Akcakaya Bioinsp TSP 2011

Akcakaya Bioinsp TSP 2011
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. Hoy, "The tympanal hearing organ of
the parasitoid fly Ormia ochracea (diptera, tachinidae, ormiini)," Cell
Tissue Res., vol. 275, no. I, pp. 63-78, 1994.
[6] D. Robert, R. N. Miles, and R. R. Hoy, "Directional hearing by
mechanical coupling in the parasitoid fly Ormia ochracea," 1. Comp.
Physiol. A, vol. 179, no. I, pp. 29--44, Jul. 1996.
[7] --, "Tympanal mechanics in the parasitoid fly Ormia ochracea:
intertympanal coupling during mechanical vibration," J. Compo Physiol.
A, vol. 183, no. 4, pp. 443--452, Oct. 1998.
[8] A. C. Mason, M. L. Oshinsky, and R. R. Hoy, "Hyperacute directional
hearing in a microscale auditory system," Nature, vol. 410, pp. 686-690,
Apr. 2001.
[9] R. N. Miles, D. Robert, and R. R. Hoy, "Mechanically coupled ears for
directional hearing in the parasitoid fly Ormia ochracea," 1. Acoust. Soc.
Am., vol. 98, no. 6, pp. 3059-3070, 1995.
[10] M. Akcakaya and A. Nehorai, "Peformance analysis of Ormia
ochracea's coupled ears," J. Acoust. Soc. Am., vol. 124, no. 4, pp. 21002105, Oct. 2008.
[II] S. Kay, Fundamentals 0/ statistical signal processing: estimation theory.
Upper Saddle River, NJ: Prentice Hall PTR, 1993.
[12] P. Stoica and A. Nehorai, "On the concentrated stochastic likelihood
function in array signal processing," Circ., Syst., and Sig. Proc., vol. 14,
no. 5, pp. 669-674, Sept. 1995.
[13] --, "Performance study of conditional and unconditional direction­
of-arrival estimation," IEEE Trans. Acoust., Speech, Signal Process.,
vol. 38, no. 10, pp. 1783-1795, Oct. 1990.
[14] A. Nehorai and E. Paldi, "Vector-sensor array processing for electromag­
netic source localization," IEEE Trans. Signal Process., pp. 376-398,
Feb 1994.
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