Biologically Inspired Coupled Antenna Array for Direction of Arrival Estimation Murat Akcakaya*, Student Member, IEEE, Carlos H. Muravchik t, Member, IEEE, and Arye Nehorai*, Fellow, IEEE * Department of Electrical and Systems Engineering Washington University in St.Louis, St.Louis, Missouri, USA Emails: { makcak2, nehorai } @ese.wustI.edu t LEICI, Departamento Electrotecnia, Universidad Nacional de La Plata, La Plata, Argentina Email: carlosm@ing.unlp.edu.ar Abstract-We propose to design a small-size antenna array having high direction of arrival (DOA) estimation performance, inspired by the Ormia ochracea's coupled ears. The female Ormia small to be detectable by the central nervous system of the fly [3]-[8]. is able to locate male crickets' call accurately, for reproduction purposes, despite the small distance between its ears compared with the incoming wavelength. This phenomenon has been explained by the mechanical coupling between the Ormia's ears, modeled by a pair of differential equations. In this paper, we first solve the differential equations governing the Ormia ochracea's ear response, and convert the response to the pre-specified radio frequencies. Using the converted response, we then implement the biologically inspired coupling as a multi-input multi-output filter on a uniform linear antenna array output. We derive the maximum likelihood estimates of source DOAs, and compute the corresponding Cramer-Rao bound on the DOA estimation error as a performance measure. We use Monte Carlo numerical examples to demonstrate the advantages of the coupling effect. I. Fig. I. INTRODUCTION The performance of an antenna array that employs time differences of arrivals is directly proportional to the size of the array's electrical aperture, such that large-aperture arrays are required to achieve better DOA estimation performance [I], [2]. However, using a large aperture array to improve performance can be costly also may not be feasible, since in many tactical and mobile applications the sensing systems are confined to small spaces, requiring small-sized arrays. In this paper, we propose a biologically inspired antenna array to achieve high DOA estimation performance with small aperture arrays. The approach is inspired by a parasitic fruit fly called Ormia ochracea. To perpetuate its species, a female Ormia ochracea must find a male field cricket using the cricket's mating call. The female Ormia has a remarkable ability to locate these crickets very accurately using binaural (two-ear) cues (interaural differences in intensity and arrival time from an incident acoustic wave). This is unexpected due to the significant mismatch between the wavelength of the cricket's call (about 7 cm) and the distance between the fly's ears (about 1.2 mm) which gives rise to cues that are extremely This work was supported by DARPA Grant No. HROOII-09-P-0007, the Department of Defense under Air Force Office of Scientific Research MURI Grant FA9550-05-1-0443, ONR Grant NOOOl40810849 and NSF Grant CCF0963742. Carlos Muravchik was funded also by ANPCyT and CIC, Argentina. 978-1-4244-9721-8/I0/$26.00 ©2010 IEEE Mechanical model of the female Ormia ochracea's ears [9]. Experimental research in [9] explains that the Ormia's localization ability arises from a mechanical coupling between its ears, modeled as a system consisting of spring and dash pots (with six key parameters: {(ci,ki): i 1,2,3}) as shown in Fig. 1. In the equivalent mechanical system, the intertympanal bridge (the cuticular connecting structure) is assumed to consist of two rigid bars connected at the pivot through a coupling spring k3 and dash-pot C3. The springs and dash-pots, located at the extreme ends of the bridge, approximate the dynamical properties of the tympanal membranes, sensory organs, and surrounding structures in the Ormia's two ears. In our previous work [10], we analyzed the localization accuracy of the Ormia's coupled ear system using a statistical approach, namely by computing the Cramer-Rao bound (CRE) [II]. We showed quantitatively that the coupling improves the accuracy of direction of arrival (DOA) estimation in the presence of interference and noise. In this paper, we develop a biologically inspired coupled antenna array. First, we solve the second order differential equations governing the Ormia's coupled ear response, and then convert this response to fit the desired radio frequencies. Using the converted response, we implement the biologically inspired coupling (BIC) as a multi-input multi-output filter and 1961 = Asilomar 2010 obtain the desired array response. We show that, compared to a standard antenna array, an antenna array with BIC has higher localization accuracy. By standard antenna array we refer to a system without the BIC. The rest of the paper is organized as follows. In Section II, we introduce the measurement model for a uniform linear antenna array with BIC. In Section II-G, we first derive the maximum likelihood estimates (MLEs) of the DOAs. Then in Section II-H, we compute the CRB on the DOA estima tion error to analyze the estimation accuracy performance. In Section III, using Monte Carlo numerical examples, we compare the biologically inspired coupled antenna array with a standard one, and demonstrate the improvement in the estimation performance due to the BIC. Finally we provide concluding remarks in Section IV. II. ANTENNA ARRAY MODEL affected, therefore we do not include it in our analysis. We will consider the effect of the combined coupling (BIC and unknown undesired) in our future work. B. Response of the Ormia's Coupled Ears To obtain the response of the Ormia's coupled ears, we solve the second-order differential equations governing the mechanical model proposed in [9] for the Ormia's ears (Fig. 1), and find the corresponding transfer function. The governing differential equations are: + [�: �;] [ �� ] [�: �;] [ t� ] [ma ma] [ t� ] + where In this section, we introduce the measurement and statistical models for the biologically inspired antenna array. • • • A. Measurement Model x(t)= A(cjJ)s(t) + e(t), at radio with the [1]. We complex t= 1, ..., N • • To solve the differential equations and obtain the transfer function (and hence the frequency response) of the system, we apply the Laplace transform' to (2) assuming zero initial values, (3) (1) where where • x(t) is Mx 1 output vector of the array, with M antennas; s(t) = [SI(t), ... , sQ (t)V is the Q x 1 input signal vector with Q as the number of the sources; A(cjJ)= [a(¢I)··· a(¢Q)] is the array response, with as the DOA of the qth source; a(¢q)= [ 1, exp ( - jwt1q) , ... , exp • • ¢q • (-jw(M - l)t1q)] for a uniform linear array; • • • t1q = Xi(t, t1), i = 1,2, are the input signals; Yi(t), i = 1,2, are the displacements of each ear; and mo, k's, and c's are the effective mass, spring and dash pot constants, respectively. We develop a coupled multiple-antenna array frequencies inspired by the Ormia's ears. We start measurement model of a standard antenna array assume narrow-band incoming signal and write the envelope of the measurements as • (2) d sin ¢q with d • • as the distance between each antenna (w� focus on 2-D direction finding); v is the speed of signal propagation in the medium; e(t) = e a(t) + ee (t) is additive noise, such that ea(t) and ee (t) are the amplifier and environment noise com ponents, respectively. Next, we implement the BIC as a filtering procedure and obtain the biologically inspired array response. We first obtain the frequency response of the Ormia's ears, and modify it to fit the desired radio frequencies. We then employ the converted response as a two-input two-output filter and obtain a two antenna array with the BIC. We also generalize the BIC concept to multiple-antenna arrays. Note that in practice, systems with closely spaced antennas typically undergo undesired electromagnetic coupling among the antennas. In our work, to simplify the analysis we assume that the calibration is achieved beforehand and hence we ignore the effect of the undesired coupling. Assuming known calibration, the performance of the system should not be Yl(s) and Y2(s) are the Laplace transforms of Yl(t) and Y2(t), XI(s) and X2(s) are the Laplace transforms of XI(t) and X2(t), Dl(S) = mos2 + (Cl + C3)S + kl + k3 and D2(S) mos2 + (C2 + C3)S + k2 + k3, N(s)= C3S + k3 (coupling effect), P(s) = D1(s)D2(s) - N2(s) is the characteristic func tion. We obtain the Laplace transform of the impulse responses associated with (2) by substituting 1, Xl(t) 6(t) ---> Xl(S) X2(t) Xl(t -�) ---> X2(S) e-Sd• Then the system responses are = = = = (D2(S) - N(s)e-S�)/p(s), (Dl(s)e-S� - N(s))/P(s). (4) For s= jw, we obtain the frequency responses of the Ormia's coupled ears. Using these responses, in [10], we demonstrated that the coupling amplifies the amplitude and phase differences between the responses of the Ormia's two ears. Note that these equations represent a two-input two-output filter system (see also Section II-E). lSee our approach in [lO] for the state-space solution of the Ormia's ear responses. In this work, we focus on the Laplace transform solution which simplifies the solution. 1962 C. Converting to Desired Radio Frequencies for Array Re sponse Design We now modify the frequency response of the Ormia's ears to fit the desired radio frequencies by re-computing the poles of the transfer function in (3), the roots of P( s ) Dl ( s ) D2 ( s ) - N2 ( s ) O. We shift the resonance frequencies of the system by changing the imaginary parts of the poles. This corresponds to changing the system parameters, namely mass, spring and dash-pot constants defined in the analogous mechanical model (see (2)). We scale the resonant frequency locations (controlling the imaginary parts of the poles) and the real parts of the poles using different constants. Therefore, we keep the real parts as free parameters. In our future work, we will optimize the coupling using the real parts of the poles without modifying the resonant frequencies. = = D. Biologically Inspired Array Processing We first consider two-antenna array for BIC implemen tation. For an antenna array with two identical antennas, Dl(jW) D2(jW) D(jw), we apply the BIC filter to the measurements in (1) and obtain in the frequency domain = Fig. 2. Two-input two-output filter representation of the Ormia's coupled ears' response. E. Filter Interpretation In this section, we explain the physical effects of the biologically inspired coupling on the linear antenna array through a filter interpretation. • = (5) where . Hr(Jw) • N(jW) -1 N(jw) D(jw) [D(jW) = ] Assuming incoming narrow-band signal W We 27r Ie where Ie is the carrier frequency, we obtain the measurement model in the time domain for biologically inspired antenna array as = [�����] = H,(jwe) = [�����] = Hr(jwe)A(cp)s(t) + e(t), F. Statistical Assumptions (6) where e(t) Hr(jwe)e(t) ea(t) + H,(jwe)ee(t), such that only the environment noise is affected by the BIC The implicit assumption here is that the array will be designed with coupling between the antennas and/or by filtering to achieve a response matrix, H,(jwe)A(CP). We next extend the model in (6) to M identical antennas. We assume each antenna is coupled to its immediate neigh boring antennas in the array, i.e., each antenna (except for the first and the last antennas) is coupled to two antennas. We will consider other possible coupling configurations in our future work. Therefore we generalize the two-input two-output filter by using it in a tridiagonal M x M matrix form: = [Y�'(�)l = YM(t) where • H,(jw,)-' = Hr(jwe) = [�g�;l 0 [X.l·(�)l xM(t) = Hr(jwe)A(cp)s(t) + e(t) N( W,) 0 N(jw,) D(jw,) N(jw,) �g�:l � 0 From now on to simplify the notation (7) �I o 0 N(jw,) D(jw,) Hr(jw) = Hr. . • The mechanical coupling is represented as a two-input two-output filter (Fig. 2), amplifying the differences be tween the outputs of the system, see [10]. Since the mechanical coupling amplifies the amplitude and phase differences between the frequency responses of the Ormia's ears [10], it effectively creates larger distance between successive antennas, a virtual array with a larger aperture. Applying the BIC to the antenna array, we generate a virtual array with a larger aperture. Larger aperture improves the DOA estimation performance (providing higher estimation accuracy). We introduce our statistical assumptions on the measure ment model. We assume, in (1), • • cp [<PI, ... , <PQ] T is the Q x 1 vector of deterministic unknown DOA parameters; s(t) is a Gaussian input signal vector, E[s(t)] 0, E[s(t)S(t')H] Pbtt' and E[s(t)S(t')T] 0, with P as the Q x Q unknown source covariance matrix, and for t, t' 1, ... , N btt' 1 when t t' and zero otherwise; e(t) is Gaussian distributed and E[e(t)] 0, (a;I + a;HrHf)btt' and E[e(t)e(t')H] E[e(t)e(t')T] 0, such that a; and a; are the unknown variances of amplifier and environment noise, respectively; s(t) and e(t') are uncorrelated for all t and t'. = = = = • = = = = • G. Maximum Likelihood Estimation The maximum likelihood estimator of the DOA is defined as the value that maximizes the likelihood function (see (10)). It is asymptotically optimal, namely it is unbiased and it attains the CRB of minimum variance [II]. Following the statistical assumptions in Section II-F, we write the probability density function of the measurements as 1963 where .i,j=1,... ,Q+1 U=P [:tH(O)A(O)P+0-2I] -1 AH(O)A(O)P, N II p[y(t);cfJ,P,o-�,0-;] t=l • where = E[y(t)y(t)H] = A(cfJ)PA(cfJ)H + o-;:E(p), with ( :E p)=pI+HrH[I, and p=0-;"/0-;; A(cfJ)=HrA(cfJ). We obtain :E(p)-1/2, then we define jj= :E(p)-1/2y, and A(O)= :E(p)-1/2A(cfJ), where 0= [p,cfJ]T is (Q+1) x 1 • R • vector of unknown parameters. Next, we rewrite (8) • N N exp CRB(O)= ;l {Re (btr [ (1 @ U) [::J (DHII-L D) T)] ) r1 (15) b where, • [jj(t)R-1jj(t)1�) • • • R E[jj(t)jj(t)H] A(O)PA(O)H = where = +0-;1. Then taking the logarithm of (9) and considering it as a function of the unknown parameters, we obtain the log likelihood function as LF(O,P, ; )= -N [Mln(7r)+InlRI +tr(R-1R)] , (10) where R= 1J L� jj(t)jj(t)H, is the sample covariance, and l • • • • • • 0: P P(O)=A(O)tR(A(O)t)H - &;(O)(A(O)HA(O))-l, &;(0)=tr(II-L R)/(M -Q), A(O)t= [A(O)HA(O)]-lA(O)H, II=A(O)A(O)t, -L II =I -II. Concentrating the likelihood function using these estimates, P(O)=L[O,P(O),&;(0)], we obtain the MLE of 0 through 0= arg;;inp(O)= arg;;inln IA(O)P(O)A(O)H+&;(0)11. (11) H. Cramer-Rao Bound We analyze the array's statistical performance, i.e., accuracy in estimating the source direction, by computing the Cramer Rao bound. The CRB is the lower bound on estimation error for any unbiased estimator. We concentrate the likelihood function in (10) with respect to and 0-; and compute the CRB on the covariance matrix of any unbiased estimator of Using the results in [13] and [14], we define P O. CRB-1(0)=N . F�(O), (12) where (13) P(O) with as defined in (11). Then we apply the Lemma C.1 and C.2 of [14], and obtain [CRB-1(0)Lj= �� Re {tr [ uDfII-L Di ] } , e (14) 1 is a Q+1 x Q+1 matrix D= [D ... DQ + ], 1 of ones, 1 btr is block trace operator, @ Kronecker product, [::J is block Schur-Hadamard product, bT is block transpose operator. Note that we modified the results in [14] to account for our assumptions and filtering effect. III. 0- "tr ( · ) " is the trace operator. We follow the procedure explained in [12], such that we derive the MLEs of and 0-; as a function of BA(O) Be i · Then collecting the terms we have • }] p[jj(t);cfJ,P,P,o-;] = }] 17r�1 D.,= NUMERICAL RESULTS We compare the localization performances of the BIC and standard multiple-antenna arrays using Monte Carlo simula tions. In the following discussions, by BIC array we refer to a biologically inspired coupled antenna array. Recall that standard array is the system without the BIC. We use the following scenario: Single source with true incoming direction as ¢=55°; f=1 GHz is the frequency of operation; 5 identical dipole antennas; d=0.1A and d= 0.2A interelement distances. For the BIC uniform linear array, we demonstrate our results on estimation of direction of arrival and noise-to-interference ratio in Figs. 3, and 4. In Figs. 3(a) and 3(b) for a fixed signal to-noise ratio, SNR=-lOdB, we plot the root mean-square error (RMSE) on the maximum likelihood estimation of direction of arrival, and CRB of DOA estimation for the standard and BIC arrays with d = 0.1A and d = 0.2A interelement spacings, respectively. We observe that the CRB on DOA estimation error and RMSE of MLE are smaller for the BIC array, meaning a decrease in estimation error and an improvement in the localization performance. The MLE algorithm attains the bound asymptotically. In Fig. 4, for N = 10 time samples, we plot the CRB on DOA estimation for the standard and BIC uniform linear arrays for different SNR values and demonstrate the decrease in the minimum bound on the estimation error due to the BIC. Figs. 3 and 4 confirm that the BIC decreases the minimum bound on the estimation error and improves the performance of DOA estimation. The physical reason of the improvement in the localization performance is that the BIC works as a multi input multi-output filter, magnifying the phase differences (time differences) between the signals received at successive antennas and creating a virtual array with a larger aperture. Note that in these examples the effect of the BIC increases as the interelement spacing of the array, d, decreases. 1964 10' �Standard Array d=O.1/• ......-SIC Array d=O.1J. 500 1000 1500 Number of Time Samples (N) -15 2000 4'�----------r=-+=Standard ======C'il Array d=O.21, � MSE Standard Array MSE BIC Array -eo- eRB -5 (a) (a) --T- Signal-to-Noise Ratio (dB) ...... BIC Array d=O.27.. Standard Array -.-eRe BIC Array '0 -2 O f t � 10-lL-__ ___ __ ___-' 500 1500 2000 1000 !l 0 � � ��0 --�-1�5���-1�0���-�5���� � Number of Time Samples IN) Signal-to-Noise Ratio (dB) (b) (b) Fig. 3. RMSE in the direction estimation and corresponding CRBs vs. number of time samples for the standard (blue) and BIC (red) uniform linear arrays with different interelement spacings, d, and SNR=-IO dB. (a) d = O.L\. (b) d IV. = 0.2>' CONCLUSION We designed a multiple-antenna array with couplings bio logically inspired by the mechanically coupled ears of Ormia ochracea. First, we obtained the response of the mechanical model representing the coupling between the Ormia's ears. We then converted this response to the desired radio frequencies. We implemented the biologically inspired coupling using the converted system as a multi-input multi-output filter. Then, we derived the maximum likelihood estimates of the direction of arrival and computed the Cramer-Rao lower bound on estimation error as a performance measure. Using Monte Carlo simulations, we demonstrated the improvement in the local ization performance due to the biologically inspired coupling. In our future work, we will develop algorithms to optimize the biologically inspired coupling design, then also consider systems with unknown undesired electromagnetic coupling among the antennas, and different coupling configurations. REFERENCES [I] P. Stoica and K. Sharman, "Maximum likelihood methods for direction of-arrival estimation," IEEE Trans. on Acoust., Speech and Signal Process., vol. 38, no. 7, pp. 1132 -1143, jul 1990. [2] A. Swindlehurst and P.Stoica, "Maximum likelihood methods in radar array signal processing," Proc. IEEE, vol. 86, no. 2, pp. 421 --441, feb 1998. [3] W. Cade, "Acoustically Orienting Parasitoids: Fly Phonotaxis to Cricket Song," Science, vol. 190, pp. 1312-1313, Dec. 1975. Fig. 4. Square-root of the CRB on direction of arrival estimation vs. SNR for standard (blue), and BIC (red) uniform linear arrays with different interelement spacings, d, N=lO time samples. (a) d (b) d = 0.2>'. = 0. 1>'. [4] D. Robert, M. J. Amoroso, and R. R. Hoy, "The evolutionary conver gence of hearing in a parasitoid fly and its cricket host," Science, vol. 258, no. 5085, pp. 1135-1137, 1992. [5] D. Robert, M. P. Read, and R. R. 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