A Generalized ESPRIT Approach to Direction-of-Arrival Estimation

254 IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 3, MARCH 2005
A Generalized ESPRIT Approach to
Direction-of-Arrival Estimation
Abstract—A new spectral search-based direction-of-arrival
(DOA) estimation method is proposed that extends the idea of the
conventional ESPRIT DOA estimator to a much more general class
of array geometries than assumed by the conventional ESPRIT
technique. A computationally efficient polynomial rooting-based
search-free implementation of the proposed algorithm is also
developed.
Index Terms—DOA estimation, generalized ESPRIT.
I. INTRODUCTION
SENSOR ARRAY processing has found numerous applications
in radar, sonar, wireless communications, speech processing,
seismology, and other fields. Much of the work in array
processing has been focused on high-resolution methods for direction-of-arrival
(DOA) estimation, among which MUSIC [1],
[2] and ESPRIT [3], [4] are very popular techniques.
The ESPRIT algorithm is applicable to a particular class of
sensor arrays whose geometry is shift invariant. In the case of arrays
with multiple invariances, one- and two-dimensional extensions
of the conventional ESPRIT algorithm were developed in
[5]–[8]. In this paper, we extend the ESPRIT approach to a much
more general class of arrays that are not required to satisfy any
shift-invariance properties at all. A polynomial-rooting-based
search-free implementation of the proposed algorithm is also
developed.
II. ARRAY MODEL
Let us consider an array of sensors that consists of two
nonoverlapping subarrays of sensors each. 1 Let the first and
second subarrays be composed of the sensors with the indices
Manuscript received July 13, 2004; revised November 10, 2004. This
work was supported in part by the Wolfgang Paul Award Program of the
Alexander von Humboldt Foundation (Germany) and the German Ministry of
Education and Research, the Premier’s Research Excellence Award Program
of the Ministry of Energy, Science, and Technology (MEST) of Ontario, the
Discovery Grants Program of the Natural Sciences and Engineering Research
Council (NSERC) of Canada, and the Research Partnerships Program of
Communication and Information Technology Ontario (CITO). This paper was
presented in part at the IEEE SAM Workshop, Sitges, Spain, July 2004. The
associate editor coordinating the review of this manuscript and approving it for
publication was Dr. Raffaele Parisi.
F. Gao is with the Department of Electrical and Computer Engineering, Mc-
Master University, Hamilton, ON L8S 4K1, Canada.
A. B. Gershman is with the Department of Communication Systems, University
of Duisburg-Essen, 47057 Duisburg, Germany, on leave from the Department
of Electrical and Computer Engineering, McMaster University, Hamilton,
ON L8S 4K1, Canada.
Digital Object Identifier 10.1109/LSP.2004.842276
1 The assumption of nonoverlapping subarrays is only needed for notational
simplicity; the extension of the proposed approach to the case of overlapping
subarrays is straightforward.
Feifei Gao and Alex B. Gershman, Senior Member, IEEE
and , respectively. It is assumed that
the displacements between the th sensor of the first subarray
and the corresponding sensor of the second subarray are known
for all indices . However, these displacement
vectors may be completely arbitrary for different values of .
Apparently, such array specification is a generalization of the
conventional ESPRIT (shift invariant) case, where for any ,
the th sensor of the first subarray and the corresponding sensor
of the second subarray should be displaced by the same displacement
vector.
Assume that signals from far-field uncorrelated sources impinge
on the array. The array snapshot vector can be
written as
where is the array direction matrix, is the
vector of the source waveforms, and is the vector
of sensor noise that is assumed to be white Gaussian and to have
equal variance in each sensor. The array direction matrix can be
expressed as
where
1070-9908/$20.00 © 2005 IEEE
(1)
(2)
(3)
(4)
are the direction matrices of the first and second subarrays,
respectively, is the steering vector of the first
subarray
diag (5)
is the wavelength, are the - and -components
of the displacement vector between the th sensor of the first
subarray and the corresponding sensor of the second subarray,
and are the source DOAs with respect to the
normal direction to the -axis.
Let us write the eigendecomposition of the array covariance
matrix as [1]
where and are the diagonal matrices that contain the signaland
noise-subspace eigenvalues of , respectively, whereas
and are the orthonormal matrices composed of signal- and
(6)
(7)

GAO AND GERSHMAN: A GENERALIZED ESPRIT APPROACH TO DIRECTION-OF-ARRIVAL ESTIMATION 255
noise-subspace eigenvectors of , respectively. Here, and
stand for the Hermitian transpose and the statistical expectation,
respectively.
III. SPECTRAL SEARCH-BASED GENERALIZED ESPRIT
The matrix can be written as
where the matrices and correspond to the first and second
subarrays, respectively. Then, using the idea of conventional
ESPRIT [4], we have
where is an full-rank matrix, and
Introducing the notations
we can form a matrix
where
(8)
(9)
(10)
(11)
diag (12)
(13)
(14)
(15)
The th column of the matrix in the right-hand side of (14)
becomes equal to zero when .If , then in such a
case, the matrix will drop rank. As and are tall
matrices, we can obtain the source DOAs as the values of for
which the matrix drops rank, where is
an arbitrary full-rank matrix. A related discussion on the
choice of can be found in [11] from the traditional ESPRIT
viewpoint. To make our method consistent with the conventional
ESPRIT algorithm, is chosen. Then, the following
spectral function can be used to estimate the source DOAs:
(16)
We stress that there is some similarity between (16) and the
RARE estimator [9], [10], because the latter estimator also
makes use of a rank-dropping criterion. However, RARE represents
an extension of MUSIC rather than ESPRIT.
In the finite sample case, the covariance matrix can be estimated
as
(17)
where in the number of snapshots. The eigendecomposition
of the sample covariance matrix (17) can be written as
(18)
where and are the diagonal matrices that contain the signaland
noise-subspace eigenvalues of , respectively, whereas
and are the orthonormal matrices that contain the signal- and
noise-subspace eigenvectors of , respectively.
Using (18), the finite sample version of (16) can be written as
(19)
IV. SEARCH-FREE GENERALIZED ESPRIT
Estimator (19) involves computationally intensive spectral
search over . Let us now derive a computationally more
efficient search-free modification of the generalized ESPRIT
DOA estimator based on polynomial rooting. To develop such
an estimator, let us further specify the array geometry. Let us
assume that the -component of the th displacement is
zero for all . Furthermore, without any loss of
generality, let us assume that (note that the
sensors can be always enumerated in such a fashion). 2 Then
Denoting ,wehave , where
(20)
diag (21)
and the denominator of (16) can be written as the following
polynomial:
(22)
If all are integers, then the signal
DOAs can be found by means of rooting the polynomial (22).
Clearly, this approach can be easily extended to the case when
there is any common multiplier that makes all the values of
integers.
In the finite sample case, the polynomial (22) becomes
(23)
and the signal DOAs can be estimated from the closest to the
unit circle roots of (23), similarly to the root-MUSIC estimator
[2].
In the conventional ESPRIT array geometry case, we have
, and the polynomial (23) reduces to
(24)
2 These assumptions specify the array to have parallel but not necessarily identical
displacement vectors. Note that this array specification is much more general
than that required for conventional ESPRIT, where all the displacement
vectors have to be identical.

256 IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 3, MARCH 2005
Fig. 1. DOA estimation RMSEs and CRB versus SNR.
In this case, the roots of (24) are the generalized eigenvalues
of the matrix pencil , and hence, the proposed
search-free generalized ESPRIT estimator reduces to the conventional
ESPRIT estimator [3], [4]. Note, however, that the
proposed estimator in (23) applies to a much more general class
of array geometries than conventional ESPRIT.
V. SIMULATIONS
In our simulations, we compare the performance of the proposed
spectral and search-free generalized ESPRIT estimators
with the deterministic Cramér–Rao bound (CRB) [12]. All
results are averaged over 100 simulation runs and over the
sources. The array is assumed to consist of two subarrays of
sensors each, where the first subarray is linear and
vertically oriented. Two uncorrelated equipowered sources impinge
on the array from the DOAs and
with respect to the endfire direction of the first subarray (which
corresponds to the normal direction to the -axis). The interelement
spacings between the second and first, third and
second, and fourth and third sensors of the first subarray are
, , and , respectively. The horizontal displacements
between the first, second, third, and fourth sensors of the second
subarray and the corresponding sensors of the first subarray
are , , , and , respectively.
Note that neither conventional ESPRIT nor root-MUSIC are
applicable to such array structure, but the proposed search-free
generalized ESPRIT technique can be applied. The DOA estimation
root-mean-square errors (RMSEs) of both the spectral
search-based and search-free generalized ESPRIT estimators
are shown in Fig. 1 versus the signal-to-noise ratio (SNR) for
.
Fig. 2 displays the DOA estimation RMSEs of the same estimators
versus the number of snapshots . In this figure, SNR
dB.
From Figs. 1 and 2, we see that the search-free generalized
ESPRIT estimator has a lower threshold in SNR and number
of snapshots than spectral search-based generalized ESPRIT.
Fig. 2. DOA estimation RMSEs and CRB versus u.
This observation is in agreement with the results of [13], where
it has been shown that the threshold performance of polynomial
rooting-based DOA estimation methods can be expected
to be better than that of their spectral search-based counterparts
because polynomial rooting methods are insensitive to radial
errors with respect to the unit circle (of course, provided that
there is no subspace swap effect).
It can be also seen from Figs. 1 and 2 that with high SNRs
and a large number of snapshots, the performances of both estimators
are reasonably close to the deterministic CRB.
VI. CONCLUSIONS
A new DOA estimation method has been proposed that extends
the concept of the conventional ESPRIT estimator to a
more general class of array geometries that are not necessarily
shift invariant. A computationally efficient search-free implementation
of the proposed algorithm is developed.
REFERENCES
[1] R. O. Schmidt, “A Signal Subspace Approach to Multiple Source Location
and Spectral Estimation,” Ph.D. dissertation, Stanford Univ., Stanford,
CA, May 1981.
[2] A. J. Barabell, “Improving the resolution performance of eigenstructurebased
direction-finding algorithms,” in Proc. ICASSP, Boston, MA, May
1983, pp. 336–339.
[3] A. Paulraj, R. Roy, and T. Kailath, “A subspace rotation approach to
signal parameter estimation,” Proc. IEEE, vol. 74, pp. 1044–1045, Jul.
1986.
[4] R. Roy and T. Kailath, “ESPRIT—estimation of signal parameters via
rotational invariance techniques,” IEEE Trans. Acoust., Speech, Signal
Process., vol. 37, no. 7, pp. 984–995, Jul. 1989.
[5] A. B. Gershman and R. A. Ugrinovsky, “Two-dimensional angle-of-arrival
estimation in arrays with unknown global geometry” (in Russian,
translated in Telecommunications and Radio Engineering), Radiotechnika,
no. 1, pp. 71–73, 1990.
[6] A. L. Swindlehurst, B. Ottersten, R. Roy, and T. Kailath, “Multiple
invariance ESPRIT,” IEEE Trans. Signal Process., vol. 40, no. 4, pp.
867–881, Apr. 1992.
[7] A. L. Swindlehurst and T. Kailath, “Algorithms for azimuth-elevation
direction finding using regular array geometries,” IEEE Trans. Aerosp.
Electron. Syst., vol. 29, no. 1, pp. 145–156, Jan. 1993.

GAO AND GERSHMAN: A GENERALIZED ESPRIT APPROACH TO DIRECTION-OF-ARRIVAL ESTIMATION 257
[8] N. D. Sidiropoulos, R. Bro, and G. B. Giannakis, “Parallel factor analysis
in sensor array processing,” IEEE Trans. Signal Process., vol. 48, no. 8,
pp. 2377–2388, Aug. 2000.
[9] M. Pesavento, A. B. Gershman, and K. M. Wong, “Direction finding in
partly calibrated sensor arrays composed of multiple subarrays,” IEEE
Trans. Signal Process., vol. 50, no. 9, pp. 2103–2115, Sep. 2002.
[10] C. M. S. See and A. B. Gershman, “Direction-of-arrival estimation in
partly calibrated subarray-based sensor arrays,” IEEE Trans. Signal
Process., vol. 52, no. 2, pp. 329–338, Feb. 2004.
[11] B. D. Rao and K. V. S. Hari, “Weighted subspace methods and spatial
smoothing: analysis and comparison,” IEEE Trans. Signal Process., vol.
41, no. 2, pp. 788–803, Feb. 1993.
[12] P. Stoica and A. Nehorai, “MUSIC, maximum likelihood and
Cramer–Rao bound,” IEEE Trans. Acoust., Speech, Signal Process.,
vol. 37, no. 5, pp. 720–741, May 1989.
[13] B. D. Rao and K. V. S. Hari, “Performance analysis of root-MUSIC,”
IEEE Trans. Acoust., Speech, Signal Process., vol. 37, no. 12, pp.
1939–1949, Dec. 1989.