IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 5, 2006 251

Autocalibration Algorithm for RobustCapon Beamforming

Shohei Kikuchi, Student Member, IEEE, Hiroyuki Tsuji, Member, IEEE, and Akira Sano, Member, IEEE

Abstract—The robust Capon beamformer (RCB) is more robustthan the standard Capon beamformer when there is uncertainty, asin the case of a sensor location error or an array model mismatch.However, the RCB suffers from mutual coupling between the arraysensors, which is difficult to neglect in practice. In this work, anautocalibration algorithm is described that improves the RCB byrecursively calibrating the uncertainty due to mutual coupling. Theproposed method is better able to estimate the angle-of-arrival andpower of the signal of interest than the RCB, as shown by numericalsimulation and practical analysis of experimental data obtained inan anechoic chamber.

Index Terms—Angle-of-arrival estimation, autocalibration,power estimation, robust Capon beamforming (RCB).

I. INTRODUCTION

B EAMFORMING is essential to array signal processing invarious fields, including radar, sonar, and wireless commu-

nications. The standard Capon beamforming (SCB) with perfectknowledge of the array steering vector shows excellent interfer-ence rejection performance [1]. In practice, however, the perfor-mance of the SCB is seriously degraded by errors in the signalangle-of-arrival (AOA), amplitude and phase between the arraysensors, i.e., array steering vector uncertainty. Diagonal loadingis commonly used to improve the robustness of the SCB [2], buta technique has not yet been reported for selecting the diagonalloading factor under condition of the steering vector uncertainty.The robust capon beamformer (RCB) proposed by Li et al. isone of the most promising methods for dealing with steeringvector uncertainty, which is related to the ellipsoidal uncertaintyset of the array steering vector [3]. This beamformer has shownexcellent power estimation performance for the signal of interest(SOI).

However, the RCB cannot compensate for robustness in thepresence of mutual coupling between the array sensors since itseffect on adjacent elements of the sensors strengthen the arrayoutput. Thus, a real-time calibration method that eliminates theeffect is needed for the RCB to be practical. In this paper, wepresent an auto-calibration algorithm that improves RCB per-formance. Specifically, we model the uncertainty due to mu-tual coupling and amplitude and phase errors. We also present anew cost function based on the Capon power expression, which

Manuscript received January 7, 2006; revised March 2, 2006. This work wassupported in part by a Grant-in-Aid for the 21st century Center of Excellencefor Optical and Electronic Device Technology for Access Network from theMinistry of Education, Culture, Sport, Science, and Technology in Japan.

The authors are with the School of Integrated Design Engineering, Keio Uni-versity, Yokohama, Japan (e-mail: kikuchi@contr.sd.keio.ac.jp).

Digital Object Identifier 10.1109/LAWP.2006.874070

plays an important role in the constrained quadratic minimiza-tion problem. The calculation of the algorithm is performed in arecursive manner, as in a Friedlander’s work [4]. The proposedmethod can identify the power of impinging signals as well asthe AOA, whose performance shows much better than the RCB.Our algorithm is highly efficient in that it can simultaneously ob-tain both the AOA, the power, and the array uncertainty, whilethe RCB is unable to estimate the AOA since the beamwidthof the AOA spectrum is relatively large [5]. Numerical simula-tions and analysis of experimental data measured in an anechoicchamber demonstrate the method’s effectiveness.

II. PROBLEM FORMULATION

Consider an array of -element uniformly spaced sensors,in which sources emit plane waves im-pinging on the array from distinct directions, . Letbe the steering vector: with

, where is the intersensor spacing, andis the signal carrier wavelength. The baseband signal repre-

sentation of the th snapshot of the array output is expressed as

(1)

where is the -th transmitted signal, an independent identi-cally distributed sequence of complex circular Gaussian randomvectors, and and are uncorrelated

. Suppose is the SOI and are theinterference signals without loss of generality. Vector isadditive white Gaussian noise (AWGN) with zero mean and un-known covariance matrix , independent of the signal sam-ples. Let denote the theoretical covariance matrix of the arrayoutput vector, a positive definite matrix is

(2)

where and denote the powers of the SOI and un-correlated interference signals, respectively. In practical appli-cations, is replaced by finite sample covariance matrix:

(3)

where is the number of snapshots.

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252 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 5, 2006

III. CAPON BEAMFORMING

A. Standard Capon Beamforming (SCB)

The SCB is formulated so as to select a weight vector thatminimizes the array output power by using the following lin-early constrained quadratic problem [1]:

subject to (4)

The weight vector is derived by solving (4)

(5)

and the power of the SOI is derived using

(6)

B. Robust Capon Beamforming (RCB)

Since the performance of the SCB in the presence of asteering vector error (e.g., arising from an array calibrationerror) deteriorates, Li et al. proposed the robust Capon beam-former (RCB) [3] to estimate the power of the SOI moreaccurately. The RCB is an extension of the SCB in that thesteering vector is assumed to have ellipsoidal uncertainty

(7)

where the presumed array response vector, , and a positive def-inite matrix are given. An RCB is obtained to solve the fol-lowing quadratic optimization problem under an ellipsoidal con-straint [3]:

subject to (8)

where in (7) without loss of generality. This problemis solved using the Lagrange multiplier method, which mini-mizes the cost function

(9)

where is the Lagrange multiplier, and is calculated usingthe differentiation of (9) in terms of . The RCB estimates of thesteering vector of the SOI

(10)

where consists of the eigenvectors of , and is a diagonalmatrix whose diagonal elements are the eigenvalues of , whichis obtained by eigenvalue decomposition [6]

(11)

The optimal weight vector of the RCB is obtained by substi-tuting into of (5)

(12)

Note that the RCB weight vector corresponds to the beamformerfor diagonal loading [2]. Thus, the power of the SOI corre-sponding to (6) is expressed as

(13)

where eliminates the scaling ambiguity.

IV. AUTO-CALIBRATION ALGORITHM

The RCB achieves robust SOI power estimation even whenthere is uncertainty in the SOI steering vector and strong inter-ference. However, the effect of mutual coupling between arraysensors is considerable in practice, and it degrades the powerestimation performance. Therefore, the mutual coupling effectshould also be included in the uncertainty set. In this section, wedescribe our auto-calibration algorithm for improving the RCB.It accurately estimates not only the power but also the AOA ofimpinging signals. The basic idea is that uncertainty, includingthe mutual coupling effect, is modeled in a manner similar toa Friedlander [4], and parameters of uncertainty are recursivelyidentified by solving the quadratic optimization problem underan ellipsoidal constraint as in the RCB. The proposed algorithmdramatically improves the AOA estimation performance of theRCB, which alone is unable to estimate the AOA because itsAOA spectrum has wide peaks. Our algorithm can accuratelyestimate both the power and AOA of the SOI even in the pres-ence of the effect of mutual coupling as well as steering vectoruncertainty.

A. Data Model With Uncertainty

First, we remodel the received signal by taking into consider-ation the mutual coupling effect and the uncertainty of the arraysensors. Similar to Friedlander’s approach [4], the receivedsignal model in (1) is modified

(14)

where is the diag-onal matrix of the error due to sensor location uncertainty, and

and denote the amplitude and phase errors, respectively.A complex matrix represents the magnitude of mutual cou-pling between sensors. Since the mutual coupling coefficientsare inversely proportional to the distance between the elements,uniform linear arrays (ULAs) considered through the numericalexamples make the matrix a banded Toeplitz matrix [4]. Notethat steering vector in (1) is replaced by in (14), where

is given. The theoretical covariance matrix corresponding to(2) is then

(15)

The problem of interest is to estimate the power, , andAOA, , by recursively identifying the uncertainty ma-trices, and , from only the received signal.

KIKUCHI et al.: AUTOCALIBRATION ALGORITHM FOR ROBUST CAPON BEAMFORMING 253

B. Proposed Algorithm

The proposed algorithm is formulated as a constrainedquadratic minimization problem. A new cost function is givenfrom the Capon power expression, which is minimized withrespect to , and

(16)

The key idea of the proposed method is that and can beidentified in manners similar to those with RCB when we as-sume these matrices include ellipsoidal uncertainty such as (7).Equation (16) is recursively minimized by the following.

Step 0) Initialize , and calculate samplecovariance matrix using (3), where denotesthe th iteration.

Step 1) Estimate from the peaks of theCapon AOA spectrum:

(17)

where .Step 2) Estimate using the following minimization,

which is a modification of the cost function (16)with respect to vector derivedfrom the components of :

(18)

where

Then, the th estimate of is obtained as

(19)

Note that the constraint in (18) is equivalent to thatof the RCB in (8) and that it works as the uncertaintyset of the amplitude and phase.

Step 3) Estimate in a fashion similar to that for estimatingwith a constraint equivalent to (18)

subject to (20)

where

Repeat Step 1) to Step 3) for integers until conver-gence is achieved.

Fig. 1. AOA spectra of SCB, RCB, and proposed method. M = 6 antennaelements, and N = 256 snapshots.

Step 4) Estimate the th signal power

(21)

where .By the definition of the cost function in (16) based on theCapon beamformer, our algorithm identifies both the AOA andpower of distinct signals, while only AOA is estimated bythe MUSIC-like approach of Friedlander [4].

V. NUMERICAL EXAMPLES

A. Simulation Results

The performances of the proposed and conventional methodswere first compared through a numerical simulation consid-ering mutual coupling. Narrowband sources were assumed tobe received at a six-element ULA from three distinct directions,

, and 35 . Their powers were set to 30, 25, and 35 dB,respectively. The number of sensors was , and thenumber of snapshots was . It was assumed that theerrors in amplitude and phase came from uniform distribu-tions, and , and was abanded Toeplitz matrix whose first column consisted of thevector . Fig. 1 shows theAOA and power estimation results for the SCB, RCB, and theproposed method. Obviously, the proposed method performedmuch better for both AOA and power estimation. Fig. 2 showsthe beam pattern of an antenna array when the signal from0 was regarded as the SOI and the other two sources wereinterference. The proposed method formed a beam to the SOIand steered nulls toward the interference sources, while theSCB and RCB failed. Fig. 3 depicts the estimated output signalto interference and noise ratio (SINR) when the noise powerwas fixed at 0 and the SOI power was varied between 30and 20 dB. The proposed method was better at eliminatinginterference even in the presence of mutual coupling. Finally,we investigated the convergence speed since the computationalburden of the proposed method strongly depends on the numberof iterations of the steps described in Section IV.B. Fig. 4

254 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 5, 2006

Fig. 2. Beam pattern of array when signal from 0 was SOI.M = 6 antennaelements, and N = 256 snapshots.

Fig. 3. Input power of SOI versus output SINR. M = 6 antenna elements,and N = 256 snapshots.

Fig. 4. Estimated power for each iteration. M = 6 antenna elements, andN = 256 snapshots.

performs that the method produced accurate estimates after acouple of iterations. The results shown in Figs. 3 and 4 were an-alyzed for 1000 independent trials in total. These results showthat the proposed method can effectively estimate the AOA and

Fig. 5. Experimental setup in anechoic chamber.

Fig. 6. Array beam pattern for experimental data analysis.M = 10 antennaelements, and N = 1024 snapshots.

power even if the array sensors suffer from uncertainty due tomutual coupling.

B. Experimental Results

We also conducted an experimental analysis using the datameasured in an anechoic chamber. Fig. 5 shows the basic exper-imental setup and the main specifications. The receive antennahad a ten-element ULA consisting of patch antennaswith 0.8-wavelength sensor spacing. Narrowband signals mod-ulated by -shift QPSK were transmitted from two separatesignal generators , and sample snapshot was 1024.Note that the signals from two generators could be regarded asuncorrelated sources. The carrier frequency and bandwidth ofthe transmitted signals were set at 1.74 GHz and 200 kHz, re-spectively, and the distance between the receiver and transmitter(Tx1) was 10 m, which the plane wave assumption was avail-able. Fig. 6 illustrates the beamforming results when the mainbeam was directed toward Tx1 ( ). The RCB failed to steer anull toward the interference, while the SCB could not so muchas make a main beam to the direction of the SOI. The resultsconfirm that the effect of mutual coupling cannot be neglectedin the analysis and that the proposed method worked well evenin the presence of mutual coupling. That is, it formed a beam to-ward the SOI ( ) and steered nulls to the interference ( ).

KIKUCHI et al.: AUTOCALIBRATION ALGORITHM FOR ROBUST CAPON BEAMFORMING 255

VI. CONCLUSION

We have described the new autocalibration algorithm that im-proves the RCB even in the presence of mutual coupling be-tween array sensors. The uncertainty of the steering vector ismodeled using two matrices, and the power and AOA of thesignal of interest are estimated by recursively identifying theuncertainty under a ellipsoidal constraint. The validity of theproposed algorithm was demonstrated by numerical simulationand experimental data analysis.

REFERENCES

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[3] J. Li, P. Stoica, and Z. Wang, “On robust Capon beamforming and diag-onal loading,” IEEE Trans. Signal Processing, vol. 51, pp. 1702–1715,Jul. 2003.

[4] B. Friedlander and A. J. Weiss, “Direction finding in the presence ofmutual coupling,” IEEE Trans. Antennas Propagt., vol. 39, no. 3, pp.273–184, Mar. 1991.

[5] J. Li, P. Stoica, and Z. Wang, “Doubly constrained robust Caponbeamformer,” IEEE Trans. Signal Process., vol. 52, pp. 2407–2423,Sep. 2004.

[6] G. H. Golub and C. F. Van Loan, Matrix Computations, 2nd ed. Balti-more, MD: John Hopkins University Press, 1989.