This repository includes MATLAB codes evaluating Pareto-optimal sets of antenna directivity and sensitivity. The methodology is detailed in [1], and this repository shows how to evaluate results presented in Fig. 5, Fig. 6 and Fig. 13 in that paper. The array made of 21 thin perfectly conducting strip dipoles is used as an example.
[1] K. Schab, L. Jelinek, M. Capek, and M. Gustafsson, ``Directivity estimates and tolerance tradeoffs in the presence of random perturbations,'' in review.
In order to evaluate the Pareto-optimal set in directivity and sensitivity, the electromagnetic description of the underlying structure must be performed. This is done by two electromagnetic solvers: ATOM (method of moments) and HFSS (finite element method). The output of this stage is stored in a *.mat file containing:
- OPsweep{1,1}.zPort: impedance matrix of the array
- OPsweep{1,1}.sPort: scattering matrix of the array
- OPsweep{1,1}.R0: radiation matrix of the array
- OPsweep{1,1}.F: far-field matrix of the array in the main direction (both polarizations)
- OPsweep{1,1}.FthPatt: embedded element patterns in theta-polarization
- OPsweep{1,1}.FphPatt: embedded element patterns in phi-polarization
- angleVec: angles for the plot of embedded element patterns
- a: radius of the smallest sphere circumscribing the array
- fList: frequency
- kList: wavenumber
- lambdaList: wavelength
- nDOF: number of degrees of freedom (number of dipoles)
In the case of the ATOM solver (http://www.antennatoolbox.com), the results are obtained from Galerkin's method applied to the electric field integral equation (Method of Moments). The geometry of the dipole array is shown in data/Fig5geometry_ATOM.fig which should be zoomed in to observe the details. The structure is made of a triangular mesh with 2688 triangles. Red edges show delta-gap ports to which 50 Ohm ideal transmission lines are connected. The dipoles are perfectly conducting. The dipoles are placed along the z-axis, and the array is created along the x-axis. The radiation in the endfire direction (along the x-axis) is optimized. The embedded element patterns are evaluated in the xy-plane for y > 0. The embedded element pattern corresponds to unit power wave at the given port and transmission line loading at other ports. The solution outputs are stored in data/opdata-MeshedHalfWaveDipolesATOM.mat.
In the case of HFSS, the simulation setup is the same as in the previous case, but the frequency-domain finite element method was used. The raw data exported from HFSS solver can be found in data/hfss-data/. The MATLAB parser A_parseHFSSdata.m reforms the data into a standardized structure stored in data/opdata-MeshedHalfWaveDipolesHFSS.mat.
The Pareto-optimal set is evaluated using the FunBo package (http://www.antennatoolbox.com/fundamentalBounds), part of which is included in the repository in the "bin" folder and must be included in MATLAB paths, including its subdirectories. The evaluation is executed by B_calculatePareto.m, where the user must choose which input file to take (ATOM or HFSS):
% tag = 'MeshedHalfWaveDipolesATOM';
tag = 'MeshedHalfWaveDipolesHFSS';
The script generates three figures showing the Pareto-optimal set in D-\chi with different axis normalizations (Fig. 6 in the paper) and one figure showing embedded element patterns (Fig. 5 in the paper).
The Pareto-optimal data are stored in data/pareto-MeshedHalfWaveDipolesATOM.mat or data/pareto-MeshedHalfWaveDipolesHFSS.mat.
The Monte Carlo analysis uses random samples of the array feeding perturbed with a given error according to the proportional, uncorrelated, zero-mean model discussed in [1].
The resulting data are stored in data/montecarlo-MeshedHalfWaveDipolesATOM.mat or data/montecarlo-MeshedHalfWaveDipolesHFSS.mat.
The results of the Monte Carlo samples can also be visualized as the perturbations of the radiation pattern. This is provided by the script D_visualizePerturbedPatterns.m, where nominal, perturbed, and ensemble mean patterns are plotted. This figure is not included in [1] but provides insight into the random trials within the Monte Carlo ensemble.
The important part of the theory developed in the manuscript is the anticipated directivity estimates visualized in Fig. 13. This estimate depends on two parameters (sensitivity and uniformity) and provides an accurate predictor of mean directivity regardless of the nominal directivity in the direction of interest, i.e., it is accurate for both main-beam statistics as well as predicting sidelobe levels. These results can be generated by executing E_generalDirectivityEstimate.m.