## Computational EM Modeling Techniques

### Model-Based Parameter Estimation

..: References :..

1-)

**The Simultaneous Interpolation of Antenna Radiation Patterns in Both the Spatial and Frequency Domains Using Model-Based Parameter Estimation **
*by Douglas H. Werner and Rene J. Allard*
__ABSTRACT__: The Padé rational function fitting model commonly
used for model-based parameter estimation (MBPE) in the frequency domain
is enhanced to include spatial dependence in the numerator and denominator
coefficients. This allows the function to interpolate an antenna radiated
electric field pattern in both the frequency and spatial domains simultaneously,
such that a single set of coefficients can be used to accurately reconstruct
an entire radiation pattern at any frequency in the fitting-model range.
A simple procedure is introduced for transforming interpolated electric
fields into gain patterns using input impedance versus frequency curves
also obtained via MBPE. The utility of this method is demonstrated by
applying it to a dipole antenna over a frequency range of 150–950
MHz and using a polynomial representation in

*θ*
for the coefficient spatial dependence. It is also used to estimate radiation
patterns for a three-element Yagi array between the frequencies of 470 and 500 MHz using a binomial
representation for the spatial variation that includes terms dependent
on

θ as well as

Φ. The use of this method for interpolating
radiation patterns has at least two significant advantages; one being
large compression ratios for the amount of data that must be stored to
accurately reproduce patterns and the other being a significant decrease
in the amount of time required for modeling problems with large computational
domains.

2-)

** The Model-Based Parameter Estimation of Antenna Radiation Patterns Using Windowed Interpolation and Spherical Harmonics**
*by Rene J. Allard and Douglas H. Werner*
__ABSTRACT__: Techniques previously reported for interpolating antenna
radiation patterns simultaneously in both the spatial and frequency domains
are reexamined here in order to develop new algorithms with improved efficiency
and speed while maintaining comparable accuracy. To this end, interpolation
in the frequency domain is constrained to a windowed scheme whereby a
series of reduced-order Padé rational function fitting models of
only three or four sampling frequencies are used to interpolate over a
relatively large bandwidth. The use of this piece-wise technique is shown
to reduce the complexity of determining the unknown coefficients without
any significant loss in the resulting interpolation accuracy or increase
in the required number of sampling frequencies. In this paper, exact analytical
expressions are found for the unknown coefficients that allow the spatial
domain interpolation to be performed entirely separate from the frequency
domain interpolation. The technique is applied to the case of a 0.5 meter
dipole modeled from 150 to 950 MHz, and for

θ from 0 to 90 , using fitting
windows with three and four sampling frequencies. The results are compared
to those obtained using a previously developed generalized simultaneous
interpolation procedure. In the spatial domain, spherical harmonics are
introduced as model-based fitting functions for antenna radiation patterns.
Specifically, interpolation procedures are developed using zonal harmonics
and tesseral harmonics to model one- and two-dimensional far-field radiation
patterns, respectively. The case of a z-oriented 0.5 meter dipole is considered
to demonstrate the efficiency and accuracy obtained
by applying these physically-based fitting models in the spatial domain.
Zonal harmonics are used to interpolate far-field radiation patterns of
this antenna for

θ from 0 to 180 for several different frequencies over
the range from 150 to 950 MHz. This example demonstrates the attractiveness
of using spherical harmonics functions to accurately and efficiently interpolate
radiation patterns in the spatial domain.

3-)

**A Model-Based Parameter Estimation Technique for Wide-Band Interpolation of Periodic Moment Method Impedance Matrices With Application to
Genetic Algorithm Optimization of Frequency Selective Surfaces **
*by Ling Li, Douglas H. Werner, Jeremy A. Bossard, and Theresa S. Mayer*
IEEE Transactions on Antennas and Propagation, VOL. 54. NO. 3, pp. 908-924, March 2006.

__ABSTRACT:__ A model-based parameter estimation (MBPE) technique
is introduced in this paper for efficiently interpolating periodic
moment method (PMM) impedance matrices over a wide
frequency band. In the model, only the Floquet harmonics that
strongly affect the frequency band of interest are employed to approximate
the matrix elements, while the contributions from all
other higher-order harmonics are compactly represented by two
additional terms. The derivation of the model is physics-based, and
the objective is to find the coefficients of the terms in the model by
utilizing the values of the impedance matrix elements calculated
via PMM at only a few frequency points. The number and position
of these fitting points can be pre-determined from the cutoff frequencies
of the Floquet harmonics, which allows the MBPE interpolation
process in this case to be completely automated. In other
words, the number and position of the sampling points are only
dictated by the periodicity of the frequency selective surface (FSS)
structure and the frequency range of interest. Unlike many of the
other scattering parameter-based techniques, the shape and the
resonances in the response of the FSS do not have any impact on
the construction of the interpolation model. This makes it particularly
useful in genetic algorithm (GA) aided FSS design, since
for a fixed periodicity and frequency range the MBPE interpolation
is independent of the scattering response of candidate FSS designs.
Several examples of the new PMM-MBPE approach are presented
including one in which it is used to considerably speed up
the GA-based design process for a reconfigurable FSS.

4-)

**A Model-Based Parameter Estimation Technique for Wideband Interpolation of Periodic Moment Method Impedance Matrices**
*by Ling Li, Douglas H. Werner, and Jeremy Bossard*
__ABSTRACT:__ There are several computational electromagnetic modeling techniques that
have been developed specifically for the analysis and design of Frequency
Selective Surfaces (FSS) [1]. One of the most popular and widely used of these
techniques is the Periodic Method of Moments (PMM) [2]. The application of the
PMM to the analysis of FSS geometries at a single frequency involves the construction and solution of a
linear system of equations of the form Zi = v, where Z is the impedance matrix, i is the vector that
contains the unknown coefficients of the basis functions used to expand the surface current, and v is the known
excitation vector. For simulations where wideband analysis is required, the entire
system has to be reconstructed and solved for each frequency sampling point in
the band of interest. Therefore, finding a way to reduce the time involved in calculating the Z matrix
is extremely important. In this paper, we introduce a Model-Based Parameter Estimation (MBPE) technique [3] that
can be used to accelerate the computation of the PMM impedance matrix Z over wide frequency ranges for efficient
analysis and design of FSS.

5-)

**Improved Model-Based Parameter Estimation Approach for Accelerated Periodic Method of Moments Solutions With
Application to the Analysis of Convoluted Frequency Selected Surfaces and Metamaterials**
*by Xiande Wang, and Douglas H. Werner*
IEEE Transactions on Antennas and Propagation, Vol. 58, Issue 1, pp. 122 - 131, January 2010.

__ABSTRACT:__ An improved “smart” interpolation approach known as model-based parameter estimation (MBPE) is applied to the
wide-band interpolation of periodic method of moments (PMM) impedance matrices for normal and oblique incidence cases. Prior
to interpolation, easy to calculate but hard to interpolate, phase terms are removed fromthe impedance matrices.
An efficient spectral-domain PMM formulation is introduced for the accelerated analysis of frequency selective surface
(FSS) problems with a large number of unknowns, employing a one dimensional

*O(N * log

*N)* FFT-based method
to speed up the computation of matrix-vector products within the bi-conjugate gradient (BCG) iterative solver,
which is made possible by the asymmetric multilevel block-Toeplitz structure of the impedance-matrix. The MBPE interpolation algorithm
provides a faster matrix fill time than the brute force method and is comparable or even faster than the 2-D FFT-based method
for a large number of unknowns. It also has the advantage that it can be applied to non-uniform gridding cases. The accuracy and
efficiency of the proposed techniques for large FSS problems are demonstrated by several design examples for both the normal and
oblique incidence cases.We also apply this efficient analysis tool to the design of multiband single-layer FSS filters and
artificial magnetic conductors (AMC) comprised of a 2-D periodic arrangement of convoluted metallic strips in the shape of a
Hilbert curve. The multiband properties of the Hilbert curve FSS filters are studied for different iteration orders
(i.e., different degrees of space-filling).

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