankel integrals. The limitations on the validity of a previously published approximate expression are also discussed.

Lossy transmission line simulation based on closed-form triangle impulse responses

Analytical frequency-domain expressions for single and coupled transmission lines with triangular input waveforms are first developed. The inverse Fourier transform is then used to obtain an expression for the time-domain triangle impulse responses for frequency-independent transmission line parameters. The integral associated with the inverse Fourier transform is solved analytically using a differential-equation-based technique. Closed-form expressions for the triangle impulse responses are given in the form of incomplete Lipschitz-Hankel integrals (ILHI) of the first kind. The ILHI can be efficiently calculated using existing algorithms. Combining these closed-form expressions for the triangle impulse responses with a time-domain convolution method using a triangle impulse as a basis function, provides an accurate and efficient simulation method for very lossy transmission lines embedded within linear and nonlinear circuits.

On certain generalized incomplete gamma functions

Recently, Chaudhry and Zubair have introduced a generalized incomplete gamma function Γ( v, x; z) which reduces to the incomplete gamma function Γ( v, x) when its variable z vanishes. We show that Γ( v, x; z) may be written essentially as a single Kampé de Fériet function which in turn may be expressed as a linear combination of two incomplete Weber integrals. Then by using properties of the latter integrals we deduce additional representations for Γ( v, x; z). In particular, we show that Γ( v, x; z) is essentially completely determined by a finite number of modified Bessel functions for all v ≠ 0 provided we know the values of the two incomplete Weber integrals when 0 < Re v ⩽ 1. When v = 0 we derive connections between the generalized incomplete gamma function and incomplete Lipschitz-Hankel integrals, and indicate that there exist connections with other special functions.

An accurate and efficient analysis for transient plane waves obliquely incident on a conductive half space (TM case)

We develop a method for the analytical evaluation of the inverse Laplace transform representations for transient transverse magnetic (TM) plane wave obliquely incident on a conductive half-space. We assume that the permittivity and conductivity of the dispersive half-space are independent of frequency. The time-domain expressions for the reflected and transmitted waves are first represented as inverse Laplace transforms. The transient fields are then shown to consist of two canonical integrals. The canonical integrals, in turn, are solved analytically, thereby yielding close-form solutions involving incomplete Lipschitz-Hankel integrals (ILHIs). The ILHIs are computed numerically using efficient convergent and asymptotic series expansions, thus enabling the efficient computation of the transient fields. The exact, closed-form expressions are verified by comparing with previously published results and with results obtained using standard numerical integration and fast Fourier transform (FFT) algorithms. An asymptotic series representation for the ILHIs is also employed to obtain a relatively simple late-time approximation for the transient fields. This approximate late-time expression is shown to accurately model the fields over a large portion of its time history.

An accurate and efficient analysis for transient plane waves obliquely incident on a conductive half space (TE case)

A method which allows for the analytical evaluation of the inverse Laplace transform representations for a transient TE plane wave, obliquely incident on a conductive half-space, is discussed. We assume that the permittivity and conductivity of the dispersive half-space are independent of frequency. Starting with the equations for the transmitted wave in the Laplace domain, the corresponding time-domain expressions are first represented as inverse Laplace transforms. The transient fields are shown to consist of two canonical integrals f(/spl beta/) and c(/spl beta/). The canonical integrals, in turn, are solved analytically, thereby yielding solutions involving incomplete Lipschitz-Hankel integrals (ILHTs). The ILHIs are computed numerically using efficient convergent and asymptotic series expansions, thus enabling the efficient computation of the transient fields. The solutions are verified by comparing with previously published results and with results obtained using standard numerical integration and fast Fourier transform (FFT) algorithms.

A Comment on "Propagation of ultrawide-band electromagnetic pulses through dispersive media" [and reply

In the above paper (see ibid., vol.37, p.192, 1995) the authors went to great lengths to derive an explicit solution in terms of incomplete Lipschitz-Hankel integrals (ILHI's) to the propagation problem of a double exponential pulse in a homogeneous medium. Technically, He has no doubt on the correctness of their ILHI representation. However, He doubts their claim about the computational efficiency of the ILHI representation. Dvorak and Dudley reply to the Comment.

Propagation of ultra-wide-band electromagnetic pulses through dispersive media

We develop an efficient method for the analysis of ultra-wide-band (UWB) electromagnetic pulses (e.g., double-exponential pulse) propagating through a waveguide or cold plasma (i.e., the ionosphere). First we show that the inverse Fourier-transform representations for the electric and magnetic fields satisfy second order, nonhomogeneous, ordinary, differential equations. These differential equations are solved analytically, thereby yielding closed-form expressions involving incomplete Lipschitz-Hankel integrals (ILHIs). The ILHIs are computed using efficient convergent and asymptotic series expansions. We demonstrate the usefulness of the ILHI expressions by comparing them with the fast Fourier-transform technique (FFT). Because of the long tails associated with UWB pulses, a large number of sample points are required in the FFT, to avoid aliasing errors. In contrast, the ILHI expressions provide accurate and efficient numerical results, regardless of the number of points computed. An asymptotic series representation for the ILHIs is also employed, to obtain a relatively simple, late-time approximation for the transient fields. This approximate late-time expression is shown to accurately model the waveform over a large portion of its time history. >

Applications for incomplete Lipschitz-Hankel integrals in electromagnetics

Incomplete Lipschitz-Hankel integrals (ILHIs) are a class of special functions that appear in the analytical solutions for numerous canonical problems in electromagnetics. The reason that they appear so often in electromagnetics is because they provide solutions to the wave equation. The intent of the article is to provide an overview of the research that has been carried out in this area. After providing a brief introduction to the theory of ILHIs, applications for these special functions in problems involving diffraction, dispersion, sources radiating in layered media, and traveling-wave sources radiating in a homogeneous space, are discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Exact, closed-form expressions for transient fields in homogeneously filled waveguides

It is well known that transient electromagnetic waves in waveguides exhibit dispersion. Exact, closed-form expressions, which involve Bessel functions of the first kind, have been derived for the impulse response of a waveguide, but exact, closed-form expressions for more complex pulses are absent from the literature. In this paper, it is demonstrated that incomplete Lipschitz-Hankel integrals can be used to represent transient pulses in homogeneously filled waveguides. A continuous wave pulse is investigated in this paper, however, this technique can also be applied to a number of other transient waveforms. The resulting expressions are verified by numerically integrating the pulse distribution multiplied by the known impulse response.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Exact, closed-form field expressions for two-dimensional, traveling-wave current strips

In this paper, exact, closed-form expressions are derived for the electromagnetic fields associated with two-dimensional, traveling-wave current strips radiating into a homogeneous space. The field expressions involve incomplete Lipschitz-Hankel integrals of the Hankel form. Convergent and asymptotic series expansions for the incomplete Lipschitz-Hankel integrals allow the fields to be computed more efficiently than was previously possible using numerical integration. Two-dimensional, traveling-wave current strips are important because they appear in problems involving physical optic scattering from strips, diffraction through slits, and radiation from parallel plate waveguides. The problem of physical optic scattering from a strip is investigated in this paper. >

Exact, closed-form field expressions for traveling-wave current filaments in homogeneous space

Exact, closed-form field expressions are derived for the canonical problem of electromagnetic radiation by a semi-infinite traveling-wave current filament in free space. In order to avoid time consuming numerical integration, the fields are represented in terms of complementary incomplete Lipschitz-Hankel integrals, whose series expansions allow for accurate and efficient computation of the field components everywhere in space. The field expressions are also shown to reduce to elementary functions in many special cases, e.g., in the far-zone, in the vicinity of the current filament, and when the propagation constant along the wire is that of free space. Representative plots for the electric field components are presented for values of the propagation constant corresponding to both fast and slow-wave propagation. Since a superposition of two semi-infinite current filaments yields a finite-length current filament, the method and the results obtained in this paper can be readily applied to a number of proble...

Incomplete Lipschitz-Hankel integrals of struve functions

Representations for incomplete Lipschitz-Hankel integrals of Struve functions are given in closed form by using Kampé de Fériet functions whose double series representations converge everywhere.

Numerical computation of 2D sommerfeld integrals— A novel asymptotic extraction technique

The accurate and efficient computation of the elements in the impedance matrix is a crucial step in the application of Galerkin's method to the analysis of planar structures. As was demonstrated in a previous paper, it is possible to decompose the angular integral, in the polar representation for the 2D Sommerfeld integrals, in terms of incomplete Lipschitz-Hankel integrals (ILHIs) when piecewise sinusoidal basis functions are employed. Since Bessel series expansions can be used to compute these ILHIs, a numerical integration of the inner angular integral is not required. This technique provides an efficient method for the computation of the inner angular integral; however, the outer semi-infinite integral still converges very slowly when a real axis integration is applied. Therefore, it is very difficult to compute the impedance elements accurately and efficiently. In this paper, it is shown that this problem can be overcome by using the ILHI representation for the angular integral to develop a novel asymptotic extraction technique for the outer semi-infinite integral. The usefulness of this asymptotic extraction technique is demonstrated by applying it to the analysis of a printed strip dipole antenna in a layered medium.

Numerical computation of 2D sommerfeld integrals— Deccomposition of the angular integral

Spectral domain techniques are frequently used in conjunction with Galerkin's method to obtain the current distribution on planar structures. When this technique is employed, a large percentage of the computation time is spent filling the impedance matrix. Therefore, it is important to develop accurate and efficient numerical techniques for the computation of the impedance elements, which can be written as two-dimensional (2D) Sommerfeld integrals. Once the current distribution has been found, then the near-zone electric field distribution can be obtained by computing another set of 2D Sommerfeld integrals. The computational efficiency of the 2D Sommerfeld integrals can be improved in two ways. The first method, which is discussed in this paper, involves finding a new way to compute the inner angular integral in the polar representation of these integrals. It turns out that the angular integral can be decomposed into a finite number of incomplete Lipschitz-Hankel integrals, which in turn can be calculated using series expansions. Therefore, the angular integral can be computed by summing a series instead of applying a standard numerical integration algorithm. This new technique is found to be more accurate and efficient when piecewise-sinusoidal basis functions are used to analyze a printed strip dipole antenna in a layered medium. The incomplete Lipschitz-Hankel integral representation for the angular integral is then used in another paper to develop a novel asymptotic extraction technique for the outer semi-infinite integral.

Reduction formulas for kampé de Fériet functions associated with certain classes of incomplete Lipschitz—Hankel type integrals of cylindrical functions

Reduction formulas and integral representations for Kampé de Fériet functions F p:2;1 q:1;0 which are associated with certain classes of incomplete integrals of Bessel or cylindrical functions (including, for example, incomplete Lipschitz—Hankel integrals) are obtained. In addition, more general series identities related to these Kampé de Fériet functions are derived and a new derivation of Lerch's theorem for the partial sums of the coefficients of a special case of Gauss's hypergeometric function is provided.