Modern Characterization of Electromagnetic Systems and its Associated Metrology. Magdalena Salazar-Palma

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