# Abstract

Sparse grid collocation methods are used for uncertainty quantification in electromagnetic propagation problems. The first application here involves waves propagating in lossy media with uncertain permittivities and permeabilities, for which several cases with increasing random-space dimensionality are exemplified. The objective in the second application is to compute expected signal strength above flat Earth surface at ranges far from transmitter location, where randomness is present due to uncertain refractivity of the atmosphere. Two different sparse grid algorithms are demonstrated throughout the paper, and the deterministic evaluators are accessed as a black box by the sparse grid algorithms. The difficult task of a priori method adaption is overcome by a first order check that enforces the black box solver only a few times. Through the results considered, strengths of the two algorithms are differentiated depending on the characteristics of the randomness. The advantage of a new routine used in this work is emphasized by showing its numerical contribution to the field estimation problem in a specific example. By means of the sparse grid methods used, we quantitatively provide the relative importance of each RV in contributing to the changes in the field distribution received at the observation range.

# Abstract

Certain fundamental errors that appeared in a recent paper in the IEEE Transactions on Antennas and Propagation, August 2013 have been pointed out and the correct integral representation with respect to the vertical wavenumber of a VED radiating in the presence of a conducting half-space is derived.

# Abstract

A new Volterra integral equation of the second kind with square integrable kernel is derived for paraxial propagation of radiowaves over a gently varying, perfectly conducting rough surface. The integral equation is solved exactly in terms of a infinite series and the necessary and sufficient conditions for the solution to exist and converge are established. Super exponential convergence of the Neumann series for arbitrary surface slope is established through asymptotic analysis. Expressions are derived for the determination of the number of terms needed to achieve a given accuracy, the latter depending on the parameters of the rough surface, the frequency of operation and the maximum range. Numerical results with truncated series are compared with that obtained by solving the integral equation numerically for a sinusoidal surface, Gaussian hill, and a random rough surface with Pierson-Moskowitz spectrum.

# Abstract

Kac's conjecture relating the solution of wave and telegraph equations in higher dimensions through a Poisson-process-driven random time is established through the concepts of stochastic calculus. New expression is derived for the probability density function of the random time. We demonstrate how the relationship between the solution of a lossy wave- and that of a lossless wave equation can be exploited to derive some statistical identities. Relevance of the results presented to the study of pulse propagation in a dispersive medium characterized by a Lorentz or Drude model is discussed and new evolution equations for 2-D Maxwell's equations are presented for the Drude medium. It is shown that the computational time required for updating the electric field using the stochastic technique is expected to go up as O(?t).

# Abstract

By considering the 2D problem of electrical line sources radiating in the presence of perfectly conducting cylinders and decomposing the real power flow on a circumscribing observation circle separating the transmit nodes from the receive nodes, simple formulas are derived for the electromagnetic degrees of freedom in scattering environments for a network of nodes communicating with each other. The locations, magnitudes, and phases of the line sources are assumed to be independent and identically distributed (i.i.d.) random variables. Similarly, the locations of the scatterers in the region outside the observation circle are assumed to be i.i.d. random variables. The exact scattering problem is cast in the form of an integral equation, where the Fourier coefficients of the scatter current density are the unknowns. Based on the Born approximation that is valid for mild scatter densities and asymptotic analysis, a closed form expression is derived for the number of degrees of freedom in scattering environments. The benefit of observing near-fields in the determination of degrees of freedom is included in the numerical examples considered. If the power per source and/or the number of sources within the circumscribing circle are made to increase algebraically with the size of the circle, it is shown that scattering environments can offer much higher degrees of freedom than what are available in free-space.

# Abstract

Domain truncation by transparent boundary conditions for open problems where parabolic equation is utilized to govern wave propagation are in general computationally costly.We utilize two approximations to a convolution-in-space type discrete boundary condition to reduce the cost, while maintaining accuracy in far range solutions. Perfectly matched layer adapted to the Crank-Nicolson finite difference scheme is also verified for a 2-D model problem, where implemented results and stability analyses for different approaches are compared.

# Abstract

Adaptive correction of the excitation coefficients of a phased array achieved through dithering the magnitudes and phases of the element coefficients and sensing the fields through a near-zone probe is demonstrated by considering a linear array. Knowledge of the reference signal generated by the desired array at the near-zone probe is assumed. Deviations in the coefficients of the actual array from the desired array are corrected adaptively and simultaneously by means of a gradient based algorithm. Requirements for the algorithm to converge, its performance with and without a receiver noise and the effect of the dither parameters are studied. The effect of element mutual coupling on the performance of the array is also demonstrated by considering an array of half-wave dipoles.

# Abstract

We discuss the use of the parabolic equation (PE) along with the alternate direction implicit (ADI) method in predicting the loss for three specialized tunnel cases: curved tunnels, branched tunnels, and rough-walled tunnels. This paper builds on previous work which discusses the use of the ADI-PE in modeling transmission loss in smooth, straight tunnels. For each specialized tunnel case, the ADI-PE formulation is presented along with necessary boundary conditions and tunnel geometry limitations. To complete the study, examples are presented where the ADI-PE numerical results for the curved and rough-walled tunnel are compared to known analytical models and experimental data, and the branched tunnel data is compared to the numerical solutions produced by HFSS.

# Abstract

The dependence of the communication capacity of multiple-input multiple-output (MIMO) wireless systems on the average received signal-to-noise ratio (SNR), assuming the channel is unknown at the transmitter and perfectly known at the receiver, is studied through full wave electromagnetic tools. Although it is commonly accepted that the capacity of a MIMO system increases linearly at high SNRs when plotted versus the SNR expressed in dB, the fact that the number of effective degrees of freedom (DOF) of the system increases with SNR in many practical environments calls this conclusion into question for reasonably high SNRs. Based on a full wave electromagnetic investigation, we are able to analytically predict and then confirm a significant region on the MIMO capacity curve where the capacity grows quadratically when plotted versus the SNR in dB. This gives analytical insight into a portion of the capacity curve that may previously be (incorrectly) attributed to the concavity of the logarithm function rather than the increase in electromagnetic degrees of freedom. The quadratic, rather than linear, growth of capacity suggests that it may be worthwhile to invest more transmit power to achieve higher performance gains. However, to fully take advantage of this second order benefit, the numbers of antennas at the transmitter and the receiver must be close to or slightly larger than the wavevector-aperture-product (WAP) of the corresponding EM system.

# Abstract

The angular correlation of received fields in multipath environments is studied. The focus is put on two-dimensional (2-D) cases, and the frequently used uncorrelated scattering assumption is tested through full-wave Monte Carlo simulations. The results show that this assumption is valid for the discrete finite spectra of the received waves when the scattering objects in the environments are distributed in complete randomness, either when they are surrounding the transmit/receive regions or in clusters away from them. The correlation among wave components from different angles increases only when the randomness of the scatterer deployment is reduced.

# Abstract

Alternate direction implicit (ADI) method is used to study radio wave propagation in tunnels using the parabolic equation (PE). We formulate the ADI technique for use in tunnels with rectangular, circular and arched cross sections and with lossy walls. The electrical parameters of the lossy walls are characterized by an equivalent surface impedance. A vector PE is also formulated for use in tunnels with lossy walls. It is shown that the ADI is more computationally efficient than the Crank Nicolson method. However, boundary conditions become more difficult to model. The boundary conditions of the ADI intermediate planes are given the same boundary conditions as the physical plane and the overall accuracy is reduced. Also, when implementing the ADI in tunnels with circular cross sections the order at which the line by line decomposition occurs becomes important. To validate the ADI-PE, we show simulation results for tunnel test cases with known analytical solutions. Furthermore, the ADI-PE is used to simulate real tunnels in order to compare with experimental data. It is shown that the PE models the electric fields most accurately in real tunnels at large distances, where the lower order modes dominate.

# Abstract

The four-state random walk (4RW) model, wherein the particle is endowed with two states of spin and two states of directional motion in each space coordinate, permits a stochastic solution of the Schrödinger equation (or the equivalent parabolic equation) without resorting to the usual analytical continuation in complex space of the particle trajectories. Analytical expressions are derived here for the various transitional probabilities in a 4RW by employing generating functions and eigenfunction expansions when the particle moves on a 1+1 space-time lattice with two-point boundary conditions. The most general case of dissimilar boundaries with partially reflecting boundary conditions is treated in this paper. The transitional probabilities are all expressed in terms of a finite summation involving trigonometric functions and/or Chebyshev polynomials of the second kind that are characteristics of diffusion and Schrödinger equations, respectively, in the 4RW model. Results for the special case of perfectly absorbing boundaries are compared to numerical values obtained by directly counting paths in the random walk simulations.

# Abstract

A statistical multipath propagation model is developed in the double-angular domain to include effects of both angle of departure and angle of arrival. The model is characterized by a discrete stochastic double-angular spectrum of the Green's function between two volumes that communicate with each other in the presence of cluster scattering. The whole modeling process is a combination of theories, assumptions, and numerical verification by rigorous full wave electromagnetic simulations. Notably, an uncorrelated scattering assumption for received fields is extended to the double-angular domain, and the transport theory plays a key role defining the statistical properties of the spectrum of the Green's function. The spectrum is eventually modeled as a random matrix whose entries are uncorrelated zero-mean complex Laplacian random variables. The real and imaginary parts of the matrix entries are independent and identically distributed, and the variance profile follows the reduced incident intensity of the waves through random media. The parameters of the model are completely obtained from bulk statistical knowledge of the scattering environments, and the resulting model demonstrates good performance on EM DOF and MIMO capacity estimation.

# Abstract

Transparent boundary condition in a 2D-space is presented for the four-state random walk (4RW) model that is used in treating the standard parabolic equation by stochastic methods. The boundary condition is exact for the discrete 4RW model, is of explicit type, and relates the field in the spectral domain at the boundary point in terms of the field at a previous interior point via a spectral transfer function. In the spatial domain, the domain of influence for the boundary condition is directly proportional to the "time" elapsed. By performing various approximations to the transfer function, several approximate absorbing boundary conditions can be derived that have much more limited domain of influence.

# Abstract

The diffusion and Schr¨odinger propagators have been known to coexist on a lattice when a particle undergoing random walk is endowed with two states of spin in addition to the two states of direction in a 1+1 spacetime dimension. In this paper we derive explicit expressions for the various transitional probabilities by employing generating functions and transform methods. The transitional probabilities are all expressed in terms of a one-dimensional integral involving trigonometric functions and/or Chebyshev polynomials of the first and second kind from which the spacetime continuum limits of the diffusion equation and Schr¨odinger equation follow directly.

# Abstract

The diffusion behavior of electromagnetic (EM) waves in two dimensional (2-D) multipath media is studied through integral equation based full wave Monte Carlo simulations. The influences of some physical factors are explored, among which the area density of the embedded obstacles manifests itself to be the most important one in determining wave diffusion. A lossy system starts to behave diffusively when the area density approximately exceeds 5%, and the diffusion equations are generally applicable for predicting power decay. At low densities, the power-distance relation of the waves appears to follow power laws. The sizes and shapes of the obstacles have a secondary effect on the diffusion of waves. Whenever a system contains small objects or objects with reflecting sides, the waves therein are more diffusive and the diffusion equation approximates the reality more accurately. Absorption loss decreases wave diffusion in general, but our results show that the diffusion equation for a system with very lossy but small obstacles can work very well for predicting power decay.

# Abstract

Topology control problems are associated with assignment of power levels to nodes of a wireless network so that the resulting graph topology satisfies certain properties. In this paper we consider the problem of power-efficient topology control with switched beam directional antennas taking into account their non-uniform radiation pattern within the beamwidth. Previous work in the area have all assumed a uniform gain model with these antennas which renders antenna orientation insignificant as a parameter in topology control algorithms. We present algorithms that take into account a model of non-uniform gain with the objectives of minimizing the total power and maximum power to keep the network connected. We consider two cases: one where the antenna orientation is assumed given and another where the antenna orientation needs to be derived as well. For the first case, we present optimal and approximation algorithms for constructing power-efficient topologies . For the second case, we prove the problem to be NP-complete and present heuristic solutions along with approximation bounds. Through comparison of the two cases by simulation, significant reductions are shown in the maximum as well as total power required to keep the network connected for the second case, thus demonstrating the benefits of using antenna orientation as parameter in topology construction.

# Abstract

The electromagnetic (EM) degrees of freedom (DOF) of a noise limited system in two dimensions with random multiple scattering is evaluated numerically following a rigorous DOF theory first developed by Miller and Piestun for optical systems. The received EM fields are efficiently calculated by fast multipole method (FMM), and the ensemble average of the DOF number is obtained through Monte Carlo simulation technique. The results show that the average EM DOF number is strongly dependent on the sizes of the transmit volume, the receive volume, and the scattering region. In particular, the average number of DOF generally increases with both the transmit and receive volumes. However this increase is a non-linear process and will not continue indefinitely. As the transmit volume or the receive volume expands, an upper-bound of the average DOF number is expected due to noise effects. Due to the lack of criteria for choosing a critical parameter involved in Miller and Piestun's original DOF definition, a modified definition is also considered. Even though the modified definition is SNR dependent, it provides a clearer physical meaning of the DOF. In addition, the simulations also suggest that it might not be appropriate to ignore the influence of the SNR on the DOF number when the system concerned is of general form and relative small, where no critical point of the channel quality can be identified. A logarithmic dependence of the DOF number on the total source power is demonstrated for such systems.

# Abstract

New analytical shadow loss model is presented that takes into account absorption and scattering of incident waves by obstacles. The input parameters for the model are the obstacle occupational density, obstacle number density, and mean geometric cross section of the obstacles. The model yields a closed form expression for the mean excess loss in a typical office environment. Comparison with measured excess loss in NLOS situations is shown to demonstrate the utility of the model. The model is expected to be valid when the density of obstacles is sufficiently large.