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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, NO 10, VOL. 51, OCTOBER 2004

Asymptotic SNR of Scalar and Vector
Minimum-Variance Beamformers for
Neuromagnetic Source Reconstruction
Kensuke Sekihara*, Member, IEEE, Srikantan S. Nagarajan, David Poeppel, and Alec Marantz

Abstract—To reconstruct neuromagnetic sources, the minimum-variance beamformer has been extended to incorporate thethree-dimensional vector nature of the sources, and two types ofextensions—the scalar- and vector-type extensions—have been
proposed. This paper discusses the asymptotic signal-to-noise
ratio (SNR) of the outputs of these two types of beamformers. We
first show that these two types of beamformers give exactly thesame output power and output SNR if the beamformer pointing
direction is optimized. We then compare the output SNR of the
beamformer with optimum direction to that of the conventional
vector beamformer formulation where the beamformer pointing
direction is not optimized. The comparison shows that the beamformer with optimum direction gives an output SNR superior tothat of the conventional vector beamformer. Numerical examples
validating the results of the analysis are presented.
Index Terms—Biomagnetism, inverse problems, magnetoencephalography, minimum-variance beamformer, neural signal
processing.

I. INTRODUCTIONTHE SEARCH FOR efficient algorithms for reconstructing
spatio-temporal brain activities from neuromagnetic measurements has attracted great interest [1]. One promising algorithm is the minimum-variance beamformer, which was originally developed in the field of array signal processing suchas in radar, sonar, and seismic exploration, and has been successfully applied to neuromagnetic source reconstruction problems [2]–[4]. However, the minimum-variance beamformer in
its original form [5] cannot be directly applied to neuromagnetic
source reconstruction. This is because the neural source distribution is an electrophysiological current distribution, which is athree-dimensional (3-D) vector quantity, so the method should
be extended to incorporate the 3-D vector nature of the sources.

So far, two types of extensions of the minimum variancebeamformer that incorporate source orientation have been
proposed in the literature. One extension, called the scalar-type
beamformer, uses a beamformer weight that depends not only
on the location but also on the direction of a source [6]. The
other type of extension, called the vector-type beamformer,
uses a set of three weights where each weight detects one component in the three orthogonal directions [7], [8]. A previous
study [9] showed that the scalar-type beamformer can attain
twofold better output SNR compared to the vector formulation.
In this paper, we show that the scalar and the vector formulations give exactly the same asymptotic output SNR if the
beamformer pointing direction is chosen to maximize the beamformer outputs. We then analyze the performance of this beamformer with optimum pointing direction, and compare its performance to that of the conventional vector beamformer formulation where the beamformer pointing direction is not optimized.
A theoretical analysis is given for an isolated source and fortwo closely located sources. The results of the analysis showthat significant SNR degradation can arise in the conventional
formulation, and numerical experiments validate these results.
Following a brief review on the two types of extensions
in Section II, this paper presents our theoretical analysis in
Sections III–V. Section VI presents numerical examples that
illustrate the results of our analysis. Throughout this paper,
plain italics indicate scalars, lower-case boldface italics indicate
vectors, and upper-case boldface italics indicate matrices. The
eigenvalues are numbered in decreasing order.
II. SCALAR AND VECTOR MINIMUM-VARIANCE
BEAMFORMER FORMULATIONS
A. Definitions

Manuscript received July 9, 2003; revised December 21, 2003. The work of K.
Sekihara was supported by Grants-in-Aid from the Ministry of Education, Science, Culture and Sports in Japan (C13680948 and C16500296).. The work of S.
S. Nagarajan was supported by the Whitaker Foundation and by the National Institute of Health (P41RR12553-03 and R01-DC004855-01A1). The work of D.
Poeppel was supported by the National Institute of Health (NIH-R01-DC5660).
Asterisk indicates corresponding author.
*K. Sekihara is with the Department of Electronic Systems and Engineering,Tokyo Metropolitan Institute of Technology, Tokyo 191-0065, Japan (e-mail:ksekiha@cc.tmit.ac.jp).
S. S. Nagarajan is with the Department of Radiology, University of California,
San Francisco, CA 94143 USA.
D. Poeppel is with the Department of Linguistics and Biology, University of
Maryland, College Park, MD 20742 USA.
A. Marantz is with the Department of Linguistics and Philosophy, Massachusetts Institute of Technology, Cambridge, MA 02139 USA.
Digital Object Identifier 10.1109/TBME.2004.827926

th
We define the magnetic field measured by thedetector coil at timeas
, and a column vector
as a set of measured
is the total number of sensor coils and sudata whereindicates the matrix transpose. The spatial loperscript
cation is represented by a 3-D vector :. Thesecond-order moment matrix of the measurement is denoted, i.e.,, whereindicates the ensemble
average, which is replaced with the time average over a cerholds,
istain time window in practice. Whenalso equal to the covariance matrix of the measurement. The. The source orientation issource magnitude is denoted
defined as a 3-D column vector

0018-9294/04$20.00 © 2004 IEEE

SEKIHARA et al.: ASYMPTOTIC SNR OF SCALAR AND VECTOR MINIMUM-VARIANCE BEAMFORMERS

whose component (where equals , , or ) is equal tothe cosine of the angle between the direction of the source
and the direction. The 3-D source vector is expressed as.
as the output of the th sensor. The output
We defineandis induced by the unit-magnitude source located at
is defined
pointing in the direction. The column vector. The array responseasmatrix, which represents the sensitivity of the whole sensor. The array
array at , is defined as, whereresponse vector in the direction is defined as. The array response matrix
and thearray response vector
are often, respectively, called thelead field matrix and the lead field vector in the field of thebiomagnetic imaging.
B. Scalar-Type Minimum-Variance Beamformer

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components of a source current vector. (Note that any of threeorthogonal directions can be used, instead of the , , and directions.) A set of weights for a vector-extended minimum-variance beamformer is derived using the optimization [7], [8](4)is a weight matrix defined as;is the identity matrix; andindicates the trace operation. The resultant weight matrix is
given by [8]
where

(5)
Using this weight matrix, the , , andsource current vector are estimated fromcomponents of the(6)

We focus on the technique referred to as the minimum-variance beamformer [5] for reconstructing neuromagnetic sources.
Since the neuromagnetic source is a 3-D vector quantity, theoriginal minimum-variance beamformer formulation should be
extended to incorporate the 3-D vector nature of sources. Twotypes of extensions have been proposed. The scalar extension
by minimizing
under thederives the weight
constraint of. The explicit form of the weightis expressed as(1)
Note that the weight in (1) depends not only on the spatial location but also on the direction , and therefore the information.
regarding the source orientation is needed to calculate
Using the weight in (1), the output source power (the powerof the reconstructed source) is obtained fromis the component (, or ) of the estiwheremated source vector. The output source power is conventionallygiven by [8](7)When using the above equation, the white-noise power gain is, as is shown in Appendix I. Thus,
given by,the output SNR corresponding to the output power in (7),is expressed in [9]

(8)

(2)
We also define the value

III. EQUIVALENCE OF THE TWO TYPES OF BEAMFORMERSUNDER THE OPTIMUM POINTING ORIENTATION

such that
(3)

where
is the variance of the input noise. Because the squareof the weight norm
is the white noise power gain, as isis equal to the reconstructed source
shown in Appendix I,
power divided by the reconstructed noise power, and this value
is customarily called the output SNR1. It should be noted that theexpression similar to (3) has been reported in the fields of power
spectrum estimation [10] and of antenna-array processing [11].
C. Vector-Type Minimum-Variance Beamformer
The other type of extension, called the vector-type beamformer, allows simultaneous estimation of the source orientation
as well as the source magnitude. It uses a set of three weight vec,
, and, which estimate the , , andtors,
1This value is in fact equal to the output SNR plus 1 because it is the output
signal power plus output noise power divided by the output noise power.

This section shows that the scalar-type beamformer in (1) andthe vector-type beamformer in (5) give exactly the same outputs
if the beamformer orientation is optimized. As mentioned previously, the scalar-type beamformer requires the determinationof the source orientation at each spatial location . One way
to determine the optimum orientation at each is to choose theorientation that gives the maximum power output, i.e., is determined by using(9)
We define the eigenvalues and eigenvectors of the 3 3 matrixasand , where, 2, and 3. We assume that the eigenvalues are numbered in decreasing order. It
is well known, based on the Rayleigh-Ritz formula, that giving
is equal to , which is
the minimum of
the eigenvector corresponding to the minimum eigenvalue of1728

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, NO 10, VOL. 51, OCTOBER 2004

[12], [13]. The maximum output source poweris then expressed as

eigenvector corresponding to the minimum eigenvalue of (14),opt
is expressed as
and thus
opt(10)
We define this value of the output power as opt , namely,.
opt
We next show that the vector beamformer also attains opt by
maximizing the beamformer output. For the vector-type beamformer, the output source power in the direction is expressed. Using (5), the maximizaastion of the output power leads to

(11)

(15)

is invertible, the generalBecause the matrix
ized eigenproblem in (14) can be changed to the following conventional eigenproblem:(16)
and can also be obtained as the minimum eigenvalue of (16).
In the vector beamformer formulation, the output SNR maxiopt, which is obtained
mized with respect to is defined as
by using
opt

Because the relationship(17)
holds, the optimum obtained in maximizing the right-handside of (11) is equal to the eigenvector corresponding to the, whichmaximum eigenvalue of the matrix
is equal to. Accordingly, we obtain the optimum direction, i.e.,. Thus,as the eigenvector corresponding tothe output power is expressed as

opt

where

We define the generalized eigenvalues and eigenvectors ofin the metricasand , i.e.,andsatisfy(18)(12)

This equation indicates that either type of beamformer formulation attains opt when the beamformer pointing direction is set
to the direction that gives the maximum output.
We then show that the scalar and vector beamformer formulations also give exactly the same output SNRs (the values)
when the beamformer pointing direction is optimized. In thescalar beamformer formulation, the output SNR maximized
opt, which is obtained bywith respect to is defined as
opt

(13)

opt
is expressed
The optimum in (17) is equal to , andopt
as. It is easy to see that the generalized
eigenproblem in (18) can be changed to the conventional eigenproblem(19)
and is obtained as the minimum eigenvalue in (19).
Comparing the matrix in (19) with that in (16), one can seethat both of these matrices consist of the product of the same twoand, and only
matrices,the order of multiplication differs. Therefore, the eigenvalue
opt
in (19) is equal to in (16) [12], andis rewritten as
opt

We define the generalized eigenvalues and eigenvectors ofin the metric
as
and ,
i.e.,
and
satisfy(14)
where, 2, and 3. Then, it is shown in Appendix II (andin [14]) that the optimum in (13) is given as , which is the

(20)

opt
opt
Becauseis exactly the same as, it can be concluded that the scalar and the vector formulations can attainthe same output SNR when the direction that gives the maximum SNR is chosen. We denote this output SNR obtained withthe optimum beamformer direction as opt , namely, opt.

SEKIHARA et al.: ASYMPTOTIC SNR OF SCALAR AND VECTOR MINIMUM-VARIANCE BEAMFORMERS

IV. ASYMPTOTIC OUTPUT SNR OF BEAMFORMERS WITH
OPTIMUM ORIENTATIONIn Section III, we show that when the beamformer pointing
direction is optimized, both the vector and the scalar beamformer formulations attain the output SNR opt . In thissection, we derive the explicit form of opt . Here, we omit
for simplicity unless any confuthe notations of andsion arises. We define for later use the generalized cosine
andin the metricasbetween two column vectors. Whenis equal to the identity matrix, this cosine is simply denoted.
We first assume the simplest case where a single source whose
exists at , and define the lead fieldorientation is equal to. The
vector for this single source as , such thatpower of the source is defined as . The power of the input
noise is again denoted . The covariance matrix and its inverse
are then expressed as(21)(where1729

sources create similar lead field vectors, but is nearly equal tozero when their lead field vectors are very different. We define
the power of the first source as , and that of the second source
as . The covariance matrix is then expressed as(25)
and as. GenWe define asholds for a large sensor array anderally, the relationshipthe value of is very close to 1. (Numerical examples of thesevalues are presented in Section VI.) As shown in Appendix III,, as follows:
it is straightforward to derive opt
opt

(26)

Equation (26) shows that the output SNR is directly affectedby the spatial correlation, and only when the relationshipholds, the input SNR is preserved in the beamformer reconstruction process.
V. ASYMPTOTIC OUTPUT SNR OF VECTOR BEAMFORMERWITHOUT ORIENTATION OPTIMIZATION

is again the power of the input noise) and
(22)

where
and
. This is sometimes referred to as the input power SNR [15]. This is usuallymuch greater than 1 for large sensor arrays such as those usedin recent neuromagnetic measurements, and is approximately
equal to 1. (Numerical examples of these values are given in
Section VI.)
In the scalar beamformer formulation, when the beamformer
is expressed,
pointing direction is set to , the output SNR
using (3), as

In Section IV, we derived the explicit form of opt , which is
the output SNR attained either by the scalar or the vector beamformer formulations when the beamformer pointing direction is
optimized. On the other hand, as mentioned in Section II-C, thevector beamformer conventionally does not optimize the beamformer pointing direction, and (8) is used for calculating theoutput SNR. In this section, we derive the explicit form of theoutput SNR when (8) is used.
We first assume a case where a single source exists, and use
the expression in (22) for the inverse of the covariance matrix.
(whereWe use the three orthogonal directions , , and(, 2, 3) is the three eigenvectors of) tocalculate the weight matrix , and define such that(
, 2, 3). Then, the output power obtained using (7) is, which can be expressed asdenoted conv
conv

(23)
It can be seen that
Setting equal toopt

reaches its maximum whenin (23), opt
is expressed as

.
(27)
(24)

Thus, the input SNR and the output SNR are equal and there is
no SNR degradation in the beamformer reconstruction.
We next analyze a case where two sources exist. The locationsof the sources are denoted and , and their orientations are
and . We define the lead field vectors for the twodenotedand.
sources as and , such that
Here, we define the spatial correlation [16] between the two, which is nearly equal to 1 when the twosources as

Let us define the output SNR obtained using (8) as conv
The denominator in (8) can be expressed as

.

(28)

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, NO 10, VOL. 51, OCTOBER 2004

where we use the relationshipBy further assuming the use of the spherically symmetric homogeneous conductor model, we can next derive an expression
when two sources exist. We use the same notafor conv
tions defined in Section IV. After some lengthy calculations, (apart of which is shown in Appendix III), we finally obtain
(33)

conv
where

(34)

(29)

where
, 2, and subAssuming
stituting (27) and (28) into (8), we finally obtainand where we have (35), as shown at the bottom of the page. The
is given in (47). In deriving (33)–(35),definition ofor
we use several appropriate approximations such as. However, it is not straightforward to evaluate
and using (34) and (35). In Section VI, we numerically determine these values and show that is much smaller than 1 but
is considerably larger than 1 in typical neuromagnetic measurement conditions.

conv
VI. NUMERICAL EXAMPLES
Assuming that, convship(30), and using the relationis finally expressed as

conv
(31)
is one third of the inputThis equation indicates that conv
, where the input SNR is
SNR. This is in contrast to opt
preserved.
When the spherically symmetric homogeneous conductor
model [17] is used for the forward calculation, the source
current vector is expressed in the two tangential components.
is anmatrix, andAs a result, the lead field matrix
is a 2 2 matrix. In such cases, it is easy toshow that (31) changes to
conv

(32)

is half of the input
The above equation indicates that conv
SNR. This result is in accordance with that obtained by Vrba andRobinson [9], who assumed a special source-sensor configuration where a single source exists directly below the center of arotationally-symmetric sensor array.

Numerical examples are presented here to illustrate several
results of the analysis in Sections II–V. A sensor alignment of
the 148-sensor array from Magnes 2500™ (4D Neuroimaging
Inc., San Diego) whole-head neuromagnetometer was used. The
coordinate origin was set at the center of the sensor coil locatedat the center of the coil array. The direction was defined as thatfrom the posterior to the anterior; the direction was defined asthat from the right to the left hemispheres; and direction was
defined as that perpendicular to the surface of the coil at theorigin.
Two point sources were assumed to exist atand, i.e., they were 1.6 cm apart and located.
6 cm below the center of the sensor array on the plane
The source-sensor configuration and the coordinate system areillustrated in Fig. 1. The simulated time courses assigned to theand, are also shown infirst and the second sources,
Fig. 1. The cross correlation coefficient between these two timecourses is approximately equal to 0.086, and they are nearlyandareorthogonal to each other. The amplitudes of, i.e., the two sources
adjusted to have the relationshiphave equal powers.
The lead field vectors of the sensor array for these sources,
and , were calculated by using the spherically homo.
geneous conductor model with its center set at
is calculated using
The simulated magnetic recording. Simulated sensor noise uncorrelated
among sensor channels was added so that the ratio between theto theaverage power of the signal magnetic field(35)

SEKIHARA et al.: ASYMPTOTIC SNR OF SCALAR AND VECTOR MINIMUM-VARIANCE BEAMFORMERS

Fig. 2.
(

1731

Cross-sectional view of the reconstructed first source along the line

x = 0; z = 06) cm for the single-source experiments. The solid line

shows Zopt and the broken line shows Zconv . The SNR of the simulated data
hkbk i=(M ) was set to 1, resulting in equal to 148.

Fig. 1. Coordinate system and source-sensor configuration used in the
numerical experiments. The coordinate origin was set at the center of the sensor
coil located at the center of the array. The plane at x = 0 cm is shown. The
two point sources, shown by the small filled circles, were located on the line
(x = 0; z = 6) cm. The time courses assigned to the two sources are shownin the two upper panels.

0

noise poweris equal to 1. Here,is the total number of
sensors and it is equal to 148 in our numerical experiments.
Typical simulated recording with this SNR is shown in thelower panel in Fig. 1.
First, we simulated a case where a single source exists. We
to the first source, but we set theassigned the time course. In this case,second-source time course to zero, i.e.,
the input SNR for the first source, , is equal to(148) andresults in 0.998. The cross sections of the reconstructed firstare shown in Fig. 2.
source along the line
Here, the reconstruction with opt is shown by the solid line,
and that with conv is shown by the broken line. The peak valueof opt is equal to 147.5, and that of conv is equal to 70.5.
These results show that the peak from opt is nearly twice ashigh as the peak from conv , and this is in accordance with theresults in Section V.
We then simulated a case where the two sources are both active by assigning the time courses
and, shown in
Fig. 1, to the first and the second sources. The orientation of theand.
two sources were set equal tois nearly equal
The spatial correlation represented by. Simulated sensor noise
to zero in this caseis equal to one. The
was added so that the ratio
input SNRs for the first and second sources, and , are both
equal to 131, and the values of and are both equal to 0.992.
The cross sections of the reconstructed results along the line
are shown in Fig. 3. The reconstruction

with opt is shown by the solid line, and that with conv isshown by the broken line.
The peak intensities of opt were found to be 131 for the
first and the second sources, and those of conv were foundto be 19. Thus, the peak-intensity ratio of conv to opt is. On the othercalculated to be conv opt
andin this case are 0.047 andhand, since the values of
6.04, respectively, the SNR reduction factorin (33) is approximately equal to. Therefore, theintensity reduction of conv can be well explained by the factor, and these results clearly validate (33).
VII. CONCLUSION
In conclusion, this paper proves that the scalar and vector
minimum-variance beamformer formulations give exactly thesame output power and output SNR, if the beamformer pointing
direction is optimized. We then compare the theoretical output
SNR of a beamformer with optimum pointing direction to the
output from the conventional vector beamformer formulation
without an optimized pointing direction. The comparison shows
significant SNR degradation with the conventional vector beamformer formulation.
The primary purpose of this paper was to show that boththe scalar and vector types of beamformers potentially attainthe same SNR performances, although the conventional way of
formulating the vector beamformer can cause significant SNR
degradation. However, it is not within the scope of this paper
to discuss the superiority of one of the two formulations over
the other. This is because the quality of the source reconstruction results are determined not only by the asymptotic SNR but
also by various factors such as spatial resolution, source correlation or the influence from background physiological noise.
Therefore, To determine the superiority, a thorough investigation would be required, not only of the theoretical SNR but alsoof other performance measures including the spatial resolution
and the robustness to various causes of errors such as mentioned
above. Such investigations are currently being conducted withresults to be published in the near future.

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, NO 10, VOL. 51, OCTOBER 20040
hk k iFig. 3. Cross-sectional view of the reconstruction results along the line (x = 0; z = 6) cm for the experiments where the two sources are both active. The
solid line shows opt and the broken line shows conv . The SNR of the simulated data b =(M ) was set to 1, and this results in both and equal to131. The source orientations of the two sources were set to (0:91; 0:42) and (0:91; 0:42), resulting in cos (f ; g ) 0.

Z

Z

0

APPENDIX IThis Appendix shows that the white-noise power gain isequal tofor the scalar beamformer and it is equal
for the conventional vector beamformer.
tois contained in the
We assume that the additive noiseis expressed asmeasurement, i.e., the measurement, where is the number
is the white Gaussian noiseof sources. We assume that
uncorrelated among sensor channels. The variance of the noise
is denoted . Then, the output noise for the scalar beamformer
is expressed as. The variance of the output noise is,thus, equal to(36)
For the vector beamformer, using (6), the output noise is ex. Therefore, the variance of the output
pressed asnoise is expressed as

(37)
where

indicates the trace operation.
APPENDIX II

We defineas an

is not affected
Since the value of the ratioby the norm of , we set the norm of so as to satisfy the. Then, the minimization problem in
relationship(38) is rewritten as(40)
We change this constrained minimization problem to an unconstrained minimization problem by introducing the Lagrange
such thatmultiplier , i.e., we define the function(41).
The minimization in (40) is equivalent to minimizing, we calculate the derivatives
To obtain the minimum of(42)
By setting these derivatives to zero, we can derive the relationships,
and. Therefore, the minimum
value ofis equal to, which is the minimum eigen, and that attains this minimum value isvalue of.
equal to the eigenvector
APPENDIX III

and aspositive definite matrices, andcolumn vector. This Appendix shows that
(38)

where
and
are the minimum eigenvalue and its
corresponding eigenvector of the following generalized eigen
problem:This Appendix provides several supplementary formulae thatare the basis of the analysis for the two-source case in Sections IV and V. We define the lead field vector for the first source
as and its power as . The lead field vector for the secondsource is defined as and its power as . The power of the
input noise is denoted . When two sources exist, the covariance matrix is expressed as (25). Its inverse is expressed as

(43)
(39)

SEKIHARA et al.: ASYMPTOTIC SNR OF SCALAR AND VECTOR MINIMUM-VARIANCE BEAMFORMERS

1733

(50)

;
;
; and
(43), we can obtain the following equations:

where

;
. Using(44)

and the denominator can be expressed as in (50), shown at the
top of the page. We assume that the beamformer can correctly detect the source orientation at and thus is equal to . We also
assume that the norm of the lead field vector is similar between. Using these apthe two tangential directions, i.e.,
propriate approximations, we can derive (33) from (49) and (50).
ACKNOWLEDGMENTandThe authors wish to thank Dr. S. Robinson and J. Vrba for
their insightful discussions.
(45)
REFERENCES
where
(46)
and

(47)
is an arbitrary lead field vector, andwhen.
Using (44) and (45), the output SNR for the scalar beamformer is expressed as
Note that(48)

The optimum output SNR is obtained using this equation by setting equal to , and it results in (26). To derive (33), we choose
the two orthogonal directions as the directions , and , which
are the eigenvectors ofwhen the spherical homogeneous conductor is used. We then calculate the numerator
and the denominator of (8). The numerator can be expressed as(49)[1] S. Baillet, J. C. Mosher, and R. M. Leahy, “Electromagnetic brain mapping,” IEEE Signal Processing Mag., vol. 18, pp. 14–30, 2001.
[2] I. Hashimoto, K. Sakuma, T. Kimura, Y. Iguchi, and K. Sekihara, “Serial
activation of distinct cytoarchitectonic areas of the human SI cortex after
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[3] I. Hashimoto, T. Kimura, M. Tanosaki, Y. Iguchi, and K. Sekihara,“Muscle afferent inputs from the hand activate human cerebellum
sequentially through parallel and climbing fiber systems,” J. Clin.
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[4] R. Ishii, R. Dziewas, W. Chau, P. Soros, H. Okamoto, A. Gunji, and C.
Pantev, “Current source density distribution of sleep spindles in humans
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[5] B. D. van Veen and K. M. Buckley, “Beamforming: A versatile approachto spatial filtering,” IEEE ASSP Mag., vol. 5, pp. 4–24, Apr. 1988.
[6] S. E. Robinson and J. Vrba et al., “Functional neuroimaging by synthetic
aperture magnetometry (SAM),” in Recent Advances in Biomagnetism,
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[7] M. E. Spencer, R. M. Leahy, J. C. Mosher, and P. S. Lewis, “Adaptive
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and Computers, Nov. 1992, pp. 156–161.
[8] B. D. van Veen, W. van Drongelen, M. Yuchtman, and A. Suzuki, “Localization of brain electrical activity via linearly constrained minimum variance spatial filtering,” IEEE Trans. Biomed. Eng., vol. 44, pp. 867–880,
Sept. 1997.
[9] J. Vrba and S. Robinson et al., “Differences between synthetic aperturemagnetometry (SAM) and linear beamformer,” in Proc. 12th Int. Conf.
Biomagnetism, R. Hari et al., Eds., 2001, pp. 681–684.
[10] M. A. Lagunas, M. E. Santamaria, A. Gasull, and A. Moreno, “Maximum likelihood filters in spectral estimation problems,” Signal Processing, vol. 10, pp. 19–34, 1986.
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[12] H. Lütkepohl, Handbook of Matrices. New York, NY: Wiley, 1996.
[13] C. D. Meyer, Matrix Analysis and Applied Linear Algebra. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2000.
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[17] J. Sarvas, “Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem,” Phys. Med. Biol., vol. 32, pp. 11–22, 1987.

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Kensuke Sekihara (M’88) received the M.S. and
Ph.D. degrees from Tokyo Institute of Technology,
Tokyo, Japan, in 1976 and 1987, respectively.
From 1976 to 2000, he was with the Central
Research Laboratory, Hitachi, Ltd., Tokyo, Japan.
He was a Visiting Research Scientist at Stanford
University, Stanford, CA, from 1985 to 1986, and at
Basic Development, Siemens Medical Engineering,
Erlangen, Germany, from 1991 to 1992. From 1996
to 2000, he worked with the “Mind Articulation”
research project sponsored by the Japan Science andTechnology Corporation. He is currently a Professor at the Tokyo Metropolitan
Institute of Technology. His research interests include the neuromagnetic sourcereconstruction and statistical signal processing, especially its application tofunctional neuroimaging.
Dr. Sekihara is a member of the IEEE Medicine and Biology Society and theIEEE Signal Processing Society.

Srikantan S. Nagarajan received the Bachelor’s
degree in electrical engineering from the Universityof Madras, Madras, India, and the Master’s andPh.D. degrees from the Department of Biomedical
Engineering, Case Western Reserve University,
Cleveland, OH, in 1993 and 1995, respectively.
From 1995–1998, he was a post-doctoral Fellow
at the Keck Center for Integrative Neuroscience,
University of California at San Francisco (UCSF).
In 1999, he was a Research Scientist at ScientificLearning Corporation, Berkeley, CA and an Adjunct
Assistant Professor in the Department of Otolaryngology, UCSF. From 2000
to 2002, he was an Assistant Professor in the Department of Bioengineering,
University of Utah, Salt Lake City. Currently, he is the Director of the Biomagnetic Imaging Laboratory and an Assistant Professor in Residence in the
Department of Radiology, UCSF. His research interests in neural engineering
include multimodal and multiscale functional brain imaging using functional
magnetic resonance imaging, magnetoencephalography, electroencephalography, systems and computational neuroscience, and neuro-rehabilitation.

David Poeppel received the B.S. and Ph.D. degrees
in cognitive neuroscience from Massachusetts Institute of Technology, Cambridge, in 1990 and 1995,
respectively.
He is currently an Associate Professor in the
Departments of Linguistics and Biology, Universityof Maryland at College Park. His research uses
the functional neuroimaging methods magnetoencephalography, electroencephalography, positronemission tomography, and functional magnetic
resonance imaging to investigate the neural basis of
speech and language processing.

Alac Marantz received the B.A. degree in psycholinguistics from Oberlin College, Oberlin, OH, in 1978
and the Ph.D. degree in linguistics from the Massachusetts Institute of Technology (MIT), Cambridge,
MA, in 1981.
He joined the faculty of MIT in 1990, where he iscurrently the Kenan Sahin Distinguished Professorof Linguistics and Head of the Department ofLinguistics and Philosophy. His research interests
include the syntax and morphology of natural languages, linguistic universals, and the neurobiologyof language. He is currently involved in revising morphological theory withinlinguistics and in exploring magnetoencephalography techniques to uncover
how the brain process language.