54 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 12, 2013

Identification of Multiple Objects Using Their NaturalResonant Frequencies

Woojin Lee, Tapan K. Sarkar, Fellow, IEEE, Hongsik Moon, and Magdalena Salazar-Palma, Senior Member, IEEE

Abstract—A methodology for identification of multiple objectsusing their natural resonant frequencies is presented. Thismethod-ology is then applied to generate a library of poles of various objectsin the frequencydomainandcompare themto thecomputednaturalpoles of unknown objects using the typical transient temporal latetime response in the timedomain. TheCauchymethod is applied di-rectly in the frequency domain to extract the Singularity ExpansionMethod (SEM)poles, and thus a library of poles of various perfectlyconducting objects (PEC) is generated. The Matrix Pencil (MP)method is thenapplied to the late time response to compute theSEMpoles for identification in the timedomain. Simulation examples areanalyzed to illustrate the potential of this proposed methodology.

Index Terms—Cauchymethod,Matrix Pencil (MP)method, nat-ural poles, resonance, scattered electromagnetic field, SingularityExpansion Method (SEM).

I. INTRODUCTION

T HE RADAR system has long been used for detection andidentification of objects using the reflected scattered en-

ergy from a target illuminated by a radar system. The currentproblem is to detect and identify various objects of differentshapes, made of composite materials that may be buried under-ground, using the scattered electromagnetic (EM) field. Mul-tiple studies to analyze these problems have been performedusing the resonance phenomena of the EM field scattered froman object because it is possible to associate its natural resonantfrequencies for identification of an object. The Singularity Ex-pansion Method (SEM) introduced by Baum [1] was to find thenatural resonant frequencies of an object using the late time re-sponse. There are the well-known relationships between the sig-nature of the object and the late time response.If we illuminate an object by a plane wave, the backscattered

energy from the object contains information about the nature ofthe object. If we consider the mechanisms of backscattered en-ergy from an object, the backscattered energy is composed oftwo parts. The first part is the early time response that containsan impulse pertaining to the specular reflection, i.e., it is a directreflection of the incident wave from the surface of the object.The second part is the oscillating part corresponding to the latetime response, i.e., it is the surface and creeping waves (externalresonances) and cavity waves (internal resonances) related to

Manuscript received December 17, 2012; accepted December 30, 2012. Dateof publication January 11, 2013; date of current version March 12, 2013.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 Leganes, 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.2013.2237746

the shape, size, and composition of the object. In the case of aperfectly conducting object (PEC), only surface creeping waves(external resonances) exist. Therefore, one can address signa-tures (dimension, shape, constitution, etc.) of an object usingthe SEM poles. The SEM poles yield a pair of complex conju-gate poles corresponding to the damped sinusoids using a typ-ical transient temporal response from various objects (e.g., an-tennas, canonical objects, and aircrafts).Such information about the SEM poles can also be obtained

using the Cauchy method in the frequency domain. One cannotice that it is not necessary to identify the early time and thelate time regions where the SEM formulation holds. The Cauchymethod is based on the approximation of a transfer function ofa linear time invariant system (LTI) in the frequency domainusing a rational function approximation. The pole computationsare carried out using the Cauchy method by approximating thetransfer function as a ratio of two rational polynomials [2], [3].Thus, one can generate a library of poles of various objects usingthe Cauchy method [4].The objective of this letter is to illustrate that by observing

the complete impulse response, the presence of multiple objectscan be isolated in the time domain. Then theMatrix Pencil (MP)method can be applied to the late time response of this transienttemporal impulse response [5], [6]. In the time domain, it is rel-atively easier to locate the late time response. The MP methodapproximates a time-domain function by a sum of complex ex-ponentials, and this approximation is valid only for the late timeresponse. By generating the pole library using the frequency-do-main data, and the actual poles computed using the time-domaindata, we illustrate that the correlation between the two pole setsobtained using totally different methodologies provide a robustidentification procedure. The poles using responses from datagenerated in different domains can be used for comparison pur-poses [7]. In addition, the Time-Difference-of-Arrival (TDOA)technique is introduced to find the coordinates of the unknownobject [8].In this letter, we start with the MPmethod to identify multiple

unknown objects by comparing them to the generated pole li-brary. Some simulation examples are presented to illustrate thisnovel and accurate way for generating and identifying the sig-natures of various objects.

II. PROCEDURE TO IDENTIFY THE UNKNOWN OBJECT

In general, the EM transient signal of the observed late timeresponse from an object can be formulated after sampling awaveform as

for (2.1)

1536-1225/$31.00 © 2013 IEEE

LEE et al.: IDENTIFICATION OF MULTIPLE OBJECTS USING THEIR NATURAL RESONANT FREQUENCIES 55

Fig. 1. Simulation model using one transmitter, two receivers, and one PECsphere with 0.15 m diameter.

for (2.2)

where , , , , , , and are the signal and noise inthe system, sampling time, residue of the th pole, th pole ofthe system, negative damping factor of the th pole, and angularfrequency of the th pole, respectively. Such a model is validbecause the scatterer can be treated as an LTI system [9]. Thetransient response from a structure can be characterized by thebest estimates of , , and using the MP method [5], [6].Fig. 1 shows the configuration of the Higher-Order-Basis-

Based Integral Equation Solver (HOBBIES) simulation modelusing one transmitter (center antenna), two receivers (left andright antennas), and one PEC sphere located in free space [10].The specifications for each of the antennas for transmitter andreceiver are 0.15 m length and 1.5 mm radius. The diameter ofthe sphere is 0.15 m. Spacing between the transmitter and thereceiver is 2.5 m to fully minimize the effects of the antennacoupling. The target sphere is located at 12.046 m and is ori-ented by from the axis of the transmitting antenna. We ap-plied a 1-V excitation to the transmitter. The response of theobject is computed from 0.01 to 5 GHz (sampling frequency

GHz), and the number of samples used is 500. Toisolate the response of the desired objects at the receiver, thereceived signals at both the antennas are computed with andwithout the presence of the object of interest. Subtracting oneresponse from the other one can reduce the coupling betweenthe various antennas. In addition, the antenna impulse responseneeds to be deconvolved out from the computed total responsefrom both the antennas and the object. Therefore, the decon-volved response from the object at each receiver can berepresented as

(2.3)

where and are the received signalwith the object present and the received signal without the objectpresent, respectively. is the response of a receivingantenna. Fig. 2 shows the response of the PEC sphere seen bythe left and right receivers in the frequency domain using (2.3).The peak in the object response around 3.5 GHz is due to theantenna response.For applying the MP method, we need to obtain the time-do-

main response of the object from the frequency-domain datausing the inverse fast Fourier transform (IFFT). Fig. 3 displays

Fig. 2. Frequency-domain response of the 0.15-m-diameter PEC sphere ob-tained from (a) the left receiver and (b) the right receiver.

Fig. 3. Time-domain response of the 0.15-m-diameter PEC sphere obtainedfrom (a) the left receiver and (b) the right receiver.

the time-domain response of the object (for the left and right re-ceivers). The MP method then is applied to extract the naturalpoles of the unknown object using only the late time response.We apply two criteria to extract the natural poles. The first cri-terion filters the poles out having very high damping factors( ). The second criterion allows one to remove singlepoles located on the real axis, poles having positive , and poleslocated in the range GHz and GHz.Fig. 4 displays the pole library [4] along with the computed

poles of the unknown object (one PEC sphere) using the MPmethod using the data from the left and the right receivers. If onecompares these two sets of computed poles from the time-do-main response of the left and the right receivers, they are almostidentical not only for the first order pole, but also for the res-onant frequencies of the higher-order poles. It means one canextract the SEM poles with high accuracy using the late timeresponse of the detected object irrespective of using the left orthe right receiver. Thus, one can identify the unknown object asa 0.15-m-diameter PEC sphere as shown in Fig. 4(a). To eval-uate the performance of this methodology, we compute the esti-mated error of the identification accuracy following the normal-ized mean square errors (MSEs) using the resonant frequenciesas

(2.4)

where is the -norm of a vector. and are the resonancefrequency of the pole library and extracted resonance frequency,respectively.

56 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 12, 2013

Fig. 4. Pole library versus computed poles of the unknown object (one PECsphere) using the MP method (from the left and right receivers). (a) First-orderpole. (b) Resonant frequency.

From Fig. 4, one can identify the detected object as a 0.15-m-diameter PEC sphere with an approximate 98% accuracy usingdata from both the receivers. From Fig. 3, one can also identifythe location of the object with respect to a global coordinatesystem using the TDOA [8]. If we assume that the locations ofthe left receiver, the right receiver, and the object are ,

, and , respectively, as shown in Fig. 1, then

(2.5)

where

(2.6)

(2.7)

(2.8)

(2.9)

where is the radial distance of the illuminated surface of theunknown object from the transmitter. and are the dis-tances of the surface of the object from the left receiver andright receiver, respectively. and are the time delay (peakof the impulse response) of the left receiver and the right re-ceiver, respectively. is the distance from the center of the ob-ject to the illuminated surface of the object by the antennas.From (2.5)–(2.9), one can calculate the radial distance ( )

(2.10)

Fig. 5. (a) Two-spheres model. (b) One-wire and one-cone model.

TABLE IACTUAL VERSUS ESTIMATED TARGET COORDINATES (A SPHERE)

where

(2.11)

We can also estimate the displacement of the object in anglefrom the direction of the normal using the law of cosine

(2.12)

where is the azimuthal angle of inclination from thenormal. Table I presents the actual versus estimated coordinatesof the target from the origin. It has a relative error of 0.26% forthe distance ( ) and 0.44% error for the . Therefore, onecan locate the 0.15-m-diameter PEC sphere with an approxi-mate 98% accuracy at 12.002 m radial distance andazimuthal angle.

III. SIMULATION EXAMPLES

Two simulation examples are presented to illustrate the ap-plication of this methodology for detection of multiple objects.The simulation setup is the same as outlined before. We alsoapply the same criteria to extract the natural poles for the un-known objects.Fig. 5 shows the simulation models of the two-PEC-spheres

case and one-PEC-cone, one-PEC-wire case. For thetwo-PEC-spheres model, the diameters of the first sphereand the second sphere are 0.1 and 0.15 m, respectively. Thelocation of the first sphere from the origin (the location of thetransmitter) and in angle with respect to the normal joiningthe three antennas are 10.154 m and 10 . The coordinates ofthe second PEC sphere are 12.046 m radial distance andazimuthal angle. For one-PEC-cone and one-PEC-wire model,the coordinates of the 0.1-m-diameter, 0.1-m-height PEC coneand the 0.1-m-length, 1-mm-radius PEC wire from the originare 7.2469 m radial distance, 15 azimuthal angle and 8.0601 mradial distance, 7 azimuthal angle, respectively.

LEE et al.: IDENTIFICATION OF MULTIPLE OBJECTS USING THEIR NATURAL RESONANT FREQUENCIES 57

Fig. 6. Pole library versus computed poles of the unknown objects (two PECspheres) using the MP method. (a) First-order pole. (b) Resonant frequency.

TABLE IIACTUAL VERSUS ESTIMATED TARGET COORDINATES (TWO SPHERES)

TABLE IIIACTUAL VERSUS ESTIMATED TARGET COORDINATES (A CONE AND A WIRE)

Tables II and III describe the actual versus estimated coordi-nates of the unknown objects from the origin using the TDOAtechnique. Figs. 6 and 7 show comparing the pole library usingthe Cauchy method to the computed poles of the unknownobjects using the MP method. From Table II and Fig. 6, onecan locate the 0.1-m-diameter PEC sphere with approximatelya 99% accuracy in 10.133 m radial distance, 10.071 azimuthalangle, and the 0.15-m-diameter PEC sphere with approxi-mately a 95% accuracy in 12.002 m radial distance,azimuthal angle. One also can identify the PEC cone (0.1 mdiameter and 0.1 m height) with approximately a 97% accuracyin 7.2642 m ( ), 15.036 , and the PEC wire (0.1 m diameter,1 mm radius) with approximately a 99% accuracy in 8.0802 m( ), 7.0419 as shown in Table III and Fig. 7. Finally, onecan detect and identify unknown multiple objects with highaccuracy using the proposed methodology.

Fig. 7. Pole library versus computed poles of the unknown objects (one PECcone and one PECwire) using the MPmethod. (a) First-order pole. (b) Resonantfrequency.

ACKNOWLEDGMENT

The authors would like to thank the reviewers for suggestingways to improve the readability of the manuscript.

REFERENCES[1] C. E. Baum, “On the singularity expansion method for the solution of

electromagnetic interaction problems,” Air ForceWeapons Laboratory,EMP Interaction Note 88, 1971.

[2] K. Kottapalli, T. K. Sarkar, Y. Hua, E. K. Miller, and G. J. Burke,“Accurate computation of wide-band response of electromagnetic sys-tems utilizing narrow-band information,” IEEE Trans. Microw. TheoryTech., vol. 39, no. 4, pp. 682–687, Apr. 1991.

[3] J. Yang and T. K. Sarkar, “Interpolation/extrapolation of radar crosssection (RCS) data in the frequency domain using the Cauchymethod,”IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2844–2851, Oct.2007.

[4] W. Lee, T. K. Sarkar, H. Moon, and M. Salazar-Palma, “Computationof the natural poles of an object in the frequency domain using theCauchy method,” IEEE Antennas Wireless Propag. Lett., no. 11, pp.1137–1140, 2012.

[5] Y. Hua and T. K. Sarkar, “Generalized pencil-of-function method forextracting poles of an EM system from its transient response,” IEEETrans. Antennas Propag., vol. 37, no. 2, pp. 229–234, Feb. 1989.

[6] R. S. Adve, T. K. Sarkar, O. M. Pereira-Filho, and S. M. Rao, “Extrap-olation of time-domain responses from three-dimensional conductingobjects utilizing the matrix pencil technique,” IEEE Trans. AntennasPropag., vol. 45, no. 1, pp. 147–156, Jan. 1997.

[7] W. Lee, T. K. Sarkar, H. Moon, and L. Brown, “Detection and identi-fication using natural frequency of the perfect electrically conducting(PEC) sphere in the frequency domain and time domain,” inProc. IEEEAP-S/URSI Int. Symp., Jul. 2011, pp. 2334–2337.

[8] K. W. Cheung, H. C. So, W. K. Ma, and Y. T. Chan, “Least squaresalgorithms for time-of-arrival-based mobile location,” IEEE Trans.Signal Process., vol. 52, no. 4, pp. 1121–1128, Apr. 2004.

[9] A. V. Oppenheim, R.W. Schafer, and J. R. Buck, Discrete-Time SignalProcessing, 2nd ed. Upper Saddle River, NJ, USA: Prentice-Hall,1999.

[10] “HOBBIES (Higher Order Basis Based Integral Equation Solver),”2010 [Online]. Available: http://www.em-hobbies.com