Читать книгу Modern Characterization of Electromagnetic Systems and its Associated Metrology - Magdalena Salazar-Palma - Страница 10

The area of electromagnetics is an evolutionary one. In the earlier days the analysis in this area was limited to 11 separable coordinate systems for the solution of Helmholtz equations. The eleven coordinate systems are rectangular, circular cylinder, elliptic cylinder, parabolic cylinder, spherical, conical, parabolic, prolate spheroidal, oblate spheroidal, ellipsoidal and paraboloidal coordinates. However, Laplace’s equation is separable in 13 coordinate systems, the additional two being the bispherical and the toroidal coordinate systems. Outside these coordinate systems it was not possible to develop a solution for electromagnetic problems in the earlier days. However, with the advent of numerical methods this situation changed and it was possible to solve real practical problems in any system. This development took place in two distinct stages and was primarily addressed by Prof. Roger F. Harrington. In the first phase he proposed to develop the solution of an electromagnetic field problem in terms of unknown currents, both electric and magnetic and not fields by placing some equivalent currents to represent the actual sources so that these currents produce exactly the same desired fields in each region. From these currents he computed the electric and the magnetic vector potentials in any coordinate system. In the integral representation of the potentials in terms of the unknown currents, the free space Green’s function was used which simplified the formulation considerably as no complicated form of the Green’s function for any complicated environment was necessary. From the potentials, the fields, both electric and magnetic, were developed by invoking the Maxwell‐Hertz‐Heaviside equations. This made the mathematical analysis quite analytic and simplified many of the complexities related to the complicated Green’s theorem. This was the main theme in his book “Time Harmonic Electromagnetic Fields”, McGraw Hill, 1961. At the end of this book he tried to develop a variational form for all these concepts so that a numerical technique can be applied and one can solve any electromagnetic boundary value problem of interest. This theme was further developed in the second stage through his second classic book “Field Computations by Moment Methods”, Macmillan Company, 1968. In the second book he illustrated how to solve a general electromagnetic field problem. This gradual development took almost half a century to mature. In the experimental realm, unfortunately, no such progress has been made. This may be partially due to decisions taken by the past leadership of the IEEE Antennas and Propagation Society (AP‐S) who first essentially disassociated measurements from their primary focus leading antenna measurement practitioners to form the Antenna Measurements Techniques Association (AMTA) as an organization different from IEEE AP‐S. And later on even the numerical techniques part was not considered in the main theme of the IEEE Antennas and Propagation Society leading to the formation of the Applied Computational Electromagnetic Society (ACES). However, in recent times these shortcomings of the past decisions of the AP‐S leadership have been addressed.

The objective of this book is to advance the state of the art of antenna measurements and not being limited to the situation that measurements can be made in one of the separable coordinate systems just like the state of electromagnetics over half a century ago. We propose to carry out this transformation in the realm of measurement first by trying to find a set of equivalent currents just like we do in theory and then solve for these unknown currents using the Maxwell‐Hertz‐Heaviside equations via the Method of Moments popularized by Prof. Harrington. Since the expressions between the measured fields and the unknown currents are analytic and related by Maxwell‐Hertz‐Heaviside equations, the measurements can be carried out in any arbitrary geometry and not just limited to the planar, cylindrical or spherical geometries. The advantage of this new methodology as presented in this book through the topic “Source Reconstruction Method” is that the measurement of the fields need not be done using a Nyquist sampling criteria which opens up new avenues particularly in the very high frequency regime of the electromagnetic spectrum where it might be difficult to take measurement samples half a wavelength apart. Secondly as will be illustrated these measurement samples need not even be performed in any specified plane. Also because of the analytical relationship between the sources that generate the fields and the fields themselves it is possible to go beyond the Raleigh resolution limit and achieve super resolution in the diagnosis of radiating structures. In the Raleigh limit the resolution is limited by the uncertainty principle and that is determined by the length of the aperture whose Fourier transform we are looking at whereas in the super resolution system there is no such restriction. Another objective of this book is to outline a very simple procedure to recover the non‐minimum phase of any electromagnetic system using amplitude‐only data. This simple procedure is based on the principle of causality which results in the Hilbert transform relationship between the real and the imaginary parts of a transfer function of any linear time invariant system. The philosophy of model order reduction can also be implemented using the concepts of total least squares along with the singular value decomposition. This makes the ill‐posed deconvolution problem quite stable numerically. Finally, it is shown how to interpolate and extrapolate measured data including filling up the gap of missing measured near/far‐field data.

The book contains ten chapters. In Chapter 1, the mathematical preliminaries are described. In the mathematical field of numerical analysis, model order reduction is the key to processing measured data. This also enables us to interpolate and extrapolate measured data. The philosophy of model order reduction is outlined in this chapter along with the concepts of total least squares and singular value decomposition.

In Chapter 2, we present the matrix pencil method (MPM) which is a methodology to approximate a given data set by a sum of complex exponentials. The objective is to interpolate and extrapolate data and also to extract certain parameters so as to compress the data set. First the methodology is presented followed by some application in electromagnetic system characterization. The applications involve using this methodology to deembed device characteristics and obtain accurate and high resolution characterization, enhance network analyzer measurements when not enough physical bandwidth is available for measurements, minimize unwanted reflections in antenna measurements and, when performing system characterization in a non‐anechoic environment, to extract a single set of exponents representing the resonant frequency of an object when data from multiple look angles are given and compute directions of arrival estimation of signals along with their frequencies of operation. This method can also be used to speed up the calculation of the tails encountered in the evaluation of the Sommerfeld integrals and in multiple target characterization in free space from the scattered data using their characteristic external resonance which are popularly known as the singularity expansion method (SEM) poles. References to other applications, including multipath characterization of a propagating wave, characterization of the quality of power systems, in waveform analysis and imaging and speeding up computations in a time domain electromagnetic simulation. A computer program implementing the matrix pencil method is given in the appendix so that it can easily be implemented in practice.

In numerical analysis, interpolation is a method of estimating unknown data within the range of known data from the available information. Extrapolation is also the process of approximating unknown data outside the range of the known available data. In Chapter 3, we are going to look at the concept of the Cauchy method for the interpolation and extrapolation of both measured and numerically simulated data. The Cauchy method can deal with extending the efficiency of the moment method through frequency extrapolation. Interpolating results for optical computations, generation of pass band using stop band data and vice versa, efficient broadband device characterization, effect of noise on the performance of the Cauchy method and for applications to extrapolating amplitude‐only data for the far‐field or RCS interpolation/extrapolation. Using this method to generate the non‐minimum phase response from amplitude‐only data, and adaptive interpolation for sparsely sampled data is also illustrated. In addition, it has been applied to characterization of filters and extracting resonant frequencies of objects using frequency domain data. Other applications include non‐destructive evaluation of fruit status of maturity and quality of fruit juices, RCS applications and to multidimensional extrapolation. A computer program implementing the Cauchy method has been provided in the Appendix again for ease of understanding.

The previous two chapters discussed the parametric methods in the context of the principle of analytic continuation and provided its relationship to reduced rank modelling using the total least squares based singular value decomposition methodology. The problem with a parametric method is that the quality of the solution is determined by the choice of the basis functions and use of unsuitable basis functions generate bad solutions. A priori it is quite difficult to recognize what are good basis functions and what are bad basis functions even though methodologies exist in theory on how to choose good ones. The advantage of the nonparametric methods presented in Chapter 4 is that no such choices of the basis functions need to be made as the solution procedure by itself develops the nature of the solution and no a priori information is necessary. This is accomplished through the use of the Hilbert transform which exploits one of the fundamental properties of nature and that is causality. The Hilbert transform illustrates that the real and imaginary parts of any nonminimum phase transfer function for a causal system satisfy this relationship. In addition, some parametrization can also be made of this procedure which can enable one to generate a nonminimum phase function from its amplitude response and from that generate the phase response. This enables one to compute the time domain response of the system using amplitude only data barring a time delay in the response. This delay uncertainty is removed in holography as in such a procedure an amplitude and phase information is measured for a specific look angle thus eliminating the phase ambiguity. An overview of the technique along with examples are presented to illustrate this methodology. The Hilbert transform can also be used to speed up the spectral analysis of nonuniformly spaced data samples. Therefore, in this section a novel least squares methodology is applied to a finite data set using the principle of spectral estimation. This can be applied for the analysis of the far‐field pattern collected from unevenly spaced antennas. The advantage of using a non‐uniformly sampled data is that it is not necessary to satisfy the Nyquist sampling criterion as long as the average value of the sampling rate is less than the Nyquist rate. Accurate and efficient computation of the spectrum using a least squares method applied to a finite unevenly spaced data is also studied.

In Chapter 5, the source reconstruction method (SRM) is presented. It is a recent technique developed for antenna diagnostics and for carrying out near‐field (NF) to far‐field (FF) transformation. The SRM is based on the application of the electromagnetic Equivalence Principle, in which one establishes an equivalent current distribution that radiates the same fields as the actual currents induced in the antenna under test (AUT). The knowledge of the equivalent currents allows the determination of the antenna radiating elements, as well as the prediction of the AUT‐radiated fields outside the equivalent currents domain. The unique feature of the novel methodology presented is that it can resolve equivalent currents that are smaller than half a wavelength in size, thus providing super‐resolution. Furthermore, the measurement field samples can be taken at spacing greater than half a wavelength, thus going beyond the classical sampling criteria. These two distinctive features are possible due to the choice of a model‐based parameter estimation methodology where the unknown sources are approximated by a basis in the computational Method of Moment (MoM) context and, secondly, through the use of the analytic free space Green’s function. The latter condition also guarantees the invertibility of the electric field operator and provides a stable solution for the currents even when evanescent waves are present in the measurements. In addition, the use of the singular value decomposition in the solution of the matrix equations provides the user with a quantitative tool to assess the quality and the quantity of the measured data. Alternatively, the use of the iterative conjugate gradient (CG) method in solving the ill‐conditioned matrix equations for the equivalent currents can also be implemented. Two different methods are presented in this section. One that deals with the equivalent magnetic current and the second that deals with the equivalent electric current. If the formulation is sound, then either of the methodologies will provide the same far‐field when using the same near‐field data. Examples are presented to illustrate the applicability and accuracy of the proposed methodology using either of the equivalent currents and applied to experimental data. This methodology is then used for near‐field to near/far‐field transformations for arbitrary near‐field geometry to evaluate the safe distance for commercial antennas.

In Chapter 6, a fast and accurate method is presented for computing far‐field antenna patterns from planar near‐field measurements. The method utilizes near‐field data to determine equivalent magnetic current sources over a fictitious planar surface that encompasses the antenna, and these currents are used to ascertain the far fields. Under certain approximations, the currents should produce the correct far fields in all regions in front of the antenna regardless of the geometry over which the near‐field measurements are made. An electric field integral equation (EFIE) is developed to relate the near fields to the equivalent magnetic currents. Method of moments (MOM) procedure is used to transform the integral equation into a matrix one. The matrix equation is solved using the iterative conjugate gradient method (CGM), and in the case of a rectangular matrix, a least‐squares solution can still be found using this approach for the magnetic currents without explicitly computing the normal form of the equations. Near‐field to far‐field transformation for planar scanning may be efficiently performed under certain conditions by exploiting the block Toeplitz structure of the matrix and using the conjugate gradient method (CGM) and the fast Fourier transform (FFT), thereby drastically reducing computation time and storage requirements. Numerical results are presented for several antenna configurations by extrapolating the far fields using synthetic and experimental near‐field data. It is also illustrated that a single moving probe can be replaced by an array of probes to compute the equivalent magnetic currents on the surface enclosing the AUT in a single snapshot rather than tediously moving a single probe over the antenna under test to measure its near‐fields. It is demonstrated that in this methodology a probe correction even when using an array of dipole probes is not necessary. The accuracy of this methodology is studied as a function of the size of the equivalent surface placed in front of the antenna under test and the error in the estimation of the far‐field along with the possibility of using a rectangular probe array which can efficiently and accurately provide the patterns in the principal planes. This can also be used when amplitude‐only data are collected using an array of probes. Finally, it is shown that the probe correction can be useful when the size of the probes is that of a resonant antenna and it is shown then how to carry it out.

In Chapter 7, two methods for spherical near‐field to far‐field transformation are presented. The first methodology is an exact explicit analytical formulation for transforming near‐field data generated over a spherical surface to the far‐field radiation pattern. The results are validated with experimental data. A computer program involving this method is provided at the end of the chapter. The second method presents the equivalent source formulation through the SRM described earlier so that it can be deployed to the spherical scanning case where one component of the field is missing from the measurements. Again the methodology is validated using other techniques and also with experimental data.

Two deconvolution techniques are presented in Chapter 8 to illustrate how the ill‐posed deconvolution problem has been regularized. Depending on the nature of the regularization utilized which is based on the given data one can obtain a reasonably good approximate solution. The two techniques presented here have built in self‐regularizing schemes. This implies that the regularization process, which depends highly on the data, can be automated as the solution procedure continues. The first method is based on solving the ill‐posed deconvolution problem by the iterative conjugate gradient method. The second method uses the method of total least squares implemented through the singular value decomposition (SVD) technique. The methods have been applied to measured data to illustrate the nature of their performance.

Chapter 9 discusses the use of the Chebyshev polynomials for approximating functional variations arising in electromagnetics as it has some band‐limited properties not available in other polynomials. Next, the Cauchy method based on Gegenbauer polynomials for antenna near‐field extrapolation and the far‐field estimation is illustrated. Due to various physical limitations, there are often missing gaps in the antenna near‐field measurements. However, the missing data is indispensable if we want to accurately evaluate the complete far‐field pattern by using the near‐field to far‐field transformations. To address this problem, an extrapolation method based on the Cauchy method is proposed to reconstruct the missing part of the antenna near‐field measurements. As the near‐field data in this section are obtained on a spherical measurement surface, the far field of the antenna is calculated by the spherical near‐field to far‐field transformation with the extrapolated data. Some numerical results are given to demonstrate the applicability of the proposed scheme in antenna near‐field extrapolation and far‐field estimation. In addition, the performance of the Gegenbauer polynomials are compared with that of the normal Cauchy method using Polynomial expansion and the Matrix Pencil Method for using simulated missing near‐field data from a parabolic reflector antenna.

Typically, antenna pattern measurements are carried out in an anechoic chamber. However, a good anechoic chamber is very expensive to construct. Previous researches have attempted to compensate for the effects of extraneous fields measured in a non‐anechoic environment to obtain a free space radiation pattern that would be measured in an anechoic chamber. Chapter 10 illustrates a deconvolution methodology which allows the antenna measurement under a non‐anechoic test environment and retrieves the free space radiation pattern of an antenna through this measured data; thus allowing for easier and more affordable antenna measurements. This is obtained by modelling the extraneous fields as the system impulse response of the test environment and utilizing a reference antenna to extract the impulse response of the environment which is used to remove the extraneous fields for a desired antenna measured under the same environment and retrieve the ideal pattern. The advantage of this process is that it does not require calculating the time delay to gate out the reflections; therefore, it is independent of the bandwidth of the antenna and the measurement system, and there is no requirement for prior knowledge of the test environment.

This book is intended for engineers, researchers and educators who are planning to work in the field of electromagnetic system characterization and also deal with their measurement techniques and philosophy. The prerequisite to follow the materials of the book is a basic undergraduate course in the area of dynamic electromagnetic theory including antenna theory and linear algebra. Every attempt has been made to guarantee the accuracy of the content of the book. We would however appreciate readers bringing to our attention any errors that may have appeared in the final version. Errors and/or any comments may be emailed to one of the authors, at salazar@tsc.uc3m.es, mingda.zhu@live.com, hchen43@syr.edu.