IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 11, 2012 1137 Computation of the Natural Poles of an Object in the Frequency Domain Using the Cauchy Method Woojin Lee, Tapan K. Sarkar, Fellow, IEEE, Hongsik Moon, and Magdalena Salazar-Palma, Senior Member, IEEE Abstract—A methodology for the computation of the natural poles of an object in the frequency domain is presented. This methodology is then applied to compute the natural poles for perfectly conducting objects (PEC) in the frequency domain and compare the results to those obtained using the usual late time response. The main advantage of the proposed method is that there is no need to differentiate between the early time and the late time response of the object because the Cauchy method is applied to extract the Singularity Expansion Method (SEM) poles directly in the frequency domain. Simulation examples are analyzed to illustrate the potential of this method. Index Terms—Cauchy method, Matrix Pencil (MP) method, nat- ural poles, resonance, scattered electromagnetic field, Singularity Expansion Method (SEM). I. INTRODUCTION T HE RADAR system has long been used for detection of objects. From the scattered electromagnetic (EM) field from an object, it is possible to find its natural resonant frequencies for identification. The Singularity Expansion Method (SEM) introduced by Baum [1] was to find the natural resonant frequencies of an object from the late time response. The objective of this letter is to illustrate that such information about the SEM poles can also be obtained in the frequency domain where the restriction of the late time response to the SEM formulation is nonexistent. Therefore, we generate the library of poles for different objects entirely in the frequency domain using the Cauchy method in this letter. Thus, in this procedure for pole extraction, it is not necessary to identify the early time and the late time regions where the SEM formulation holds. The Cauchy method is based on the approximation of a transfer function of a Linear Time Invariant System (LTI) in the frequency domain using a rational function approxima- tion [2], [3]. This is carried out using the Cauchy method by approximating the transfer function as a ratio of two rational polynomials. This is different from the usual way of obtaining the SEM poles by applying the Matrix Pencil (MP) method to the late time response [4]–[6]. By generating the pole library Manuscript received August 06, 2012; revised September 06, 2012; accepted September 15, 2012. Date of publication September 19, 2012; date of current version October 11, 2012. W. Lee, T. K. Sarkar, and H. Moon are with the Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY 13244- 1240 USA (e-mail: wlee07@syr.edu; tksarkar@syr.edu; homoon@syr.edu). M. Salazar-Palma is with the Departmento de Teoria de la Senal y Commu- nicaciones, Universidad Carlos III de Madrid, 28911 Madrid, Spain (e-mail: m.salazar-palma@ieee.org). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LAWP.2012.2219846 using frequency-domain data and the actual poles computed using the time-domain data, we illustrate that the correlation between the two pole sets obtained using totally different methodologies provides a robust identification procedure. The poles using responses from data generated in different domains can be used for comparison purposes [7]. The unique feature of the Cauchy method is that it is not necessary to distinguish between the early and the late time re- sponses, and this may make this procedure accurate and effi- cient. Thus, one can generate a library of poles of various ob- jects using the Cauchy method. In this letter, we start with the Cauchy method to make a li- brary of poles of the object. Also, simulation examples illustrate this novel and accurate way for finding the SEM poles. II. PROCEDURE TO MAKE A LIBRARY OF POLES OF OBJECTS The Cauchy method starts by assuming that the parameter of interest that is to be extrapolated and/or interpolated, as a func- tion of frequency, can be performed using a ratio of two polyno- mials. This procedure holds for an LTI system [8]. Let us assume that the system response is an LTI system. The transfer function for an LTI system, as a rational function of frequency, can be characterized by (2.1) where the numerator and denominator polynomials are given by and , respectively. For convenience and compu- tational simplicity, we assume (2.2) where is the order of the numerator polynomial and is the order of the denominator polynomial. As seen from (2.1), the unknown coefficients and can be put into the following form: (2.3) where we have (2.4)–(2.6) as shown at the bottom of the next page. Here, the superscript denotes the transpose of a matrix. The size of the matrix is , so the solutions of and are unique only if the total number of the frequency sample points is greater than or equal to the number of unknown coefficients (2.7) 1536-1225/$31.00 © 2012 IEEE