Generalized Wideband Harmonic Imaging of Nonlinearly 

Loaded Scatterers: Theory, Analysis, and Application for 

Forward-Looking Radar Target Detection

by DaHan Liao

ARL-TR-7121 September 2014

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Army Research Laboratory

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ARL-TR-7121 September 2014

Generalized Wideband Harmonic Imaging of Nonlinearly 

Loaded Scatterers: Theory, Analysis, and Application for 

Forward-Looking Radar Target Detection

DaHan Liao

Sensors and Electron Devices Directorate, ARL

Approved for public release; distribution unlimited. 
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Generalized Wideband Harmonic Imaging of Nonlinearly Loaded Scatterers: 

Theory, Analysis, and Application for Forward-Looking Radar Target Detection

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DaHan Liao

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14. ABSTRACT

Wideband electromagnetic sensing and imaging of nonlinearly loaded scatterers is considered. Harmonic scattering theory is 

first presented, and then a generalized near-field, direct imaging functional is proposed for free-space and near-ground target 

localization within the context of forward-looking radar standoff detection exploiting sequential single-tone excitation. The 

developed scattering and imaging analysis framework is illustrated for point-like and extended targets through numerical 

experiments performed with a hybrid method-of-moments solver, in conjunction with a harmonic balance approach and an 

asymptotic field propagation technique. The steady-state harmonic scattering responses are examined in the time, frequency, 

and image domains for scatterers in free-space and half-space environments, and accurate target localization is demonstrated in 

all cases for each harmonic order considered.

15. SUBJECT TERMS

Computational electromagnetics, radar scattering and imaging, harmonic scattering, wideband, nonlinear loaded scattering

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DaHan Liao

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iii

Contents

List of Figures iv

1. Introduction 1

2. Theoretical Background 2

3. Numerical Experiments 5

3.1 Harmonic Scattering Characterization ............................................................................5

3.2 Imaging Results and Discussions....................................................................................6

4. Conclusions 16

5. References 17

Distribution List 20
iv

List of Figures

Fig. 1 Standoff sensing of nonlinearly loaded scatterer.................................................................3

Fig. 2 Harmonic imaging results for diode-loaded, free-space point-like target: a) Ψ1 (r) ; 

b) Ψ2 (r) ; c) Ψ3 (r) ; and d) Ψ4 (r) . F = [300 MHz, 1.5 GHz], transmitted power = 10 

W................................................................................................................................................7

Fig. 3 Normalized time-domain harmonic scattering responses for diode-loaded, free-space 

extended target: a) n = 1; b) n = 2; c) n = 3; and d) n = 4.........................................................9

Fig. 4 Normalized time-domain harmonic scattering responses for diode-loaded, on-surface 

extended target: a) n = 1; b) n = 2; c) n = 3; and d) n = 4........................................................10

Fig. 5 Normalized time-domain harmonic scattering responses for diode-loaded, subsurface 

extended target: a) n = 1; b) n = 2; c) n = 3; and d) n = 4.......................................................11

Fig. 6 Frequency-domain harmonic scattering responses for diode-loaded extended target 

for n = 1,2,3,4: a) free-space; b) on-surface; and c) subsurface...............................................12

Fig. 7 Harmonic imaging results for diode-loaded, free-space extended target: a) Ψ1 (r) ; b) 

Ψ2 (r) ; c) Ψ3 (r) ; and d) Ψ4 (r) . F = [300 MHz, 1.5 GHz], transmitted power = 10 

W..............................................................................................................................................13

Fig. 8 Harmonic imaging results for diode-loaded, on-surface extended target: a) Ψ1 (r) ; 

b) Ψ2 (r) ; c) Ψ3 (r) ; and d) Ψ4 (r) . F = [300 MHz, 1.5 GHz], transmitted power = 10 

W..............................................................................................................................................14

Fig. 9 Harmonic imaging results for diode-loaded, subsurface extended target: a) Ψ1 (r) ; 

b) Ψ2 (r) ; c) Ψ3 (r) ; and d) Ψ4 (r) . F = [300 MHz, 1.5 GHz], transmitted power = 10 

W..............................................................................................................................................15
1

1. Introduction

For route clearance and force protection, one technology of acute interest to the military is the 

standoff detection of targets using electromagnetic waves. An implementation of the technology 

in the form of the forward-looking radar (FLR) has been discussed in various studies1–5 within 

the context of in-road and roadside threat detection. A conventional FLR system—whether 

operating in, for example, impulse, step-frequency, or frequency-modulated continuous-wave 

(FMCW) mode—employs ultra-wideband signals to probe and—subsequently—image a scene 

by exploiting linear scattering responses. Since many prominent classes of targets encountered in 

theater are electronics-based and give off nonlinear re-radiated responses, recently there has been 

increased impetus in developing a nonlinear complement to the linear FLR. Nonlinear 

electromagnetic sensing—possessing its own unique set of implementation challenges 

notwithstanding—could potentially provide an additional mode of detection and has certain 

advantages compared to traditional radar technologies. For example, nonlinear detection can be 

leveraged to aid the extraction of a nonlinear target embedded in linear manmade or natural 

clutter; and analysis of the target harmonic re-emissions can lead to further opportunities in 

signature and image classification. The above discussions demonstrate there is the need for 

understanding the scattering properties of—as well as developing new imaging algorithms for—

nonlinear targets and hence establish the motivation for the current study.

The concept of nonlinear radar has been explored within the radio-frequency identification 

(RFID) community: associated applications range from insect tracking6 to inventory tagging,

7,8

wherein localization is usually achieved through direction-of-arrival and time-of-arrival 

techniques. The subject of nonlinear electromagnetic scattering has been investigated

extensively.

9–25 These works are pertinent to the modeling of nonlinearly loaded antennas, 

though it is conceivable that the analysis in these studies can be generalized to treat any 

nonlinearly loaded scatterer; unfortunately, the remote sensing aspect of the problem is not 

considered in detail—that is, these works do not discuss the exploitation of the harmonic 

scattering responses for target detection, localization, and imaging. A recent study26 presented a 

comprehensive numerical solver for characterizing the multistatic harmonic responses of 

nonlinearly loaded antenna targets under tone excitation in the presence of a half-space, and 

thence, proposed a subspace-based direct imaging technique for near-field location estimation of 

on-surface and buried targets. Image construction in that work is achieved not by wideband 

beamforming but rather by processing the “narrowband” scattered fields in the subspace domain 

at a single harmonic or multiple harmonics. The numerical experiment-derived imaging results 

reported26 indicate that while accurate target localization—and even “super-resolution”—can be 

obtained using an aspect-limited transceiver aperture consistent with that of the FLR, the 
2

imaging performance can be rather noise sensitive—which is typical of subspace-based methods. 

Note that the imaging algorithm described in that work necessitates the use of a fully multistatic 

scattered field matrix, which may not always be convenient to generate in practical sensing 

systems.

The current study is envisioned as a companion to the previous study,

26 with the focus on 

wideband imaging: more to the point, the objective here is to establish a wideband imaging 

framework based on the nonlinear scattering response that is directly complementary to the one 

based on the linear scattering response. The algorithm is expected to be equally applicable to all 

common sensing modes of interest—these include the fully multistatic configuration, as well as 

monostatic and bistatic ones. The presentation of this work is organized as follows. In Section 2, 

the conceptual underpinnings of nonlinear scattering theory and the harmonic imaging algorithm 

are first discussed. Then, in Section 3, numerical experiments are performed to illustrate the 

nonlinear FLR sensing problem as pertinent to step-frequency systems: viz., in Section 3.1, 

simulations are carried out using the previous scattering and propagation solver26 and, in 

Section 3.2, standoff harmonic imaging results—and interpretations thereof—are put forth for 

point-like and extended scatterers in free-space and half-space environments. Finally, in 

Section 4, a summary of this work is given.

2. Theoretical Background

The scene (Fig. 1) with the target (a nonlinearly loaded, perfectly conducting scatterer) is 

assumed to be excited by infinitesimal electric dipoles at r' transmitting time-harmonic fields at 

evenly spaced frequencies 1 2 { , ,..., }P f F ff f ∈ = . Given an induced Norton equivalent shortcircuit current I f i t a ft sc ( ) →= + sc ( ) cos 2( π θ ), the resultant voltage at the scatterer terminal is 

of the form

( ) ( ) ( ) ( ) ( )

0 0

, cos 2 , cos 2 . n

s n n

n n

v t f a ft α πθ β π θ f a nft n

∞ ∞

= =

=   + = + ∑ ∑   (1)
3

Fig. 1 Standoff sensing of nonlinearly loaded scatterer

For small signals, ( , ) ( ) n

n n β γ fa a f ≈ . The dependence of the purely real variables αn ( f a, ), 

βn ( f a, ), and γ n ( f ) on the scatterer impedance Z f s ( ) and the current-voltage (i v − )

characteristic of the load (it gvt ( ) ( ( )) = ) is implicit and hence suppressed.

For each instance of excitation at frequency f , the target generates harmonic scattered fields 

E nf s (r r , ', ) at frequencies nf ( n ≠ 0 ): for n =1, the steady-state scattered field is the sum of 

the structural mode E f s,0 (r r , ', ) —that is, the response from the scatterer when it is shortcircuited—and the re-radiation mode E f s r, (r r , ', ) ; for n >1, only the re-radiation mode 

E nf s r, (r r , ', ) exists. The re-radiation mode is evaluated by exciting the scatterer with voltage 

source V nf s ( ) at the terminal,

26 where V nf s ( ) is the phasor representation for the signal 

component of Eq. 1 with frequency nf .

For a point-like target at sr , the structural mode can be written as

E f G f fG f f s,0 (rr rr , ', , , ) = ( s s )  ,0 ( ) (r r s, ', , ) TX ( ) (2)

where TX ( f ) is the current moment of the transmitter, s,0 ( f ) is the scattering strength of the 

short-circuited target, and G f (·,·, ) is the Green’s function of the environment; the term 

Gff (r r s, ', )TX ( ) can be interpreted as equivalent to a scalar incident field E f inc s (r r, ', ) . The 

re-radiation mode can be deduced as follows. First, the short-circuit current is approximated as
4

I f fG f f sc ( ) =   e ( ) (r r s, ', , ) TX ( ) (3)

in which  e ( f ) is the scalar effective length27 of the scatterer as defined by exciting the terminal 

with a voltage source. Then from Eq. 1, it can be shown that ( ) ( ( ) ) ( ) , sc jn I f V nf f I f e s βn sc

∠ = . In 

turn, the re-radiated field is

( ) ( ) ( ) ( ) ( ( ) ) ( ) , , , , , ( ,, , ) ( ) sc jn I f E ',nf V nf G nf nf f I f e G nf nf s r s s r n sc β s r

∠

r r = r r   = r r (4)

with r (nf ) as the induced current moment at the scatterer due to a unity voltage source placed 

at the terminal. In view of Eqs. 2–4, a phase conjugation approach is proposed to obtain an 

imaging functional to locate the target at each harmonic order n :

( ) ( ) ( ) ( ) ( ) *

, ', , ', , , , n

n s ss

TX RX F

W f E nf G f G nf Ψ =   ∑∑∑   r rr r r rr (5)

where TX and RX , respectively, represent the transmitter and receiver array spaces (which are 

assumed to be collocated), W f ( ) is a spectral window function, and * denotes the conjugation 

operation. The above imaging functional is not limited to only low-power (small-signal) 

operations; it is general enough so that it can accommodate high-power (large-signal) sensing as 

well—which may be needed to detect targets with very weak nonlinear scattering.

The point spread function characteristics of Eq. 5 for free-space and near-ground (on-surface and 

shallow-buried) targets can be derived by following the procedure documented previously.28 The 

FLR sensing geometry (see Fig. 1) affords rather poor resolution in the vertical—or elevation—

direction, so the predominant issue of interest here is the behavior of the point spread function 

across the horizontal plane. In the forward-looking—or broadside—direction, assuming a 

rectangular spectral window, the first-null down-range and angular cross-range resolutions of

Ψn (r), respectively, can be shown to be

1

· ;

2

o c

r

n f

δ = ∆ (6)

1 , · ;

2

c o

n L

λ

δφ = (7)

where o c is the velocity of the electromagnetic wave in free-space; ∆f is the bandwidth of F ; 

λc o, is the wavelength at the center frequency of F ; and L is the width of the transceiver array 

aperture. The results from Eqs. 6 and 7 are valid for a monostatic sensing geometry as well as a 

bistatic configuration with two end-transmitters (i.e., one transmitter located at each end of the 

array and receiving over the entire aperture). For a fully multistatic system or a bistatic system 
5

with only one transmitter, Eq. 6 still holds, but the cross-range resolution is twice of that stated in 

Eq. 7. A tapered spectral window can be applied to suppress the sidelobes in range—at the 

expense of degrading the down-range resolution. In practice, note that depending on the nature of 

the window and the sensing configuration, the cross-range imaging behavior could also be 

affected. It is important to emphasize that the formulations for the resolution and the 

observations given above are equally valid for both free-space and near-ground target imaging.

In general, for a non-point-like—or extended—target and an arbitrary vector incident field, 

Eqs. 3 and 4 should be modified, respectively, as

I f sc ( ) = E rr inc s ( , ', · ; f f )  e ( ) (8)

and

( ) ( ( ) ) ( ) , , ', , ( ,, ;) sc jn I f

s r n sc r s nf f I f e β nf ∠ E rr = E rr (9)

where  e ( f ) is the vector effective length of the scatterer and E rr r s ( , ,nf ) is the radiated field 

pattern of the scatterer when excited with a unity voltage source. Also note that the structural 

mode for an extended target can be written as a superposition of responses from point-like 

scatterers. To obtain the most general treatment, the framework above can be expressed in terms 

of the dyadic Green’s function for the vector field problem; nevertheless, it is seen that it is 

sufficient to evaluate the imaging functional (Eq. 5) by using the dominant field and Green’s 

function components of interaction.

3. Numerical Experiments

Scattering and imaging simulations are performed in this section to illustrate the formalism 

established in the previous section.

3.1 Harmonic Scattering Characterization

The electromagnetic response of the scatterer is computed using a hybrid method-of-moments 

and harmonic balance approach, together with an asymptotic field propagator. The impedance of 

the scatterer Z f s ( ) and the short-circuit current at the scatterer terminal I f sc ( ) are first 

characterized in the spectral domain by applying the free-space and half-space mixed-potential 

integral equation solver.

29 Then—in the circuit domain—a harmonic balance technique14 is 

employed for obtaining the terminal voltage response V nf s ( ) of the target connected to the 

nonlinear load. In essence, after dividing the equivalent circuit for the scatterer into linear and 

nonlinear sections at the terminal reference plane and applying the standard nodal current law, a 
6

nonlinear matrix system is established and subsequently solved with an iterative method. Finally, 

the resultant harmonic currents over the structure are deduced using the integral equation solver 

once again. Note that for the half-space problem, for the calculation of the dyadic and scalar 

Green’s functions within the integral equation solver, exact Sommerfeld integrals are used. To 

increase the computational efficiency, a tabulation-and-interpolation routine is employed for 

evaluating these integrals. On the other hand, for field scattering and propagation, the secondorder-accurate asymptotic approximations to the integrals are exploited instead;

30,31 such a 

treatment allows the second-order surface wave component to be included in the solution. The 

overall simulation framework outlined above closely follows the one delineated previously,26 

where more details and formulations can be found.

3.2 Imaging Results and Discussions

A free-space, point-like target is examined first using the framework presented in the above 

sections. The target is a small vertical dipole element with length << λ min l , where λmin is the 

minimum wavelength over the harmonic frequency bands used for imaging. The dipole is centerloaded with a diode with it I vt v ( ) = o t (exp 1 ( ( ) ) − ), 10 o I = nA, 26 t v = mV. The sensing 

geometry is as illustrated in Fig. 1: the linear array is composed of 17 equally spaced elements 

distributed over a 2-m-wide aperture; the center of the array is at (10,0,2) m, with the target 

situated at the coordinate origin. A bistatic configuration employing a single illuminator is 

assumed: the illuminator—with transmitted power of 10 W at all incident frequencies and 

located at the center of the array—is a vertical infinitesimal electric dipole operating over the 

frequency band [300 MHz, 1.5 GHz] in P = 401 steps. For image generation, the scattered 

fields up to the fourth harmonic are collected for each frequency. (Here then λmin corresponds to 

the wavelength at frequency 4 Pf .) Fig. 2 shows Ψn (r) for n =1,2,3,4 . (The derived imaging 

theory can be applied to higher orders of n , of course, but for demonstration purposes, only the 

results for up to the fourth order are included in this work.) The Green’s function G f (·,·, )

within Eq. 5 is calculated asymptotically with the previously derived formulations.

30, 31 The 

images are constructed without any spectral window and are seen to match the theoretical point 

spread function28 very well, with the resolutions as predicted by Eqs. 6 and 7; the down-range 

and cross-range imaging sidelobe artifacts are also consistent with expectations. It should be 

noted that the re-radiated field E nf s r, (r r, ', ) is of the form n 2 f + ∝ in the small-signal regime 

whereas the structural scattered mode E f s,0 (r r, ', ) is 2 ∝ f (Rayleigh scattering); these 

frequency dependencies—which can also be readily derived from Eqs. 2 and 4—have been 

compensated in generating the images in Fig. 2.

The imaging and scattering characteristics of on-surface and shallow-buried point-like targets are 

similar to those of the free-space case and therefore are not explicitly included here.
7

(a) (b)

(c) (d)

Fig. 2 Harmonic imaging results for diode-loaded, free-space point-like target: a) Ψ1 (r) ; b) Ψ2 (r) ; 

c) Ψ3 (r) ; and d) Ψ4 (r) . F = [300 MHz, 1.5 GHz], transmitted power = 10 W.

A nonlinearly loaded extended scatterer is considered next. The scatterer is a vertical dipole 

element of length l = 0.5 m, center-loaded with a diode; the parameters of the diode and the 

sensing configuration are the same as those described above. The time and frequency domain 

backscattering responses at the center array element for up to the fourth harmonic are displayed 

in Figs. 3–6 for the free-space, on-surface, and shallow-buried scenarios. The image domain 
8

results corresponding to these scenarios are shown in Figs. 7–9. For the half-space problem, the 

ground soil has relative dielectric constant = 4 r ε and conductivity σ =15 mS/m, and the target 

is positioned directly on top of the surface or buried at 3 cm depth. For the time-domain signal 

reconstruction, the step-frequency-based excitation waveform is derived by taking the inverse 

Fourier transform of a Blackman spectral window. It is seen that, in these examples, the 

dominant contribution to the first-order harmonic return is provided by the structural mode 

scattering. As a result, at least according to the frequency-domain plots, the first-order harmonic 

response is quite dissimilar from the higher-order ( n >1) ones. The reverberations seen in the 

first-order harmonic time- and image-domain signals are due to single- and multiple-order 

diffractions from the ends of the dipole;

32, 33 these diffraction effects are also noted for the 

higher-order harmonic signals, though they are not as well defined as those in the first-order 

harmonic case. (In general, a direct correspondence between the peaks of the time-domain signal 

and the maxima in the image may not be readily observable in these results because of the 

complicated diffraction effects; however, an approximate equivalence between the time and 

image domain responses does exist for free-space and above-ground targets: it can be shown that 

the time-domain response of a transmitter and receiver pair is twice the real part of the complex 

image response contribution due to that pair.) 

As evident from Eqs. 8 and 9, note that the re-radiation mode is spectrally modulated by the 

radiation characteristics of the scatterer (as well as by the properties of the nonlinear loading). In 

these examples, the “shape” of each set of higher-order harmonic frequency domain responses in 

Fig. 6 is largely determined by the functions  e ( f ) and E rr r s ( , ,nf )—the variations of both of 

which also reflect the resonance properties of the scatterer. This spectral modulation makes it 

difficult to identity a specific order of diffraction from the dipole ends in the higher-order 

harmonic time and image responses. For the free-space and above-ground target cases, the 

higher-order harmonic frequency-domain responses (Figs. 6a and 6b, for n >1) are weaker over 

the second half of the band than over the first half since the main lobe of  e ( f ) for the vertical 

dipole—which occurs near the broadside direction (or near the plane of the ground) at the lower 

frequencies—shifts to higher elevation angles as the frequency increases toward 1.5 GHz; as 

such, the induced short-circuit current I f sc ( ) is reduced over the second half of the band, where 

the small-signal regime consequently occurs and, thus, the scattering signal intensity in general 

decreases with increasing harmonic number (i.e., , ( , ', ) ( )

n

E nf I f s r ∝ sc r r ). For the buried-target 

case, as expected, the scattering signal intensities on average are much weaker as compared to 

those of the free-space and on-surface target cases, and the small-signal regime is applicable over 

the entire band. Note that the four principal local maxima observed here in the higher-order 

harmonic spectral responses (Fig. 6c) correspond to the resonance frequencies of  e ( f ).
9

Fig. 3 Normalized time-domain harmonic scattering responses for diode-loaded, free-space extended target: 

a) n = 1; b) n = 2; c) n = 3; and d) n = 4

(a) (b)

(c) (d)
10

Fig. 4 Normalized time-domain harmonic scattering responses for diode-loaded, on-surface extended target: 

a) n = 1; b) n = 2; c) n = 3; and d) n = 4

(a) (b)

(c) (d)
11

Fig. 5 Normalized time-domain harmonic scattering responses for diode-loaded, subsurface extended target: 

a) n = 1; b) n = 2; c) n = 3; and d) n = 4

(a) (b)

(c) (d)
12

Fig. 6 Frequency-domain harmonic scattering responses for diode-loaded extended 

target for n = 1,2,3,4: a) free-space; b) on-surface; and c) subsurface

(a)

(b)

(c)
13

(a) (b)

(c) (d)

Fig. 7 Harmonic imaging results for diode-loaded, free-space extended target: a) Ψ1 (r) ; b) Ψ2 (r) ; 

c) Ψ3 (r) ; and d) Ψ4 (r) . F = [300 MHz, 1.5 GHz], transmitted power = 10 W.
14

(a) (b)

(c) (d)

Fig. 8 Harmonic imaging results for diode-loaded, on-surface extended target: a) Ψ1 (r) ; b) Ψ2 (r) ; 

c) Ψ3 (r) ; and d) Ψ4 (r) . F = [300 MHz, 1.5 GHz], transmitted power = 10 W.
15

(a) (b)

(c) (d)

Fig. 9 Harmonic imaging results for diode-loaded, subsurface extended target: a) Ψ1 (r) ; b) Ψ2 (r) ; 

c) Ψ3 (r) ; and d) Ψ4 (r) . F = [300 MHz, 1.5 GHz], transmitted power = 10 W.

Although the full-band data are always processed to obtain the results in Figs. 7–9, the images 

can also be constructed with only sub-band data—that is, for instance, given that a stronger 

response is observed over the first half of the band for the free-space and above-ground cases, 

one may choose to generate the images using the signal only from that sub-band; this, of course, 

would lead to a loss in image resolution.
16

The standoff sensing method investigated herein is envisioned for use primarily in the short 

range. From asymptotic analysis, in the small-signal regime, it can be shown that the amplitude 

of the n -th harmonic backscattered field has a dependence of the form 2( 1) n ρ− + ∝ , where ρ is 

the radial distance from the transceiver to the scatterer. The faster propagation attenuation rate of 

the higher-order harmonic signal intensities as a function of distance requires high power to be 

transmitted for long range applications.

4. Conclusions

A wideband, step-frequency-based harmonic image construction method has been formulated for 

the localization of free-space and near-ground nonlinearly loaded scatterers. The nonlinear 

scattering theory in support of the method has been outlined. Numerical experiments have been 

carried out to demonstrate the underlining concepts appearing herein. First, the steady-state 

harmonic scattering responses from point-like and extended targets are characterized using a 

hybrid frequency-domain full-wave solver, in conjunction with a harmonic balance approach and 

an asymptotic propagation technique. Then, the harmonic signals are analyzed in the time, 

frequency, and image domains for targets in free-space and half-space environments. The 

numerical examples illustrate that the targets are accurately localized by employing a forwardlooking bistatic sensing configuration consisting of a single transmitter and a linear array of 

receivers.

The overall analysis framework established in this study can be applied to investigate target 

scattering and imaging responses as a function of—for example—incident (or transmitted) 

power, frequency band of operation, standoff range, and sensing geometry. Although the mode 

of excitation of interest here is sequential single-tone illumination, the scattering algorithm and 

the imaging method are expected to be readily extendable to other modes of excitation, such as 

sequential double-tone and multiple-tone illumination over a wideband.
17

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4 Ressler M, Nguyen L, Koenig F, Wong D, Smith G. The ARL synchronous impulse 

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6 Riley JR, Smith AD. Design considerations for an harmonic radar to investigate the flight of 

insects at low altitude. Comput. Electron. Agr. 2002;35:151–169.

7 Nikitin PV, Rao KVS. Harmonic scattering from passive UHF RFID tags. Proc. IEEE 

Antennas and Propagat. Soc. Symp. 2009.

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