Identification of an unknown dielectric target in a half-space using the E-pulse technique



[1] This paper describes an approach to model the impulse response of a dielectric target below a half-space using a time domain deconvolution technique. The complex natural resonances (CNR) of the target are extracted from the time domain impulse response using a matrix pencil method and can be correlated with the target's physical properties. An automated E-pulse scheme has been used to discriminate changes in the physical properties of target. The discrimination performance is calculated quantitatively by using an energy discrimination number (EDN) and discrimination ratio (DR). This technique can ultimately be used as a diagnostic tool in detection and discrimination of an unknown target inside a half-space.

1. Introduction

[2] With the advent of ultra-wideband (UWB), there has been a considerable amount of interest shown in the application of the transient or impulse response of a radar target [Rothwell and Sun, 1990; LoVetri et al., 1998]. A typical application for such a system is the use of a UWB illuminating pulse in ground penetrating radar (GPR). However, the detection and identification of buried and subsurface objects still remains a difficult problem, as there is no single sensor or diagnostic tool which can provide sufficient information about the object. When a suitably short pulse interacts with an unknown target (radar target), it produces a time-domain scattered field waveform. As the interrogating pulse passes through the target, first, specular reflections are returned from the localized scattering centres of the target producing impulse like early-time components along with partial resonances of the target, and then, the target's global resonance phenomenon occurs after the interrogating pulse has passed through the target completely. Baum has proposed a model to represent this global resonance phenomena and it is popularly known as the Singularity Expansion Method (SEM). The Singularity Expansion Method (SEM) has been used for a long time in the detection and recognition of targets in free space [Pearson and Robertson, 1980; Baum and Pearson, 1981; Van Blaricum and Mittra, 1975; Chen and Peters, 1997; Chen, 1997; Cho and Cordaro, 1980]. Whilst this method is usually used to represent the scattered signal from the object in terms of a sum of complex exponentials; the signal that SEM operates on also contains the impulse response of the system which can be obtained via a deconvolution process if the incident signal is known [Cho and Cordaro, 1980; LoVetri et al., 1998; Sarkar and Pereira, 1995; Tseng and Sarkar, 1984, 1987; Padhi et al., 2006]. Similar techniques and methodologies are proving useful in pre-processing of GPR data in tracking and identification of buried landmines [LoVetri et al., 1998]. Although the resonance behavior of the target occurs in the late time portion of the transient signal, both the early and late time portions of the signature are useful for extracting the features of the target [Chen and Peters, 1997; Chen, 1997; Cho and Cordaro, 1980; LoVetri et al., 1998]. However, we primarily focus on the late time response of the target as it contains target specific information.

[3] The primary objective of this research work is to detect and identify a dielectric target buried below a homogeneous half-space from the backscattered time domain signature. The present work is an extension of our earlier work reported by Padhi et al. [2006], where the measurements were performed to extract the CNRs of cylindrical target (PEC) placed in free space. It is well known that the transient scattering of electromagnetic waves from a conducting or dielectric target contains information about the resonance behavior of the target itself as quantified by the complex poles or complex natural resonances (CNR) of the target. The CNRs are theoretically aspect independent, depending only on the shape and electrical properties of the target such as permittivity and conductivity. When a plane wave is obliquely incident on a half-space specified by its permittivity and conductivity, the reflected field can be obtained by multiplying the incident field and the reflection coefficient [Balanis, 1989]. This multiplication relationship in the frequency domain can be interpreted as a convolution in the time domain and analytical results for the reflection coefficient in the time domain can be found in the literature [Suk, 2000]. As the target (embedded in the half-space) is excited by a short-pulse plane wave, diffraction and scattering will occur and subsequently the late-time energy will decay similar to the late-time radiation mechanism. This late-time field contains the signature both from the target and the interface medium. The identification of targets embedded in half-space environment is a difficult problem. As the dielectric contrast is too high, Born approximation cannot be used in this context. Therefore, considering the scattering mechanism of an unknown target embedded below the half-space, it is conjectured that the transient response of target itself might be recovered from the system response using a deconvolution technique, which could remove the interface reflection from the response. Many deconvolution techniques are described and applied in radar signal processing and image processing fields both in the time domain and frequency domain. However, the problems that involve nonlinear media (half-spaces) can be modeled more efficiently using a time domain approach rather than the frequency domain approach [Rao, 1999]. For example, CNR, or SEM poles, may be computed directly using time domain methods. The problem this paper addresses is therefore: given the return from the target including the interface, under what conditions can we extract the impulse response and consequently a set of CNRs that approximately characterizes the target? For the extraction purposes in this paper we use the matrix pencil method to extract the CNRs from the isolated late time impulse response of target [Sarkar and Pereira, 1995].

[4] Radar target identification or discrimination using the time domain response of a target due to an interrogating transient incident waveform (pulse) has been a major research objective since the development of the kill-pulse (K-pulse) originated by Kennaugh [1981]. A K-pulse is a synthesized excitation waveform constructed for a particular target which results in a minimization of the scattered field. Excitation with the same waveform for dissimilar targets, results in a larger scattered response. The extinction-pulse (E-pulse) is similar to the K-pulse and has been suggested as an alternative scheme for performing resonance based aspect-independent target discrimination [Rothwell et al., 1985; Ilavarasan et al., 1993; Li et al., 1998; Chen et al., 1986; Rothwell et al., 1994]. The E-pulse scheme is essentially based on annihilating the CNRs present in the scattered signal by matching the CNRs, with which the E-pulse is constructed, with the CNRs of the target. Equivalently, in the time domain, when the synthesized E-pulse is convolved with the late time target response a zero response in late time results [Mooney et al., 2000; Gallego et al., 1993; Stenholm et al., 2003; Baum, 1999a, 1999b]. The relevance of this work is that we are exploring the possibility of using resonance based methods to detect the unknown dielectric target embedded below a planar interface.

[5] This paper demonstrates the feasibility of target discrimination with an interface present by first stripping off the interface information by a deconvolution process and recovering the free space impulse response of target. The second step then differentiates the target using the standard E-pulse technique. The paper is organized as follows: the theoretical background including the standard deconvolution process and pole extraction procedures are described in section 2 and E-pulse techniques are described in section 3. The numerical results are described in section 3 and followed by conclusions in section 4.

2. Theoretical Background

[6] In this study, the finite difference time domain (FDTD) method is used for obtaining the transient signatures. The simulation setup is shown in Figure 1. A dielectric target with permittivity of ɛr2 and conductivity σ2 is embedded inside a homogeneous interface media of permittivity of ɛr1 and conductivity σ1. A plane wave excitation using a pulse doublet (first derivative of a Gaussian pulse) is used and the scattered waves are received in the forward direction.

Figure 1.

The geometry of problem. (a) Cuboid target and (b) cylindrical target. The other parameters are a = 1 cm, d = 1 cm, h = 1.0 cm, ɛr1 = 4.0, ɛr2 = 8.0–24, σ1 = 0.01, σ2 = 0.3–4 S/m, p = 2.0 cm.

2.1. Deconvolution Method

[7] For deconvolution we use the singular value decomposition (SVD) method [Rothwell and Sun, 1990]. Let h(t) represent the impulse response of target we are seeking, y1(t), the system response without the target present and y2(t) the response with the target present. Assuming the Fresnel reflection coefficient analogy, as defined in the time domain, the convolution can be written as:

equation image

[8] If an impulse excitation is used, y1(t) can be considered as the Fresnel Reflection coefficient and the reflection from the interface can be removed by the deconvolution. What remains is the response corresponding to the target inside the half-space.

[9] The aim of deconvolution is to extract the unknown h from y1 and y2. In matrix form, equation (1) can be written as:

equation image

where y1m and y2m are m discrete samples of y1 and y2, respectively. h(t) can be constructed from (2) accordingly;

equation image

Let y1 be decomposed using SVD as:

equation image

where U and V are (m × m) orthogonal matrices and S is a diagonal matrix of singular values. Since the system is ill-conditioned, the Fourier transform of V is taken and compared with the frequency content of the excitation signal. Only those singular vectors whose frequency content are contained within the excitation signal are considered in order to estimate h(t) and the rest are discarded. The solution of (3) can be written as:

equation image


equation image

[10] The estimated integrity of the impulse response can be characterized by calculating the maximum difference norm (Err) of y2est(t) and y2(t) as:

equation image

2.2. Pole Extraction

[11] The Singularity Expansion Method (SEM) is a technique of approximating the late time transient response of a scatterer as a series of damped exponentials. This is essentially adopting a circuit approach and using S-plane concepts in an EM scattering problem scenario. The technique was initially developed by Baum and has been used by others in extracting the CNRs from the transient signature [Chen and Peters, 1997]. It is shown that the late time response can be approximated by

equation image

where Am and sm (sm = αm ± ωm) are the residues and CNRs of the scatterer, respectively. The poles are represented as sm, with αm and ωm (2πfm) the mth damping coefficient and radian frequency, respectively. The early time response from the target is due to local specular reflections and other diffractions from the scatterer and does not concern us here. However, the late time response contains the CNR information and this can be extracted from the signature using the matrix pencil of functions (MPOF) approach. The details of MPOF method can be found in [Sarkar and Pereira, 1995; Yuan and Sarkar, 2006] and a brief discussion is given in Appendix A.

3. E-Pulse Technique

[12] The E-pulse technique is a radar target identification and discrimination scheme based on the aspect independent CNRs extracted from the late time of the target response. As noted previously, the E-pulse is a time domain filter for finite duration, which annihilates all the resonant modes when convolved with the target response in the late time [Rothwell et al., 1985]. Mathematically this is defined as:

equation image

or, alternatively (9) can be written as:

equation image

where e(t) is the E-pulse, r(t) is the target response defined in (8), TE is the duration of e(t), and * is the convolution operator . To quantify the performance of the E-pulse technique, the E-pulse discrimination number (EDNp,q) and the discrimination ratio (DRp,q), are used and defined in (11a)(11c);

equation image
equation image


equation image

where EDNp,p corresponds to the case where both the E-pulse and the target response are from Target p while EDNp,q corresponds to the case in which the E-pulse is from Target q and the target response is from Target p [Ilavarasan et al., 1993; Li et al., 1998; Mooney et al., 2000; Stenholm et al., 2003].

[13] The purpose of the EDN and DR parameters is to measure how much of the filtered signal c(t) is present after normalizing with respect to the energy of the E-pulse filter. For successful target discrimination, the correct target yields the smallest EDN value. Ideally, the EDN value should be zero provided the E-pulse is matched to the target producing the return. The physical interpretation of E-pulse can be expressed in the following expression:

equation image

where ef(t) is an excitatory component and nonvanishing during 0 ≤ t < Tf, the response to which is subsequently extinguished by ee(t), which follows during Tf ≤ t ≤ Te; where Tf and Te are the time duration of ef(t) and ee(t), respectively. There are two fundamental types of E-pulse, termed as ‘forced’ and ‘natural’ E-pulses. When, Tf > 0 the forcing vector is nonzero and a solution for ee(t) exists for all choice of Te. This type of E-pulse has a nonzero excitatory component and termed as ‘forced” e-pulse. On the other hand, when Tf = 0 the forcing vector vanishes and the solution exists for ee(t) matrix which corresponds to discrete eigenvalues obtained by finding the roots of the characteristic equation. Since there is no exciting component, this type of E-pulse is known as ‘natural’ E-pulse. In this work the ‘natural’ E-pulse is used. Further details on these two fundamental types of E-pulse can be found in the works of Rothwell et al. [1985, 1994].

4. Numerical Results

[14] In this work we considered two different dielectric targets for identification; a cube and a cylindrical target. The geometry of the problem is shown in Figure 1. An FDTD code was used to calculate the time domain response of the target embedded in a half-space. A plane wave (TM polarization) pulse doublet provided the excitation and the specularly scattered waves at a far-field point were recorded. The incident total pulse-width was 0.126 ns. The cell size Δx was chosen as 1 mm, the corresponding time step Δt was 1.98 ps using the Courant condition and the total time duration of the simulation was 2000 steps. In order to avoid dispersion, the criteria λ > Δx, was satisfied throughout the spectrum. The lattice was 100 × 100 × 100 cells including the associated absorbing boundary conditions which were of the Liao type. The pulse and its frequency spectrum are shown in Figure 2. As observed in Figure 2, the frequency content of the pulse is nonzero to 30 GHz. This frequency range is chosen because the target's CNRs, which are related to its size and constitutive parameters, are expected to fall in this range. Since the dominant CNR is a half wavelength resonant length, we expect 4 to 5 CNRs to fall within this frequency range.

Figure 2.

The incident pulse doublet and its frequency spectrum (pulse width 0.126 ns and time step 1.98 ps).

[15] Considering the weak scattering mechanism for the target embedded within the half-space, the transient response of target itself might be recovered from the system response using a deconvolution technique. Our procedure follows in two steps. First, by stripping off the interface information by a deconvolution process and recovering the free space impulse response of target, and then in the second step, differentiating the target using the standard E-pulse technique. For the first step, we considered two different cases (Case A and Case B) and compared the extracted impulse response of different targets; in (1) Case A: the target is embedded within the half-space and the impulse response of the target is extracted from the scattered data using the deconvolution algorithm using the data with and without the target present, and in (2) Case B: the target is in free space and the impulse response is extracted by deconvolving the incident pulse from the scattered data. In the second step, we use the standard discrimination technique such as E-pulse to discriminate the targets with different constituent parameters.

[16] In order to verify the efficacy of our algorithm, we successfully reproduced the example as reported by LoVetri et al. [1998] using both a point source and a plane wave excitation in our 3D-FDTD code. The extracted impulse response of the cube placed below the half-space is shown in Figure 3 along with the analytical data presented by LoVetri et al. [1998]. The comparison is very close and the extracted impulse response agrees reasonably well. The difference can be minimized by choosing higher singular values in SVD algorithm. We also performed bi-static RCS measurements using PEC targets to validate our deconvolution algorithm. In both cases, the complex natural resonances of the target were extracted from the transient responses and were found to be in excellent agreement with the analytical data reported by LoVetri et al. [1998] and the measured data reported by Tseng and Sarkar [1984].

Figure 3.

Comparison of the impulse response of a dielectric cube embedded in a half-space. (The details of target can be found in the work of LoVetri et al. [1998].)

[17] For bi-static RCS measurements, two ultra-wideband TEM horn antennas were used in a bi-static transmitter and receiver configuration. The TEM horns used were wideband operating from 1–18 GHz. PEC targets (sphere and cylinder) were used in the experiment. An Agilent 8510 VNA was used to measure the scattered signals. The transmitting antenna was excited through the VNA and a linear 30dB LNA was connected to the receiving antenna terminals to amplify the received signals. The operating frequency range was 45 MHz to 10 GHz in 512 steps. We expected the first 2–3 dominant modes to fall in this frequency range. The target was placed on top of a Styrofoam podium and placed in a far-field range. The scattered signals were collected both with and without the target placed on top of the stand. The frequency response data was transformed to the time domain data by using an IFFT routine and deconvolution and pole extraction processes were carried out as described in section 2. The detailed experimental procedures are outlined by Padhi and Shuley [2007] and therefore will not be repeated here.

[18] The first three CNRs for different canonical PEC targets are tabulated in Table 1. The theoretical CNR values of spherical and cylindrical targets were also calculated from tabulated data [Marin, 1974] and are also shown in the table. The measured CNRs of cylinder were compared with the published measured data in Tseng and Sarkar [1984]. In both cases, the agreement is close.

Table 1. Extracted Resonant Frequencies for Different PEC Targetsa
TargetTheory/ExperimentFm (GHz)
  • a

    Sphere A, diameter: 3.81 cm; Sphere B, diameter: 5.08 cm; Cylinder A, diameter: 1.91 cm and aspect ratio (length to radius): 0.5; Fm = Im(sm)/(2*π).

Sphere ATheory6.7910.39*
Sphere BTheory3.405.186.98
Cylinder ATheory7.288.379.60

4.1. SEM of a Dielectric Cylindrical Target

[19] Having verified our setup and algorithms for PEC targets, we proceed to verify with dielectric targets. The extraction process for obtaining the impulse response is as follows. First the scattered waves are collected at a far-field distance with and without the target present for Case A and these signals are denoted y2(t) and y1(t), respectively. The far field computed time domain response with dielectric permittivity and conductivity of the target as a parameter are shown in Figure 4. As can be seen, the specular reflection (early time response) from the scatterer is almost independent of the conductivity of the target. Minor variations with permittivity are seen only in the late-time period. The SVD method was then used to extract the impulse response. In the process the first 147 consecutive singular values were included in the estimation of the impulse response of the target. These values are chosen by correlating their frequency content with that of the illuminating pulse. The extracted impulse response of the target parameterized again by permittivity and conductivity are shown in Figure 5. The late-time response of the extracted impulse response is again sampled and input into the matrix pencil algorithm to extract the CNRs.

Figure 4.

The far-field response of the cylindrical target as a function of (a) conductivity and (b) permittivity. The incident pulse is at θ = 45°, ϕ = 0°, and TM polarized.

Figure 5.

The impulse response of a dielectric cylindrical target as a parameter of (a) conductivity and (b) permittivity.

[20] While extracting the poles, two different groups of poles are found. These two types are associated with the internal and external resonances of the dielectric structure. The external resonance poles are associated with surface waves creeping along the outer surface and very much depend on dielectric contrast between the media (ɛr2r1) and are highly damped. On the other hand, the internal resonance poles are associated with internal bouncing modes or cavity resonances and are independent of any external environment. These poles are located close to the real axis and are less damped [Baum, 1997]. For the cylindrical target, we use the closed form expressions as suggested by Gastine et al. [1967] to distinguish the internal and external resonance poles. The extracted CNRs are rank ordered using the expression in Lostanlen et al. [2002] and Kunz and Prewitt [1980] and the dominant pole with the highest energy content is selected and listed in Table 2 for Case A and Case B.

Table 2. Damping Coefficients and Poles for the Dielectric Cylindera
TargetCase ACase B
  • a

    The units for αm, 109 N/S and Fm = Im(sm)/(2*π), GHz.


[21] For Case B, the first 115 singular values were used in the extraction process and again the sampled late time responses were used to calculate the CNRs. We repeated the same process to select the dominant poles. The extracted dominant CNRs for both cases (Case A and Case B) are tabulated in Table 2. The mean value of the maximum difference norm (Err) for the estimated impulse response from (7) is 3.2 × 10−4. The extracted poles in Case B can be compared with those of Case A. In both cases, the variation of the CNRs with regard to constitutive parameters is very similar. As the conductivity increases, the damping coefficient (αm), as expected, moves further towards the left in the complex S-plane.

4.2. SEM of a Dielectric Cuboid Target

[22] Similar procedures were used in the extraction of the impulse responses for a cuboid target. The far field response and its embedded impulse response as a function of permittivity and conductivity were described by Padhi et al. [2006] and will not be repeated here. In the deconvolution process 118 and 129 singular values were chosen for Case A and Case B, respectively. The maximum error term (Err) was 1.39 × 10−6 while estimating the impulse responses.

[23] The damping coefficients, resonance frequency and the error term (Err) are tabulated in Table 3. The extracted poles in Case A are compared with those extracted for Case B and are shown in Table 3. In both cases, as expected, the variation of CNRs with regard to constitutive parameters is similar and as the conductivity increases, the damping coefficient (αm) also moves towards the left in the S-plane.

Table 3. Damping Coefficients and Poles for the Dielectric Cubea
TargetCase ACase B
  • a

    The units for αm, 109 N/S and Fm = Im(sm)/(2*π), GHz.


4.3. E-Pulse Discrimination

[24] The natural E-pulse, as outlined in section 2 is computed based on the extracted CNRs for both targets. In the calculation of the E-pulse, 10 and 25 pairs of CNRs and residues were used for the cylindrical and cuboid target, respectively.

[25] The poles with (αm > 0) are nonphysical and may be due to noise. The nonconjugate pole or unpaired complex poles appearing at the upper frequency limit as given by the Nyquist criterion are primarily due to the branch cuts associated with the interface. The nonconjugate pole pairs and all right hand sided poles were removed and the E-pulse for each target was individually constructed. The EDNp,q and DRp,q parameters were calculated using the expressions in (11), which quantify the energy level of the late time portion of the convolved signal and are listed in Tables 4 and 5. Only three sets of data i.e. ɛr = 8 and σ = 0.3, 2.0, 4.0 S/m and denoted as target (1), (2), and target (3), respectively, were considered for each cylindrical and cuboid shaped target for the E-pulse calculations.

Table 4. DR for Cylindrical and Cuboid Targets
 DR (p,q) (dB)
p q123
Table 5. DR (Cross Convolution) for Cylindrical and Cuboid Targets
r,σ)DR (p,q) (dB)

[26] The corresponding DR values obtained from expression (11b) are listed in table IV. For all cases in which the E-pulse and the target response are from the same target (shape and composition), both the numerator and denominator of (11b) are equal, resulting in 0 dB for DR(p,q). For a cylindrical target response from target (1), when convolved with the E-pulse of target (2) and (3), i.e. same shape but different composition, values of DR(p,q) = +118.46 dB and +124.24 dB are obtained, respectively, indicating a large difference from when the targets are identical. Similarly, for cuboid targets relative amplitudes of +249.46 dB and +242.69 dB respectively were obtained when the E-pulse for target (1) was convolved with the response of target (2) and target (3). These figures demonstrate that the E-pulse technique can successfully discriminate target 1 from the other 2 targets, provided that the CNRs are known a priori.

[27] To probe further, we also carried out the convolution between the cube and cylindrical target. For this exercise, we choose the cylinder and the cube as the two different targets for three different cases of permittivity and conductivity values (i.e. (ɛr, σ) = (8, 0.3), (8, 2.0) and (8, 4.0)). The cross convolution data (DR(dB)) are tabulated in Table 5. This example also demonstrates that the E-pulse algorithm can successfully discriminate between the different shapes of targets having the same material properties.

[28] It may be observed that the differences between EDNp,p (E-pulse and target response from the same target) and EDNp,q (E-pulse and target response are from different targets) are significantly high (at levels of 100 of dBs) also resulting in high values of DR. This is basically an artifact of using the same set of CNRs for constructing the E-pulse and the target response in order to test the identification capability of the technique. At the same time, both the E-pulse and target response are re-sampled to ensure that they both have the same sampling rate before convolution [Lui and Shuley, 2004], producing better E-pulse null convolutions, when the target and E-pulse are the same. Additionally, the results here are based on FDTD simulation. In an actual measurement, the accuracies of the extracted poles are limited by the measurement tool and the Signal to Noise Ratio (SNR) of the surrounding environment. It is expected that such high values of DRs are not achievable. More realistic figures could be obtained by adding a small amount of random noise to the measured target response which would normally be the case in an actual measurement. With the introduction of Gaussian noise, the E-pulse technique with DRs as parameter is capable of discriminating targets with a minimum SNR of at least 15 dB [Rothwell et al., 1985]. To verify the statement, the cross convolution between the cube and cylindrical target with (SNR = 15 dB) is shown in Table 6. As can be seen as the SNR value increases the DR value decreases and it becomes difficult to discriminate the targets.

Table 6. DR (Cross Convolution) for Cylindrical and Cuboid Targets With 15 dB SNR
r,σ)DR (p,q) (dB)

5. Conclusion

[29] This paper describes an approach to model the impulse response of a dielectric target in a half-space using a time domain deconvolution technique. An FDTD code is used to calculate the time domain responses. A singular value decomposition algorithm is used to estimate the impulse response of the target and a robust algorithm such as the matrix pencil method is used to calculate the CNRs of the scatterer. The CNRs of the target are extracted from the time domain impulse response using a matrix pencil method and can be correlated with the target's physical properties. The extracted poles are compared with the poles scattered by the target in free space. An automated E-pulse scheme has been used to discriminate the change in physical properties of target. The discrimination performance is calculated quantitatively by using an energy discrimination number (EDN) and the discrimination ratio (DR). The calculated results successfully discriminate the targets with small changes in conductivity. This approach has wide scale application in finding the signatures of dielectric target buried in half-space.

Appendix A.: Matrix Pencil of Function

[30] Matrix pencil and Prony method are popular in extracting the poles from late time transient data and details can be found in Yuan and Sarkar [2006] and Sarkar and Pereira [1995]. The matrix pencil approach is not only computationally efficient but also handles noisy data with better statistical properties in the estimation of poles as compared to polynomial methods.

[31] Rewriting the equation (8) as follows:

equation image

[32] And, after sampling with sample step ΔT, (A1) becomes;

equation image

where Ri and si are residues or complex amplitude and exponent. To solve for Ri and si, we first construct the Hankel matrix as:

equation image

and performing SVD as:

equation image

where the subscript H defines the conjugate transpose of a matrix and Ξ is a diagonal matrix with singular values σi of Y, L is the pencil parameter. The number of exponentials M can be chosen by analyzing the values of σi. Once M decided, the first M columns of U are used to build a new M × M matrix using a least square method and the eigenvalues of this matrix are the poles γi. The Ri are solved for from the following least squares problem:

equation image

[33] Finally the si can be obtained from the expression as (A6);

equation image


[34] The authors would like to acknowledge the anonymous reviewers for their comments and suggestions which helped to improve the clarity and readability of this article. This work was supported partly by the Australian Research Council under grants DP0557169.