Biologically inspired coupled antenna beampattern design

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IOP PUBLISHING BIOINSPIRATION & BIOMIMETICS

Bioinsp. Biomim. 5 (2010) 046003 (11pp) doi:10.1088/1748-3182/5/4/046003

Biologically inspired coupled antennabeampattern designMurat Akcakaya and Arye Nehorai

Department of Electrical and Systems Engineering, Washington University in St Louis, St Louis,MO 63130, USA

E-mail: makcak2@ese.wustl.edu and nehorai@ese.wustl.edu

Received 16 April 2010Accepted for publication 19 October 2010Published 10 November 2010Online at stacks.iop.org/BB/5/046003

AbstractWe propose to design a small-size transmission-coupled antenna array, and correspondingradiation pattern, having high performance inspired by the female Ormia ochracea’s coupledears. For reproduction purposes, the female Ormia is able to locate male crickets’ callaccurately despite the small distance between its ears compared with the incomingwavelength. This phenomenon has been explained by the mechanical coupling between theOrmia’s ears, which has been modeled by a pair of differential equations. In this paper, wefirst solve these differential equations governing the Ormia ochracea’s ear response, andconvert the response to the pre-specified radio frequencies. We then apply the convertedresponse of the biological coupling in the array factor of a uniform linear array composed offinite-length dipole antennas, and also include the undesired electromagnetic coupling due tothe proximity of the elements. Moreover, we propose an algorithm to optimally choose thebiologically inspired coupling for maximum array performance. In our numerical examples,we compute the radiation intensity of the designed system for binomial and uniform ordinaryend-fire arrays, and demonstrate the improvement in the half-power beamwidth, sidelobesuppression and directivity of the radiation pattern due to the biologically inspired coupling.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Localization of sources and targets with high accuracy isimportant in many civil and military applications, attractingsignificant attention over the past few decades [1–6]. Theseapplications require antenna arrays with high radiationperformance. Most existing array design methods relyon the inter-elemental time delay of the antenna array tocontrol radiation pattern (beampattern) characteristics [7]. Theperformance of the antenna array is directly proportional tothe size of the array’s electrical aperture, such that largeaperture arrays are required to achieve highly directed, narrowbeampatterns with low sidelobe levels (SLLs). However,for tactical and mobile applications, many civil and militarysensing systems are confined to small spaces, requiring small-sized arrays, which hampers their radiation performance.

In this paper, we propose a beampattern design approachto achieve high performance with small aperture arrays. Theapproach is inspired by a parasitoid tachinid fly called Ormia

ochracea. To perpetuate its species, a female Ormia must finda male field cricket using the cricket’s mating call. The femaleOrmia has a remarkable ability to locate these crickets veryaccurately using binaural (two-ear) cues (interaural differencesin intensity and arrival time from an incident acoustic wave).This is unexpected due to the significant mismatch between thewavelength of the cricket’s call (about 7 cm) and the distancebetween the fly’s ears (about 1.2 mm) which gives rise to cuesthat are extremely small to be detectable by the central nervoussystem of the fly [8–13].

Experimental research in [14] explains that the Ormia’slocalization ability arises from a mechanical coupling betweenits ears, modeled as a system consisting of spring and dash-pots (with six key parameters: {(ci, ki) : i = 1, 2, 3}) asshown in figure 1. In the equivalent mechanical system,the intertympanal bridge (the cuticular connecting structure)is assumed to consist of two rigid bars connected at thepivot through a coupling spring k3 and dash-pot c3. The

1748-3182/10/046003+11$30.00 1 © 2010 IOP Publishing Ltd Printed in the UK

Bioinsp. Biomim. 5 (2010) 046003 M Akcakaya and A Nehorai

Figure 1. Mechanical model of the female Ormia ochracea’s ears.Reproduced with permission from [14]. Copyright 1995, AcousticSociety of America.

springs and dash-pots located at the extreme ends of thebridge approximate the dynamical properties of the tympanalmembranes, sensory organs and surrounding structures in theOrmia’s two ears.

In our previous work [15], we analyzed the localizationaccuracy of the Ormia’s coupled ears using a statisticalapproach, namely by computing the Cramer–Rao bound(CRB)1 [16]. We showed quantitatively that the couplingimproves the accuracy of direction of arrival (DOA) estimationin the presence of interference and noise.

The concept of electrically small antenna arrays with highradiation performance, superdirective (supergain) arrays, isquite old [7] and has attracted antenna researchers for thelast few decades. Different methods have been proposedto achieve the superdirectivity, namely by decoupling theantennas (reducing the effects of the undesired electromagneticcoupling among the antennas) and changing the currentdistributions applied to the array elements [17–20]. In thispaper, inspired by the female Ormia’s coupled ears, we showthat applying biologically inspired coupling (BIC) amongantennas is beneficial to achieve high radiation performance.Our goal is to demonstrate the effect of the BIC on the radiationperformance. The implementation of the BIC system and theinvestigation of the relationship between the superdirectivearrays and the BIC system are left as a future work. Wewould like to note that our approach, namely employing BIC,might also be used to complement the existing superdirectivearray design methods that overcome issues, for example, like

1 The Cramer–Rao bound is a universal minimum bound on the mean-squareerror value that can be achieved by any unbiased estimation algorithm.

Figure 2. Far-field radiation geometry of M-element antenna array.

the effect of the undesired coupling on individual antennaimpedance.

The rest of the paper is organized as follows. In section 2,we introduce the array-factor design for a uniform lineararray (ULA) inspired by the Ormia. First, we solve thesecond-order differential equations governing the Ormia’scoupled ear response. Then we convert this responseto fit the desired radio frequencies. Together with theundesired electromagnetic coupling among the array elements(due to their proximity), we include the BIC in the arrayfactor. In section 3, we assume finite-length dipoles asthe antenna elements and compute the radiation intensity,and accordingly the directivity gain, half-power beamwidth(HPBW) and side lobe level (SLL) as radiation performancemeasures. In section 4, we introduce our algorithm for theoptimum coupling selection. In section 5, we compare theradiation performances of the biologically inspired coupledand standard antenna arrays, demonstrating the improvementdue to the BIC. By standard antenna array we refer to asystem without BIC. Finally we provide concluding remarks insection 6.

2. Array factor

In this section, we compute the array factor of the proposedbiologically inspired ULA. We start with the array factor of astandard ULA, positioned without loss of generality along thez-axis (see figure 2). Since we focus on systems confined insmall spaces, we also consider the undesired electromagneticcoupling between the array elements.

Under the far-field radiation and narrow-band signalassumption, we modify the ULA factor to include theundesired coupling between the elements (see also [21]):

AF(θ) =M∑

m=1

pm exp(−j (m − 1)(ω� + β)), (1)

where

• AF(θ) is the radiation pattern (desired amplitude andphase in each direction) of M-element array assumingisotropic antennas, which depends on the positions andexcitations of the sensing elements in the system;

• p = [p1, . . . , pM ]T = Cvg is the vector of the currentson the antennas;

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• vg = [v1, . . . , vM ]T is the vector of the generator(excitation) voltages at the input of the antennas;

• C is the undesired electromagnetic coupling between thearray elements, (a transformation matrix, transforminggenerator voltages to the induced currents on eachantenna);

• ω = 2πf with f as the frequency of the radiated signal;• � = d sin θ

vis the inter-element time difference;

• d is the inter-element distance;• v is the speed of signal propagation in the medium;• θ is the elevation angle (see figure 2) and• β is the excitation phase.

We compute C similar to [21] as a function of self andmutual impedances between the antennas (see also discussionsin [22] and [23]). We summarize the computation of Cin the appendix. When the mutual impedances are zero,when there is no electromagnetic coupling, C reduces to adiagonal matrix. We compute the self and mutual impedances,assuming finite-length dipole antennas as the elements ofthe array, as explained in [7, chapter 8]; see also section 5.Note that the standard literature often ignores C, which isreasonable for sufficiently large inter-elemental distances.

The usual goal of the array-factor design is to select theexcitation voltages, vg, and phase, β, to obtain a desiredradiation pattern. Our goal is to include the BIC in thearray factor for fixed vg and β values and demonstrate theimprovement in the directivity gain, HPBW and SLL of theradiation pattern.

Next we generalize (1) to include also the couplingbiologically inspired by the Ormia’s coupled ears. First weobtain the response of the Ormia’s coupled ears. We thenconvert this response to fit the desired radio frequencies, andmodify the array factor to also include BIC. Moreover, at theend of this section, we provide a physical explanation of theBIC used in the array design.

In our work on biologically inspired antenna array andDOA estimation [24], we demonstrated our approach to obtainthe frequency response of the Ormia’s ears and how to shift itto the desired radio frequencies. For the sake of completenessand clarity of the presentation of the current work, we includethese parts in sections 2.1 and 2.2.

2.1. Response of the Ormia’s coupled ears

To obtain the response of the Ormia’s coupled ears, wesolve the second-order differential equations governing themechanical model proposed in [14] for the Ormia’s ears(figure 1), and find the corresponding transfer function. Notethat these equations represent a two-input two-output filtersystem (see also section 2.4). The governing differentialequations are[

k1 + k3 k3

k3 k2 + k3

] [y1

y2

]+

[c1 + c3 c3

c3 c2 + c3

] [y1

y2

]

+

[m0

m0

] [y1

y2

]=

[x1(t,�)

x2(t,�)

], (2)

where

• xi(t,�), i = 1, 2, are the input signals;

• yi(t), i = 1, 2, are the displacements of each ear and• m0, k’s and c’s are the effective mass, spring and dash-pot

constants, respectively.

To solve the differential equations and obtain the transferfunction (and hence the frequency response) of the system,we apply the Laplace transform2 to (2) assuming zero initialvalues:[

Y1(s)

Y2(s)

]= 1/P (s)

[D2(s) −N(s)

−N(s) D1(s)

] [X1(s)

X2(s)

], (3)

where

• Y1(s) and Y2(s) are the Laplace transforms of y1(t) andy2(t),

• X1(s) and X2(s) are the Laplace transforms of x1(t) andx2(t),

• D1(s) = m0s2 + (c1 + c3)s + k1 + k3 and D2(s) =

m0s2 + (c2 + c3)s + k2 + k3,

• N(s) = c3s + k3 (coupling effect),• P(s) = D1(s)D2(s)−N2(s) is the characteristic function.

We obtain the Laplace transform of the impulse responsesassociated with (2) by substituting

x1(t) = δ(t) → X1(s) = 1,

x2(t) = x1(t − �) → X2(s) = e−s�.

Then the system responses are

H1(s,�) = (D2(s) − N(s)e−s�)/P (s),

H2(s,�) = (D1(s)e−s� − N(s))/P (s).(4)

For s = jω, we obtain the frequency responses of the Ormia’scoupled ears. See figures 3 and 4 for the amplitude and phaseresponses of the Ormia’s ears. To demonstrate the effect ofthe mechanical coupling, we compare the coupled responseof the Ormia’s ear with a response assuming zero coupling(N(s) = 0). We observe that the coupling amplifies theamplitude and phase differences between the responses of theOrmia’s two ears. The figures are obtained using the effectivemass, spring and dash-pot constants experimentally obtainedin [14].

2.2. Converting to desired radio frequencies for the arrayresponse design

We now modify the frequency response of the Ormia’s ears tofit the desired radio frequencies. We achieve this conversionby re-computing the poles of the transfer function in (3), theroots of P(s) = D1(s)D2(s)−N2(s) = 0, for the frequenciesof interest. We shift the resonance frequencies of the systemby changing the imaginary parts of the poles. This correspondsto changing the system parameters, namely mass, spring anddash-pot constants defined in the analogous mechanical model(see (2)). Note that we do not necessarily need to change theseparameters using the same scaling constant. We can scale theresonant frequency locations (controlling the imaginary partsof the poles) and the real parts of the poles using different

2 See our approach in [15] for the state-space solution of the Ormia’s earresponses. In this work, we focus on the Laplace transform solution whichsimplifies the procedure.

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constants. We will keep the real parts as free variables whichwill enable us to optimize the coupling without modifying theresonant frequencies; see section 4. Our purpose is to preservea coupling structure similar to figures 3 and 4 which amplifiesthe differences between the amplitude and phase responses ofthe system outputs. See for example figure 5 for the amplitudeand phase responses of the converted system with f = 1 GHzas the desired frequency.

We obtain the ratio between the frequency responses

H2(ω,�)

H1(ω,�)= D1(jω)e−jω� − N(jω)

D2(jω) − N(jω)e−jω�. (5)

choosing the frequency ω depending on the application (seealso section 5).

2.3. Biologically inspired coupled array factor

To apply the BIC concept to the array factor in (1), we replacethe exponential terms in (1) with the ratio in (5):

AFI(θ) =M∑

m=1

pm

(H2(ω,�, β)

H1(ω,�, β)

)(m−1)

, (6)

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phase(H1(jω))

phase(H2(jω))

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Ph

ase

(deg

rees

)

phase(H1(jω))

phase(H2(jω))

(a)

(b)

Figure 4. Phase responses of the Ormia ochracea’s two ears. (a)Coupled system. (b) Uncoupled system.

where

H2(ω,�, β)

H1(ω,�, β)= D(jω) exp(−j (ω� + β)) − N(jω)

D(jω) − N(jω) exp(−j (ω� + β)), (7)

with D(jω) and N(jω) as defined after (3), substitutings = jω. We assume identical antennas D1(jω) = D2(jω) =D(jω). The ratio in (5) generalizes the exponential terms in(1) to include the BIC.

Note that N(jω) represents the BIC, and when there isno coupling (N(jω) = 0), AFI(θ) in (6) reduces to AF(θ) in(1). In this paper, we analytically demonstrate the biologicallyinspired beampattern design. The actual implementation isleft for a future work.

2.4. Filter interpretation

In this section, we explain the physical effects of the BIC onthe linear antenna array.

• In the receiving mode, the mechanical coupling isrepresented as a two-input two-output filter (figure 6),amplifying the differences between the outputs of thesystem; see figures 3 and 4.

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0

50

Frequency (Hz)

Ph

ase

(deg

rees

)

phase(H1(jω))

phase(H2(jω))

(a)

(b)

Figure 5. (a) Amplitude and (b) phase responses of the convertedsystem.

Figure 6. The Ormia’s coupled ears as a two-input two-output filtersystem.

• Since the mechanical coupling amplifies the amplitudeand phase differences between the frequency responses ofthe Ormia’s ears [15], it effectively creates larger distancebetween successive antennas, a virtual array with a largeraperture.

• To apply the biological coupling concept in thetransmitting mode, we include the BIC in the array factor(in addition to the undesired electromagnetic coupling).Thus, we generate a virtual array with a larger aperture.Larger aperture improves the radiation performance ofthe transmitting system (providing higher directivity gain,lower HPBW and SLL).

• For higher order antenna arrays, the structure in (6)corresponds to applying the BIC between successiveantennas, such that each antenna is coupled to itsimmediate neighboring antennas. Therefore, for everytwo successive antennas, the BIC creates a virtual arraywith a larger aperture, and hence a larger aperture for theentire array.

3. Radiation intensity, directivity, HPBW and SLL

In this section, we describe our measures to analyze theradiation performance. First, taking into account the antennafactor (element factor) and the BIC, we compute the radiationintensity of the antenna array in a given direction [7]:

UI(θ, φ) = [EF(θ, φ)]2n[AFI(θ)]2

n, (8)

where

• [EF(θ, φ)]n is the normalized element factor, far-zoneelectric field of a single element (in our work we assumethat the array is formed with finite-length dipoles; see alsosection 5);

• [AFI(θ)]n is the normalized array factor;• UI(θ, φ), the radiation intensity in a given direction, is the

power radiated from an antenna array per unit solid angle.• Hence, the radiated power Prad is

Prad =∫ 2π

0

∫ π

0UI(θ, φ) sin θ dθ dφ, (9)

where θ and φ are the elevation and azimuth angles,respectively, and sin θ dθ dφ is the unit solid angle.

Using the radiation intensity, we consider the followingmeasures to analyze the performance of the beampatterndesign.

• The directivity, DI(θ, φ), is the ratio of the radiationintensity in a given direction to the average radiationintensity:

DI(θ, φ) = 4πUI(θ, φ)

Prad, (10)

where 14π

Prad is the average radiation intensity over allangles. In our work, for comparison purposes, weconsider the directivity gain in a desired direction (atelevation θ = 0◦ and azimuth φ = 90◦; see alsosection 5).

• The HPBW, in terms of the elevation angle, θ , for a fixedazimuth angle, φ. The HPBW is defined as the anglebetween two half-power directions [7].

• The SLL defined as the maximum value of the radiationpattern in any direction other than the desired one(direction other than θ = 0◦ on φ = 90◦ plane for ourcase).

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Figure 7. Power pattern of the uniform standard and inspired ordinary end-fire antenna arrays for (a) d = 0.25λ, (b) d = 0.1λ inter-elementspacings. Bottom halves of the figures (a) and (b) present the HPBW.

The directivity gain, HPBW and SLL measure howeffectively the power is directed (steered) in a given direction.For a good performance, it is desirable to have large DI(θ, φ),small SLL and narrow HPBW in a desired direction.

4. Optimization of the biologically inspired coupling

In this section, we develop a method to maximize the radiationperformance by optimizing the BIC. We first introduce theoptimization parameters, and then formulate a cost functionfor optimum radiation pattern design.

Recall from section 2.2 that we change the imaginaryparts of the poles of the characteristic function to shift theresonance frequencies of the system response while keepingthe real parts as free variables. Therefore, under the constraintsthat we explain below, we have the freedom to choose thereal parts for optimum coupling design. We compute the

poles of the system as a function of its parameters (ks, csand m0). Recalling the discussions after (3), we assumeidentical antennas (D1(s) = D2(s) = D(s), and hence k1 =k2 = k, c1 = c2 = c), and write the characteristic functionas

P(s) = D(s)2 − N(s)2

= m20((s

2 + b1s + a1)2 − (b2s + a2)

2), (11)

where a1 = (k + k3)/m0, a2 = k3/m0, b1 = (c + c3)/m0 andb2 = c3/m0. We assume that the parameters a1, a2, b1 and b2

are positive. Then we obtain the roots (the poles of the system)as

p1,2 = −r1 ±√

i1 (12)

p3,4 = −r2 ±√

i2, (13)

where r1 = 12 (b1 + b2), r2 = 1

2 (b1 − b2), i1 = 14 ((b1 + b2)

2 −4(a1 + a2)) and i2 = 1

4 ((b1 − b2)2 − 4(a1 − a2)). In order to

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Figure 8. Power pattern of the binomial standard and inspired end-fire antenna arrays for (a) d = 0.25λ, (b) d = 0.1λ inter-elementspacings. Bottom halves of the figures (a) and (b) present the HPBW.

change the resonance frequencies, we change the values of theimaginary parts such that i1 = −(2πf1)

2 and i2 = −(2πf2)2,

where f 1 and f 2 are the resonant frequencies. We set r1 andr2 as the free parameters. We have the freedom to control thereal parts as long as r1 > r2 due to the positivity assumptionon the parameters a1, a2, b1 and b2.

Next we formulate the optimization problem to improvethe radiation performance. For an antenna array, we choose thedirectivity gain in a desired direction as the utility function tobe maximized. This is a reasonable choice since the directivitygain is also related to the SLL and the HPBW of the radiationpattern. Generally it is true that the patterns with smaller SLLand HPBW values have larger directivity gain. Therefore, weformulate the problem as

maximize DI(θ, φ)

subject to r1 − r2 > 0,(14)

where θ and φ are the elevation and azimuth of the desireddirection of transmission.

5. Numerical examples

In this section, we compare the radiation performances ofthe biologically inspired coupled and standard antenna arrays.We plot the radiation pattern and compare the directivity gain,HPBWs and sidelobe attenuation of these systems using thefollowing scenario. Moreover, we demonstrate our resultson optimum BIC selection. In the following discussions, byinspired array we refer to an antenna array with BIC.

• We consider uniform (uniform excitation voltages)ordinary and binomial (binomial expansion coefficients asthe excitation voltage values) end-fire arrays [7, chapter8], maximum at θ = 0◦; then β = −w�.

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Figure 9. Power pattern of the uniform inspired and inspired with optimum coupling (OC) ordinary end-fire antenna arrays for (a)d = 0.25λ, (b) d = 0.1λ inter-element spacings. Bottom halves of the figures (a) and (b) present the HPBW.

• Frequency of interest, f = 1 GHz.• ULA composed of 20 identical dipole antennas.• The undesired coupling matrices, C for a 0.5λ-

wavelength antenna system with different inter-elementdistances (d = 0.25λ and d = 0.1λ), are calculatedaccording to [7, chapter 8] for finite-length thin-dipoleantennas.

• The antennas are located on the z-axis parallel to the y-axis; then assuming azimuth φ = 90◦ (on the y–z plane,see figure 2), the element factor for a finite-length dipoleantenna is computed as

EF(θ, 90◦) =[

cos( kl2 sin θ) − cos( kl

2 )

cos θ

], (15)

where k = 2πλ

, λ is the wavelength of the radiated signaland l is the length of each antenna.

Table 1. Directivity gains of the antenna arrays in the desireddirection θ = 0◦ and φ = 90◦ (dB).

Uniform Binomial

d = 0.25λ d = 0.1λ d = 0.25λ d = 0.1λ

Inspired array 19.22 16.92 16.86 15.19Standard array 13.96 10.77 10.81 8.08

Recall that we focus on a 2D beampattern design in termsof elevation angle, θ .

We demonstrate our results for standard and inspiredarrays in figures 7 and 8, and summarize the calculateddirectivity gains and HPBW values in tables 1 and 2,respectively. We observe that the biologically coupledinspired array with uniform excitation voltages outperformsthe uniform standard array in terms of sidelobe suppression,

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(d

B)

−15 −10 −5 0 5 10 15−3

−2

−1

0

Elevation Angle (θ)

Rel

tive

Po

wer

(d

B)

Inspired Array d=0.25λInspired Array OC d=0.25λ

−100 −80 −60 −40 −20 0 20 40 60 80 100−60

−40

−20

0

Elevation Angle (θ)

Rel

ativ

e P

ow

er (

dB

)

−20 −15 −10 −5 0 5 10 15 20−3

−2

−1

0

Elevation Angle (θ)

Rel

ativ

e P

ow

er (

dB

)

Inspired Array d=0. 1λInspired Array OC d=0.1λ

(a)

(b)

Figure 10. Power pattern of the binomial inspired and inspired with OC end-fire antenna arrays for (a) d = 0.25λ, (b) d = 0.1λinter-element spacings. Bottom halves of the figures (a) and (b) present the HPBW.

directivity and HPBW (see figure 7, and tables 1 and 2).For binomial array, in figure 8, we observe that neither thestandard nor the inspired array have sidelobes, but the inspiredarray has much narrower HPBW and hence better directivitygain (see also tables 1 and 2). The physical reason of theimprovement in the radiation performance is the BIC thatworks as a multi-input multi-output filter, magnifying theamplitude and phase differences (time differences) betweenthe outputs of the successive antennas and creating a virtualarray with a larger aperture. In the beampattern design, thevirtual array with larger aperture creates a radiation patternwith higher directivity and sidelobe supression, and lowerHPBW. Note that in these examples the effect of the BICincreases as the distance between the antennas, d, decreases.

In figures 7 and 8, for inspired array, we choose thecoupling (the real parts of the poles of the characteristicfunction) manually to obtain better radiation performance than

Table 2. HPBWs of the antenna arrays (degrees).

Uniform Binomial

d = 0.25λ d = 0.1λ d = 0.25λ d = 0.1λ

Inspired array 19◦ 28◦ 24◦ 30◦

Standard array 45◦ 62◦ 63◦ 76◦

the standard antenna array. We then select the couplingparameters optimally following the algorithm described insection 4. In figures 9 and 10, we present the results onthe comparison of the radiation patterns corresponding tothe manually (inspired coupling used in figures 7 and 8)and optimally chosen coupling parameters. In these figuresthe inspired array optimum coupling (OC) corresponds tothe inspired array with optimum coupling. We demonstratethat we further improve the radiation performance (both

9

Bioinsp. Biomim. 5 (2010) 046003 M Akcakaya and A Nehorai

Table 3. Directivity gains of the antenna arrays in the desireddirection θ = 0◦ and φ = 90◦ (dB).

Uniform Binomial

d = 0.25λ d = 0.1λ d = 0.25λ d = 0.1λ

Inspired array 19.22 16.92 16.86 15.19Inspired array OC 25.62 22.16 22.75 20.35

Table 4. HPBWs of the antenna arrays (degrees).

Uniform Binomial

d = 0.25λ d = 0.1λ d = 0.25λ d = 0.1λ

Inspired array 19◦ 28◦ 24◦ 30◦

Inspired array OC 9.8◦ 14.6◦ 14.2◦ 17.4◦

the directivity gain and HPBW) by using the optimizationalgorithm that we propose (see also tables 3 and 4).

6. Conclusion

We designed a coupled antenna array transmission systeminspired by the mechanically coupled ears of Ormia ochracea.First, we obtained the response of the mechanical modelrepresenting the coupling between the Ormia’s ears. Then,we converted this response to the desired radio frequenciesand designed the biologically inspired coupled antenna arrayby applying the converted response to the array factor of theantenna system. Since we focus on systems confined to smallspaces, we also considered the undesired electromagneticcoupling among the array elements. We computed theradiation intensity, and accordingly the directivity gain,the half-power beamwidth and the sidelobe level of thebiologically inspired and standard antenna arrays as theperformance measures. We demonstrated the improvementin the radiation performance due to the biologically inspiredcoupling. Moreover, we showed the further improvement inthe radiation performance by optimally choosing the couplingparameters. In our future work, we will focus on theimplementation of the beampattern design, 3D beampatterndesign, design of additional biologically inspired couplingbeyond the adjacent antennas, nonlinear antenna arrays(circular, etc).

Acknowledgments

Authors would like to thank Dr Carlos Muravchik of UNLP,Argentina, for his valuable comments. This work wassupported by DARPA grant no HR0011-09-P-0007, theDepartment of Defense under Air Force Office of ScientificResearch MURI grant FA9550-05-1-0443 and ONR grantN000140810849.

Appendix. Computation of the undesiredelectromagnetic coupling matrix C

An antenna in the transmitting mode can be modeled asin figure A1. Assuming M-element antenna array, and

Zii

Zgi

pi

vgi

Figure A1. Circuit model of the ith antenna element in thetransmitting mode.

considering the mutual effect of the other antennas in the array,the induced current on the ith antenna can be computed through(see also [7] and [21])

pi(Zii + Zg) = vgi −M∑k �=i

pkZik, (A.1)

where,

• pj , j = 1, . . . ,M , is the induced current on the j thantenna;

• Zjj is the self impedance of the j th antenna;• Zg is the generator impedance;• vgj is the generator voltage applied to the j th antenna and• Zjk is the mutual impedance between the j th and kth

antennas.

Assuming identical antennas and generators (identical selfand generator impedances), we can rewrite (A.1) and obtain⎡⎢⎢⎣

Z11 + Zg Z12 · · · Z1M

Z21 Z22 + Zg · · · Z2M

· · ·ZM1 ZM2 · · · ZMM + Zg

⎤⎥⎥⎦

⎡⎢⎢⎢⎢⎣

p1

···

pM

⎤⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎣

vg1

···

vgM

⎤⎥⎥⎥⎥⎦ . (A.2)

Then recalling from section 2 that p = Cv, we get

C =

⎡⎢⎢⎣

Z11 + Zg Z12 · · · Z1M

Z21 Z22 + Zg · · · Z2M

· · ·ZM1 ZM2 · · · ZMM + Zg

⎤⎥⎥⎦

−1

(A.3)

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