Unitary Root-MUSIC with a Real-Valued Eigendecomposition: A Theoretical and Experimental Performance Study
1306
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 5, MAY 2000
Unitary Root-MUSIC with a Real-Valued
Eigendecomposition: A Theoretical and Experimental
Performance Study
Marius Pesavento, Alex B. Gershman, Senior Member, IEEE, and Martin Haardt, Senior Member, IEEE
Abstract—A real-valued (unitary) formulation of the popular
root-MUSIC direction-of-arrival (DOA) estimation technique is
considered. This Unitary root-MUSIC algorithm reduces the computational complexity in the eigenanalysis stage of root-MUSIC
because it exploits the eigendecomposition of a real-valued covariance matrix. The asymptotic performance of Unitary root-MUSIC
is analyzed and compared with that of conventional root-MUSIC.
The results of this comparison show identical asymptotic properties of both algorithms in the case of uncorrelated sources and
a better performance of Unitary root-MUSIC in scenarios with
partially correlated or fully coherent sources. Additionally, our
simulations and the results of sonar and ultrasonic real data
processing demonstrate an improved threshold performance of
Unitary root-MUSIC relative to conventional root-MUSIC. It can
be then recommended that, as a rule, the Unitary root-MUSIC
technique should be preferred by the user to the conventional
root-MUSIC algorithm.
Index Terms—Array processing, direction-of-arrival estimation,
Unitary root-MUSIC.
I. INTRODUCTION
HE PROBLEM of reducing the computational complexity
of eigenstructure methods via real-valued formulations
has recently drawn a considerable attention in the literature
[1]–[7]. In [1], Huarng and Yeh presented a real-valued
variant of spectral MUSIC based on a unitary transformation.
In [2], Zoltowski et al. developed real-valued beamspace
root-MUSIC. Linebarger et al. [3] proposed efficient algorithms for real-valued SVD updating. Based on a unitary
transformation [1], Haardt and Nossek developed a unitary
ESPRIT algorithm [4]—an efficient variant of ESPRIT [8]
exploiting the centro-Hermitian property of the forward–backward covariance matrix. Later, Haardt formulated efficient
multidimensional extensions of the unitary ESPRIT technique
[5]. In [6], Gershman and Haardt employed Unitary ESPRIT
with an eigenvalue-based reliability test to develop a new
algorithm based on the estimator bank scheme [9]. In [7],
Gershman and Stoica developed a unitary variant of MODE.
T
Manuscript received January 15, 1999; revised October 19, 1999. This work
was supported in part by the Natural Sciences and Engineering Research
Council (NSERC) of Canada. The associate editor coordinating the review of
this paper and approving it for publication was Dr. Brian M. Sadler.
M. Pesavento and A. B. Gershman are with the Communications Research
Laboratory, Department of Electrical and Computer Engineering, McMaster
University, Hamilton, Ont., Canada L8S 4K1 (e-mail: [email protected]).
M. Haardt is with Siemens AG, Munich, Germany.
Publisher Item Identifier S 1053-587X(00)03298-0.
In this paper, a unitary formulation of the root-MUSIC
algorithm [10] is considered. First, the exact equivalence of
this algorithm to the forward–backward (FB) root-MUSIC is
proven. We stress, however, that the implementations of these
two techniques are different. In particular, Unitary root-MUSIC
has a simpler implementation than FB root-MUSIC because
the former exploits the eigendecomposition of a real-valued
matrix. Then, the asymptotic performance of the Unitary
root-MUSIC algorithm is studied. We show that in situations
with uncorrelated signal sources, the asymptotic performances
of the Unitary and conventional root-MUSIC algorithms are
identical. However, in correlated (or coherent) source scenarios, the asymptotic performance of Unitary root-MUSIC is
shown to be better than that of conventional root-MUSIC. This
performance improvement is due to the forward–backward
averaging effect. In the final part of our paper, simulation and
experimental results are presented. In particular, the conventional and Unitary root-MUSIC techniques are tested using real
sonar and ultrasonic sensor array data.
Sometimes, forward–backward averaging has been found
to cause a certain degradation of asymptotic performances
of direction finding techniques [14]. Interestingly, Unitary
root-MUSIC does not suffer from such a degradation. Additionally, our simulations demonstrate that Unitary root-MUSIC
outperforms conventional root-MUSIC [10] in the threshold
domain. Similar results on the threshold performance were observed in the real data processing experiments. As both Unitary
and conventional root-MUSIC algorithms are applicable to
the same array configuration [uniform linear arrays (ULA’s)],
we conclude that, as a rule, Unitary root-MUSIC should be
preferred by the user to conventional root-MUSIC.
II. ARRAY SIGNAL MODEL AND COMPLEX-VALUED
COVARIANCE MATRICES
Let a ULA be composed of
sensors, and let it re) narrowband sources impinging from the
ceive (
. Assume that there are
snapshots
directions
available. The
array observation
vector is modeled as [2], [13]
(1)
where
direction vectors
1053-587X/00$10.00 © 2000 IEEE
is the
matrix of the signal
(2)
PESAVENTO et al.: UNITARY ROOT-MUSIC WITH A REAL-VALUED EIGENDECOMPOSITION
is the
1307
where is any unitary, column conjugate symmetric
matrix. According to [3], any matrix
is column conjugate
symmetric if
steering vector. In addition
vector of source waveforms;
vector of white sensor noise;
wavelength;
interelement spacing;
transpose.
(12)
For example, the following sparse matrices [1], [3], [4]
The conventional (forward-only) estimate of the covariance matrix [2], [11]
(3)
(13)
is given by [11], [12]
(4)
where
can be chosen for arrays with an even and odd number of sen.
sors, respectively, where the vector
Interestingly, the matrix (11) can be also obtained via the forward-only covariance matrix
source waveform covariance matrix;
identity matrix;
noise variance;
Hermitian transpose.
The matrix
Re
(14)
is centro-Hermitian if
IV. UNITARY ROOT-MUSIC
(5)
where is the exchange matrix with ones on its antidiagonal
and zeros elsewhere, and
stands for complex conjugate. The
matrix (3) is known to be centro-Hermitian if and only if is a
diagonal matrix, i.e., when the signal sources are uncorrelated
[4]. However, to “double” the number of snapshots and decorrelate possibly correlated source pairs in the case of an arbitrary
matrix , the centro-Hermitian property is sometimes forced by
means of the so-called FB averaging [1], [3], [4]
Let the eigendecompositions of the matrices (3), (7), and (10)
be defined in a standard way [3]
(15)
(16)
(17)
where
(6)
The FB matrix (6) is often used in various direction-finding
methods [1], [3]–[7], [14], [17], [18] instead of the conventional
(forward-only) sample covariance matrix (4). It can be readily
shown that
diag
diag
(7)
where
diag
(8)
(9)
and the subscripts and stand for signal- and noise-subspace,
respectively. In turn, the eigendecompositions of the sample covariance matrices (4), (6), and (11) can be defined as
We introduce the real-valued covariance matrix as [1], [3], [4]
(18)
(19)
(20)
diag
III. REAL-VALUED COVARIANCE MATRIX
(10)
and the real-valued sample covariance matrix as
(11)
where
1308
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 5, MAY 2000
where
diag
diag
(35)
diag
diag
diag
and
diag
Writing the characteristic equation for the matrix (6) as
, we obtain that
(21)
Equation (21) can be identified as the characteristic one for the
real-valued covariance matrix (11). Hence, the eigenvectors and
eigenvalues of the matrices (6) and (11) are related as
(22)
(23)
The conventional root-MUSIC polynomial is given by
(24)
(25)
where
(26)
, and
. Similarly to (24) and (25),
the FB root-MUSIC polynomial can be used:
(27)
-
(28)
A simple manipulation with the use of (22) yields
V. PERFORMANCE ANALYSIS
-
-
Let us term the polynomial (31) as the Unitary root-MUSIC
polynomial since it exploits the eigendecomposition of the realvalued matrix (11) instead of that of the complex matrices (4)
or (6). From (29)–(31), it is clear that the FB and Unitary rootMUSIC polynomials are identical. Hence, the performance of
Unitary root-MUSIC does not depend on a particular choice
of the unitary column conjugate symmetric matrix , although
(13) seems to be a very good choice because the matrix (11)
can be obtained from the matrix (6) with a very low computational cost. However, although both polynomials
and are proven to be identical, the computation
is a simpler operation [if
of the coefficients of implemented via the real-valued eigendecomposition (20)] because the real-valued eigendecomposition has a reduced computational complexity than the complex one, approximately by
a factor of four [2]. Since the polynomial rooting is a much simpler operation than the eigendecomposition, we can conclude
that the overall computational cost of Unitary root-MUSIC is
about four times lower than that of conventional root-MUSIC
as in (24) and (25). We also stress that this conclusion cannot be
extended to the unitary spectral MUSIC technique [1] because
the main computational cost of spectral MUSIC is due to the
exhaustive spectral search rather than the eigendecomposition.
We end this section with a few remarks on a substantial
difference between the Unitary root-MUSIC algorithm (31)
and the popular beamspace root-MUSIC technique presented
by Zoltowski et al. [2]. The latter algorithm is entirely based
on the centered array assumption (which is not easy to satisfy
in certain practical applications), whereas the unitary algorithm
(31) does not require such an assumption. Furthermore, to
come up with the real-valued computations, the algorithm [2]
ignores the complex part of the beamspace covariance matrix,
thereby ignoring some useful information contained in its
complex part. Finally, since the Unitary root-MUSIC algorithm
is equivalent in the performance to FB root-MUSIC, it can be
applied to scenarios with pairwise coherent sources, whereas
beamspace root-MUSIC is inapplicable to such scenarios.
(29)
(30)
(31)
where the manifold
(32)
should be exploited for the polynomial rooting in (30). The relationship between the former and new manifolds follows from
the expression for the real-valued true covariance matrix (10)
(33)
(34)
The asymptotic performance of conventional spectral
MUSIC has been studied by many authors, e.g., Kaveh and
Barabell [15], Stoica and Nehorai [16], and others. The performance of conventional root-MUSIC has been studied by Rao
and Hari [13]. FB spectral MUSIC has been analyzed by Pillai
and Kwon [17] and Rao and Hari [18]. To our knowledge, in
the existing literature, there is no performance analysis of either
Unitary or FB root-MUSIC. It should be noted that the FB
covariance matrix (6) has different statistical properties than
the forward-only matrix (4) and, therefore, in the general case,
the performance of FB direction-finding algorithms may differ
severely from that of their forward-only prototypes [14].
In this section, we derive closed-form expressions for
the large sample mean-square estimation error of Unitary
root-MUSIC.
PESAVENTO et al.: UNITARY ROOT-MUSIC WITH A REAL-VALUED EIGENDECOMPOSITION
1309
Using the results of Lemma 1, after manipulations outlined
in Appendix A, we obtain that
Let us introduce the “eigenvector error” vector
(36)
Then, the following lemma holds [5], [19].
Lemma 1: The signal-subspace eigenvector estimation errors (36) are asymptotically (for a large number of snapshots
) jointly Gaussian distributed with zero means and covariance
matrices given by
Re
(45)
In Appendix B, it is proven that
Cov
(46)
Therefore, we can rewrite (40) as
(37)
where
(47)
(38)
Im
Using (22) and (23), we obtain the final expression for the DOA
estimation MSE
(39)
, and
denotes the Kronecker delta.
Applying directly the results derived in [13], we obtain that
the DOA estimation mean square error (MSE) of Unitary rootMUSIC is given by
(40)
where
(41)
and
Re
Re
(42)
where
(48)
. Comparing (48) with the results
where
derived by Rao and Hari [13, Eqs. (26) and (28)], we obtain
that the only difference between the MSE’s of the conventional
and Unitary root-MUSIC algorithms is that our expression (48)
for the Unitary root-MUSIC MSE contains the eigenvectors
and eigenvalues of the FB covariance matrix (7), whereas the
aforementioned expressions in [13] contain the eigenvectors
and eigenvalues of the conventional covariance matrix (3).
From (7)–(9), it follows that in uncorrelated source scenarios
, and therefore, we conclude that the asymptotic
performances of the Unitary and conventional root-MUSIC
algorithms are identical for uncorrelated sources. However,
in situations with partially correlated or coherent sources, the
asymptotic performance of Unitary root-MUSIC is better than
that of conventional root-MUSIC due to the forward–backward
averaging effect. Although this fact is not proven analytically
for the general case of arbitrary correlation, it is clearly demonstrated in the next section by means of comparison of analytical
and simulation results.
VI. SIMULATIONS
(43)
(44)
In our simulations, we assumed a ULA of ten omnidirectional
sensors with the half-wavelength interelement spacing and two
equally powered narrowband sources with the DOA’s
and
relative to the broadside. A total of 1000 independent simulation runs have been performed to obtain each simulated point. In all our examples, the experimental DOA esti-
1310
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 5, MAY 2000
mation root-mean-square errors (RMSE’s) have been compared
with the stochastic Cramér–Rao bound (CRB) [20] and with the
results of our theoretical (asymptotic) analysis.
In the first example, we assumed two uncorrelated sources
snapshots. Fig. 1 displays the results for conand
ventional and Unitary root-MUSIC versus the SNR. Note that
in this figure, the theoretical performance and CRB curves are
nearly identical (i.e., they merge into one curve).
In the second example, we assumed two uncorrelated sources
dB. Fig. 2 shows the results for the
with the fixed SNR
conventional and Unitary root-MUSIC algorithms versus the
number of snapshots . Again (as in Fig. 1), the theoretical and
CRB curves merge here into one curve.
has been taken, and two corIn the third example,
related sources have been assumed with the absolute value of
the correlation coefficient equal to 0.99 and zero intersource
phase in the first sensor. The results for conventional and Unitary root-MUSIC versus the SNR are plotted in Fig. 3.
In the fourth example, we assumed two correlated sources
dB has
with the same parameters again, and the fixed SNR
been taken. Fig. 4 displays the results for the conventional and
Unitary root-MUSIC techniques versus the number of snapshots
.
In the last example, we assumed the fixed SNR
dB and
and considered two correlated sources. Fig. 5 shows
the results for conventional and Unitary root-MUSIC versus the
absolute value of the correlation coefficient.
From Figs. 1–5, we observe that there is an excellent correspondence of the theoretical curves and experimental points at a
high SNR and a large . In the first two examples with the uncorrelated sources (Figs. 1 and 2), the asymptotic performances
of both algorithms are identical. However, from the last three examples with the correlated sources (Figs. 3–5), we see that the
Unitary root-MUSIC algorithm has a superior asymptotic performance. In particular, Fig. 5 demonstrates the difference in the
performances of the Unitary and conventional root-MUSIC algorithms in the situation where the sources become more and
more correlated. Moreover, from our figures, it clearly follows
that both for correlated and uncorrelated source scenarios, Unitary root-MUSIC has a lower SNR threshold and better finitesample performance than conventional root-MUSIC.
VII. REAL DATA EXAMPLES
To compare the conventional and Unitary root-MUSIC algorithms further, experimental sonar and ultrasonic ULA data have
been used. In this section, we describe the data sets and present
the results of real data processing.
A. Sonar Data
The experimental site was located in the Bornholm Deep,
which is to the east from Bornholm island in the Baltic Sea.
The experiments were conducted by STN Atlas Elektronik,
Bremen, Germany, in October 1983. A horizontal ULA of
15 hydrophones was towed by a surface ship. The strongest
moving broadband source was given by another (cooperating)
ship. The sources originating from at least two other (noncooperating) moving ships have been also detected [21]. The
Fig. 1. Comparison of the performances of the conventional and Unitary
root-MUSIC algorithms versus the SNR in the first example. Note that the
theoretical and CRB curves merge.
Fig. 2. Comparison of the performances of the conventional and Unitary
root-MUSIC algorithms versus
in the second example. Note that the
theoretical and CRB curves merge.
N
sea depth at the experimental site was about 60–65 m. The
towed receiving array had the interelement spacing
m, and the sampling frequency was
Hz after
lowpass filtering with a cut-off of 256 Hz. The narrowband
snapshots with the 4-Hz bandwidth have been formed from this
Hz.
broadband data after a DFT at the frequency bin
The sequence of covariance matrices has been estimated using
nonoverlapping sliding windows with 4 s duration each.
A record of 10-min duration (i.e., with 150 sliding windows)
has been chosen. More details on this experiment and experimental data parameters may be found in [21]. In particular, certain array calibration errors may be expected (which occur because of underwater perturbations of the towed receiving array).
Fig. 6(a) and (b) shows the results of real sonar data processing using conventional and Unitary root-MUSIC, respectively. From this figure, we observe that both algorithms have
PESAVENTO et al.: UNITARY ROOT-MUSIC WITH A REAL-VALUED EIGENDECOMPOSITION
Fig. 3. Comparison of the performances of the conventional and Unitary
root-MUSIC algorithms versus the SNR in the third example.
1311
Fig. 5. Comparison of the performances of the conventional and Unitary
root-MUSIC algorithms versus the absolute value of correlation coefficient in
the fifth example.
Fig. 4. Comparison of the performances of the conventional and Unitary
in the fourth example.
root-MUSIC algorithms versus
N
nearly identical performance. In particular, both of them are able
to resolve closely spaced sources. Although some performance
degradation of the unitary algorithm relative to the conventional
one may be expected in the presence of array calibration errors,
our experimental results demonstrate to us that this degradation
is negligible.
B. Ultrasonic Data
The employed ultrasonic data were recorded at University
of Wyoming Source Tracking Array Testbed (UW STAT) [22].
These narrowband six-element array data are available on the
World Wide Web at http://www.eng.uwyo.edu/electrical/array.html and have been used as benchmark
data in [7], [22], and [23]. They have the carrier frequency of
40 kHz and the signal bandwidth of 200 Hz. The receiving
has been used.
ULA with the interelement spacing
Fig. 6. Comparison of the conventional and Unitary root-MUSIC algorithms
using real sonar data.
The data set no. 2 with one stationary source and one moving
(constant velocity) source has been used. A rectangular (maxsnapshots has been
imal-overlap) sliding window with
employed to estimate the source trajectories. The results for conventional and Unitary root-MUSIC are displayed in Fig. 7(a)
and (b), respectively. The true source trajectories are shown as
well. From these plots, we see that both algorithms have serious
problems when the sources become closely spaced. However, it
can be observed that Unitary root-MUSIC has better threshold
performance than conventional root-MUSIC. In particular, the
conventional root-MUSIC algorithm has the breakdown problems approximately between 570–735 snapshots, whereas for
the Unitary root-MUSIC technique this “breakdown interval”
is concentrated approximately between 570–705 snapshots. In
other words, for Unitary root-MUSIC, this interval is narrower
1312
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 5, MAY 2000
(50)
(51)
Inserting (37) into (50) and exploiting the property
we obtain that
,
Fig. 7. Comparison of the conventional and Unitary root-MUSIC algorithms
using real ultrasonic data.
than for the conventional one. This fact demonstrates the superior threshold performance of Unitary root-MUSIC.
(52)
VIII. CONCLUSION
The real-valued (unitary) formulation of the root-MUSIC algorithm has been addressed. Unitary and FB root-MUSIC polynomials have been shown to be exactly identical, but the Unitary
root-MUSIC technique has been proven to enjoy simpler implementation than the conventional and FB root-MUSIC techniques. Closed-form expressions for the large sample meansquare estimation errors of Unitary root-MUSIC have been derived. These expressions have been compared with the wellknown results for conventional root-MUSIC, and it has been
shown that Unitary root-MUSIC performs asymptotically similarly to or better than conventional root-MUSIC. Additionally,
our simulations and real sonar and ultrasonic data examples
have verified the better threshold performance of Unitary rootMUSIC. As both the Unitary and conventional root-MUSIC algorithms are applicable to the same array configuration (ULA),
it can be concluded that, as a rule, Unitary root-MUSIC should
be preferred by the user to conventional root-MUSIC.
APPENDIX A
PROOF OF (45)
For the sake of notational simplicity, denote
. Then
Re
where
It is straightforward to show that the function (38) has the following invariance properties:
(53)
Taking into account (53), rearranging the elements in (52), and
exchanging the indexes in this equation, we obtain that
(54)
and therefore, (52) can be simplified to
(55)
After similar manipulations, (51) can be simplified to
and
(49)
(56)
Inserting (55) and (56) into (49), we prove (45).
APPENDIX B
PROOF OF (46)
From (26), (32), and (41), it follows that
(57)
PESAVENTO et al.: UNITARY ROOT-MUSIC WITH A REAL-VALUED EIGENDECOMPOSITION
where
diag
(58)
From (26), we have
(59)
In addition, according to (12) and (22)
(60)
Using (12), (22), (32), (57), (59), and (60), and denoting
and
, we can transform the left side of (46) in
the following manner:
(61)
. Now, to complete the proof of (46), it remains
where
to be shown that
(62)
To prove (62), let us use (12) and (57) and the following identities:
(63)
(64)
Then, we have
1313
[3] D. A. Linebarger, R. D. DeGroat, and E. M. Dowling, “Efficient direction-finding methods employing forward–backward averaging,” IEEE
Trans. Signal Processing, vol. 42, pp. 2136–2145, Aug. 1994.
[4] M. Haardt and J. A. Nossek, “Unitary ESPRIT: How to obtain increased
estimation accuracy with a reduced computational burden,” IEEE Trans.
Signal Processing, vol. 43, pp. 1232–1242, May 1995.
[5] M. Haardt, “Efficient one-, two-, and multidimensional high-resolution
array signal processing,” Ph.D. dissertation, Tech. Univ. Munich, Munich, Germany, 1997.
[6] A. B. Gershman and M. Haardt, “Improving the performance of Unitary
ESPRIT via pseudo-noise resampling,” IEEE Trans. Signal Processing,
vol. 47, pp. 2305–2308, Aug. 1999.
[7] A. B. Gershman and P. Stoica, “On unitary and forward–backward
MODE,” Digital Signal Process., vol. 9, no. 2, pp. 67–75, Apr. 1999.
[8] R. Roy and T. Kailath, “ESPRIT—Estimation of signal parameters via
rotational invariance techniques,” IEEE Trans. Acoust., Speech, Signal
Processing, vol. 37, pp. 984–995, July 1989.
[9] A. B. Gershman, “Pseudo-randomly generated estimator banks: A new
tool for improving the threshold performance of direction finding,” IEEE
Trans. Signal Processing, vol. 46, pp. 1351–1364, May 1998.
[10] A. J. Barabell, “Improving the resolution performance of eigenstructure-based direction-finding algorithms,” in Proc. ICASSP, Boston, MA,
May 1983, pp. 336–339.
[11] P. Stoica and K. C. Sharman, “Novel eigenanalysis method for direction
estimation,” Proc. Inst. Elect. Eng. F, vol. 137, pp. 19–26, Feb. 1990.
, “Maximum likelihood methods for direction-of-arrival estima[12]
tion,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 38, pp.
1132–1143, July 1990.
[13] B. D. Rao and K. V. S. Hari, “Performance analysis of root-MUSIC,”
IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, pp. 1939–1949,
Dec. 1989.
[14] P. Stoica and M. Jansson, “On forward–backward MODE for array
signal processing,” Digital Signal Process., vol. 7, no. 4, pp. 239–252,
Oct. 1997.
[15] M. Kaveh and A. Barabell, “The statistical performance of MUSIC and
the minimum-norm algorithms in resolving plane waves in noise,” IEEE
Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 331–340,
Apr. 1986.
[16] P. Stoica and A. Nehorai, “MUSIC, maximum likelihood and
Cramér–Rao bound,” IEEE Trans. Acoust., Speech, Signal Processing,
vol. 37, pp. 720–741, May 1989.
[17] S. U. Pillai and B. H. Kwon, “Performance analysis of MUSIC-type high
resolution estimators for direction finding in correlated and coherent
scenes,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, pp.
1176–1189, Aug. 1989.
[18] B. D. Rao and K. V. S. Hari, “Weighted subspace methods and spatial smoothing: Analysis and comparison,” IEEE Trans. Acoust., Speech,
Signal Processing, vol. 41, pp. 788–803, Feb. 1993.
[19] G. M. Kautz and M. Zoltowski, “Performance analysis of MUSIC employing conjugate symmetric beamformers,” IEEE Trans. Signal Processing, vol. 43, pp. 737–748, Mar. 1995.
[20] P. Stoica and A. Nehorai, “Performance study of conditional and unconditional direction-of-arrival estimation,” IEEE Trans. Acoust., Speech,
Signal Processing, vol. 38, pp. 1783–1795, Oct. 1990.
[21] C. F. Mecklenbräuker, A. B. Gershman, and J. F. Böhme, “Broadband
ML-approach to environmental parameter estimation in shallow ocean
at low SNR,” Signal Process., to be published.
[22] J. W. Pierre, E. D. Scott, and M. P. Hays, “A sensor array testbed for
source tracking algorithms,” in Proc. ICASSP, Munich, Germany, 1997,
pp. 3769–3772.
[23] A. B. Gershman and J. F. Böhme, “A pseudo-noise approach to direction
finding,” Signal Process., vol. 71, pp. 1–13, Nov. 1998.
(65)
With (65), the proof of (46) is completed.
REFERENCES
[1] K. C. Huarng and C. C. Yeh, “A unitary transformation method for angle
of arrival estimation,” IEEE Trans. Acoust., Speech, Signal Processing,
vol. 39, pp. 975–977, Apr. 1991.
[2] M. D. Zoltowski, G. M. Kautz, and S. D. Silverstein, “Beamspace rootMUSIC,” IEEE Trans. Signal Processing, vol. 41, pp. 344–364, Jan.
1993.
Marius Pesavento was born in Werl, Germany,
in 1973. He received the M.S. (Diploma) degree
in electrical engineering from Ruhr University,
Bochum, Germany, in 1999.
He is currently with the Department of Electrical
and Computer Engineering, McMaster University,
Hamilton, Ont., Canada. His research interests
include statistical signal and array processing and
adaptive beamforming.
1314
Alex B. Gershman (M’97–SM’98) was born in
Nizhny Novgorod, Russia, in 1962. He received the
M.S. (Diploma) and Ph.D. degrees in radiophysics
from the Nizhny Novgorod State University, in 1984
and 1990, respectively.
From 1984 to 1989, he was with the Radiotechnical and Radiophysical Institutes, Nizhny
Novgorod. From 1989 to 1997, he was with the
Institute of Applied Physics of Russian Academy
of Science, Nizhny Novgorod, as a Senior Research
Scientist. From the summer of 1994 until the beginning of 1995, he held a Visiting Research Fellow position at the Swiss Federal
Institute of Technology, Lausanne. From 1995 to 1997, he held an Alexander
von Humboldt Fellow position at the Ruhr University, Bochum, Germany.
From 1997 to 1999, he was a Member of Staff at the Department of Electrical
Engineering at Ruhr University. Since 1999, he has been with the Department
of Electrical and Computer Engineering, McMaster University, Hamilton, Ont.,
Canada, as an Associate Professor. His research interests are in the area of
signal processing and include statistical signal and array processing, adaptive
beamforming and smart antennas, signal parameter estimation and detection,
spectral analysis, high-order statistics, and signal processing applications to
underwater acoustics, wireless communications, seismology, and radar. He has
published more than 100 journal and conference papers on these topics.
Dr. Gershman was a recipient of the 1993 International Union of Radio
Science (URSI) Young Scientist Award, the 1994 Outstanding Young Scientist
Presidential Fellowship (Russia), the 1994 Swiss Academy of Engineering
Science and Branco Weiss Fellowships (Switzerland), and the 1995–1996
Alexander von Humboldt Fellowship (Germany). He is listed in several Who’s
Who in the World and Who’s Who in Science and Engineering editions. He
is an Associate Editor of IEEE TRANSACTIONS ON SIGNAL PROCESSING and
a Member of the Sensor Array and Multichannel Signal Processing Technical
Committee of the IEEE Signal Processing Society.
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 5, MAY 2000
Martin Haardt (S’90–M’98–SM’99) was born
in Germany on October 4, 1967. He received the
Dipl.-Ing. (M.S.) degree from the Ruhr University,
Bochum, Germany, in 1991 and the Dr.-Ing. (Ph.D.)
degree from the Munich University of Technology,
Munich, Germany, in 1996, both in electrical
engineering.
From 1989 to 1990, he was a Visiting Scholar at
Purdue University, West Lafayette, IN, sponsored
by a fellowship of the German Academic Exchange
Service (DAAD). From 1991 to 1993, he worked
for Corporate Research and Development, Siemens AG, Munich, Germany,
conducting research in the areas of image and biomedical signal processing.
From 1993 to 1996, he was a Research Associate with the Institute of Network
Theory and Circuit Design at the Munich University of Technology, where
his research was focused on smart antennas. In 1997, he joined Siemens
Mobile Networks, Munich, where he was responsible for strategic research
for third-generation mobile radio systems. Since 1998, he has been serving as
Director of International and University Projects in the Chief Technical Office
of Siemens Communication on Air. He also teaches at the Munich University
of Technology and is a contributing author to The Digital Signal Processing
Handbook, (Boca Raton, FL: CRC and Piscataway, NJ: IEEE, 1998). His
current research interests include wireless communications, array and digital
signal processing, and numerical linear algebra.
Dr. Haardt received the Rohde and Schwarz Outstanding Dissertation Award
for his Ph.D. dissertation in 1997. In 1998, he was the co-recipient of the Mannesmann Mobilfunk Innovation Award for outstanding research in mobile communications and received the Best Paper Award from the Information Technology Society (ITG) of the Association of Electrical Engineering, Electronics,
and Information Technology (VDE), Germany. He is a Member of VDE.
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 5, MAY 2000
Unitary Root-MUSIC with a Real-Valued
Eigendecomposition: A Theoretical and Experimental
Performance Study
Marius Pesavento, Alex B. Gershman, Senior Member, IEEE, and Martin Haardt, Senior Member, IEEE
Abstract—A real-valued (unitary) formulation of the popular
root-MUSIC direction-of-arrival (DOA) estimation technique is
considered. This Unitary root-MUSIC algorithm reduces the computational complexity in the eigenanalysis stage of root-MUSIC
because it exploits the eigendecomposition of a real-valued covariance matrix. The asymptotic performance of Unitary root-MUSIC
is analyzed and compared with that of conventional root-MUSIC.
The results of this comparison show identical asymptotic properties of both algorithms in the case of uncorrelated sources and
a better performance of Unitary root-MUSIC in scenarios with
partially correlated or fully coherent sources. Additionally, our
simulations and the results of sonar and ultrasonic real data
processing demonstrate an improved threshold performance of
Unitary root-MUSIC relative to conventional root-MUSIC. It can
be then recommended that, as a rule, the Unitary root-MUSIC
technique should be preferred by the user to the conventional
root-MUSIC algorithm.
Index Terms—Array processing, direction-of-arrival estimation,
Unitary root-MUSIC.
I. INTRODUCTION
HE PROBLEM of reducing the computational complexity
of eigenstructure methods via real-valued formulations
has recently drawn a considerable attention in the literature
[1]–[7]. In [1], Huarng and Yeh presented a real-valued
variant of spectral MUSIC based on a unitary transformation.
In [2], Zoltowski et al. developed real-valued beamspace
root-MUSIC. Linebarger et al. [3] proposed efficient algorithms for real-valued SVD updating. Based on a unitary
transformation [1], Haardt and Nossek developed a unitary
ESPRIT algorithm [4]—an efficient variant of ESPRIT [8]
exploiting the centro-Hermitian property of the forward–backward covariance matrix. Later, Haardt formulated efficient
multidimensional extensions of the unitary ESPRIT technique
[5]. In [6], Gershman and Haardt employed Unitary ESPRIT
with an eigenvalue-based reliability test to develop a new
algorithm based on the estimator bank scheme [9]. In [7],
Gershman and Stoica developed a unitary variant of MODE.
T
Manuscript received January 15, 1999; revised October 19, 1999. This work
was supported in part by the Natural Sciences and Engineering Research
Council (NSERC) of Canada. The associate editor coordinating the review of
this paper and approving it for publication was Dr. Brian M. Sadler.
M. Pesavento and A. B. Gershman are with the Communications Research
Laboratory, Department of Electrical and Computer Engineering, McMaster
University, Hamilton, Ont., Canada L8S 4K1 (e-mail: [email protected]).
M. Haardt is with Siemens AG, Munich, Germany.
Publisher Item Identifier S 1053-587X(00)03298-0.
In this paper, a unitary formulation of the root-MUSIC
algorithm [10] is considered. First, the exact equivalence of
this algorithm to the forward–backward (FB) root-MUSIC is
proven. We stress, however, that the implementations of these
two techniques are different. In particular, Unitary root-MUSIC
has a simpler implementation than FB root-MUSIC because
the former exploits the eigendecomposition of a real-valued
matrix. Then, the asymptotic performance of the Unitary
root-MUSIC algorithm is studied. We show that in situations
with uncorrelated signal sources, the asymptotic performances
of the Unitary and conventional root-MUSIC algorithms are
identical. However, in correlated (or coherent) source scenarios, the asymptotic performance of Unitary root-MUSIC is
shown to be better than that of conventional root-MUSIC. This
performance improvement is due to the forward–backward
averaging effect. In the final part of our paper, simulation and
experimental results are presented. In particular, the conventional and Unitary root-MUSIC techniques are tested using real
sonar and ultrasonic sensor array data.
Sometimes, forward–backward averaging has been found
to cause a certain degradation of asymptotic performances
of direction finding techniques [14]. Interestingly, Unitary
root-MUSIC does not suffer from such a degradation. Additionally, our simulations demonstrate that Unitary root-MUSIC
outperforms conventional root-MUSIC [10] in the threshold
domain. Similar results on the threshold performance were observed in the real data processing experiments. As both Unitary
and conventional root-MUSIC algorithms are applicable to
the same array configuration [uniform linear arrays (ULA’s)],
we conclude that, as a rule, Unitary root-MUSIC should be
preferred by the user to conventional root-MUSIC.
II. ARRAY SIGNAL MODEL AND COMPLEX-VALUED
COVARIANCE MATRICES
Let a ULA be composed of
sensors, and let it re) narrowband sources impinging from the
ceive (
. Assume that there are
snapshots
directions
available. The
array observation
vector is modeled as [2], [13]
(1)
where
direction vectors
1053-587X/00$10.00 © 2000 IEEE
is the
matrix of the signal
(2)
PESAVENTO et al.: UNITARY ROOT-MUSIC WITH A REAL-VALUED EIGENDECOMPOSITION
is the
1307
where is any unitary, column conjugate symmetric
matrix. According to [3], any matrix
is column conjugate
symmetric if
steering vector. In addition
vector of source waveforms;
vector of white sensor noise;
wavelength;
interelement spacing;
transpose.
(12)
For example, the following sparse matrices [1], [3], [4]
The conventional (forward-only) estimate of the covariance matrix [2], [11]
(3)
(13)
is given by [11], [12]
(4)
where
can be chosen for arrays with an even and odd number of sen.
sors, respectively, where the vector
Interestingly, the matrix (11) can be also obtained via the forward-only covariance matrix
source waveform covariance matrix;
identity matrix;
noise variance;
Hermitian transpose.
The matrix
Re
(14)
is centro-Hermitian if
IV. UNITARY ROOT-MUSIC
(5)
where is the exchange matrix with ones on its antidiagonal
and zeros elsewhere, and
stands for complex conjugate. The
matrix (3) is known to be centro-Hermitian if and only if is a
diagonal matrix, i.e., when the signal sources are uncorrelated
[4]. However, to “double” the number of snapshots and decorrelate possibly correlated source pairs in the case of an arbitrary
matrix , the centro-Hermitian property is sometimes forced by
means of the so-called FB averaging [1], [3], [4]
Let the eigendecompositions of the matrices (3), (7), and (10)
be defined in a standard way [3]
(15)
(16)
(17)
where
(6)
The FB matrix (6) is often used in various direction-finding
methods [1], [3]–[7], [14], [17], [18] instead of the conventional
(forward-only) sample covariance matrix (4). It can be readily
shown that
diag
diag
(7)
where
diag
(8)
(9)
and the subscripts and stand for signal- and noise-subspace,
respectively. In turn, the eigendecompositions of the sample covariance matrices (4), (6), and (11) can be defined as
We introduce the real-valued covariance matrix as [1], [3], [4]
(18)
(19)
(20)
diag
III. REAL-VALUED COVARIANCE MATRIX
(10)
and the real-valued sample covariance matrix as
(11)
where
1308
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 5, MAY 2000
where
diag
diag
(35)
diag
diag
diag
and
diag
Writing the characteristic equation for the matrix (6) as
, we obtain that
(21)
Equation (21) can be identified as the characteristic one for the
real-valued covariance matrix (11). Hence, the eigenvectors and
eigenvalues of the matrices (6) and (11) are related as
(22)
(23)
The conventional root-MUSIC polynomial is given by
(24)
(25)
where
(26)
, and
. Similarly to (24) and (25),
the FB root-MUSIC polynomial can be used:
(27)
-
(28)
A simple manipulation with the use of (22) yields
V. PERFORMANCE ANALYSIS
-
-
Let us term the polynomial (31) as the Unitary root-MUSIC
polynomial since it exploits the eigendecomposition of the realvalued matrix (11) instead of that of the complex matrices (4)
or (6). From (29)–(31), it is clear that the FB and Unitary rootMUSIC polynomials are identical. Hence, the performance of
Unitary root-MUSIC does not depend on a particular choice
of the unitary column conjugate symmetric matrix , although
(13) seems to be a very good choice because the matrix (11)
can be obtained from the matrix (6) with a very low computational cost. However, although both polynomials
and are proven to be identical, the computation
is a simpler operation [if
of the coefficients of implemented via the real-valued eigendecomposition (20)] because the real-valued eigendecomposition has a reduced computational complexity than the complex one, approximately by
a factor of four [2]. Since the polynomial rooting is a much simpler operation than the eigendecomposition, we can conclude
that the overall computational cost of Unitary root-MUSIC is
about four times lower than that of conventional root-MUSIC
as in (24) and (25). We also stress that this conclusion cannot be
extended to the unitary spectral MUSIC technique [1] because
the main computational cost of spectral MUSIC is due to the
exhaustive spectral search rather than the eigendecomposition.
We end this section with a few remarks on a substantial
difference between the Unitary root-MUSIC algorithm (31)
and the popular beamspace root-MUSIC technique presented
by Zoltowski et al. [2]. The latter algorithm is entirely based
on the centered array assumption (which is not easy to satisfy
in certain practical applications), whereas the unitary algorithm
(31) does not require such an assumption. Furthermore, to
come up with the real-valued computations, the algorithm [2]
ignores the complex part of the beamspace covariance matrix,
thereby ignoring some useful information contained in its
complex part. Finally, since the Unitary root-MUSIC algorithm
is equivalent in the performance to FB root-MUSIC, it can be
applied to scenarios with pairwise coherent sources, whereas
beamspace root-MUSIC is inapplicable to such scenarios.
(29)
(30)
(31)
where the manifold
(32)
should be exploited for the polynomial rooting in (30). The relationship between the former and new manifolds follows from
the expression for the real-valued true covariance matrix (10)
(33)
(34)
The asymptotic performance of conventional spectral
MUSIC has been studied by many authors, e.g., Kaveh and
Barabell [15], Stoica and Nehorai [16], and others. The performance of conventional root-MUSIC has been studied by Rao
and Hari [13]. FB spectral MUSIC has been analyzed by Pillai
and Kwon [17] and Rao and Hari [18]. To our knowledge, in
the existing literature, there is no performance analysis of either
Unitary or FB root-MUSIC. It should be noted that the FB
covariance matrix (6) has different statistical properties than
the forward-only matrix (4) and, therefore, in the general case,
the performance of FB direction-finding algorithms may differ
severely from that of their forward-only prototypes [14].
In this section, we derive closed-form expressions for
the large sample mean-square estimation error of Unitary
root-MUSIC.
PESAVENTO et al.: UNITARY ROOT-MUSIC WITH A REAL-VALUED EIGENDECOMPOSITION
1309
Using the results of Lemma 1, after manipulations outlined
in Appendix A, we obtain that
Let us introduce the “eigenvector error” vector
(36)
Then, the following lemma holds [5], [19].
Lemma 1: The signal-subspace eigenvector estimation errors (36) are asymptotically (for a large number of snapshots
) jointly Gaussian distributed with zero means and covariance
matrices given by
Re
(45)
In Appendix B, it is proven that
Cov
(46)
Therefore, we can rewrite (40) as
(37)
where
(47)
(38)
Im
Using (22) and (23), we obtain the final expression for the DOA
estimation MSE
(39)
, and
denotes the Kronecker delta.
Applying directly the results derived in [13], we obtain that
the DOA estimation mean square error (MSE) of Unitary rootMUSIC is given by
(40)
where
(41)
and
Re
Re
(42)
where
(48)
. Comparing (48) with the results
where
derived by Rao and Hari [13, Eqs. (26) and (28)], we obtain
that the only difference between the MSE’s of the conventional
and Unitary root-MUSIC algorithms is that our expression (48)
for the Unitary root-MUSIC MSE contains the eigenvectors
and eigenvalues of the FB covariance matrix (7), whereas the
aforementioned expressions in [13] contain the eigenvectors
and eigenvalues of the conventional covariance matrix (3).
From (7)–(9), it follows that in uncorrelated source scenarios
, and therefore, we conclude that the asymptotic
performances of the Unitary and conventional root-MUSIC
algorithms are identical for uncorrelated sources. However,
in situations with partially correlated or coherent sources, the
asymptotic performance of Unitary root-MUSIC is better than
that of conventional root-MUSIC due to the forward–backward
averaging effect. Although this fact is not proven analytically
for the general case of arbitrary correlation, it is clearly demonstrated in the next section by means of comparison of analytical
and simulation results.
VI. SIMULATIONS
(43)
(44)
In our simulations, we assumed a ULA of ten omnidirectional
sensors with the half-wavelength interelement spacing and two
equally powered narrowband sources with the DOA’s
and
relative to the broadside. A total of 1000 independent simulation runs have been performed to obtain each simulated point. In all our examples, the experimental DOA esti-
1310
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 5, MAY 2000
mation root-mean-square errors (RMSE’s) have been compared
with the stochastic Cramér–Rao bound (CRB) [20] and with the
results of our theoretical (asymptotic) analysis.
In the first example, we assumed two uncorrelated sources
snapshots. Fig. 1 displays the results for conand
ventional and Unitary root-MUSIC versus the SNR. Note that
in this figure, the theoretical performance and CRB curves are
nearly identical (i.e., they merge into one curve).
In the second example, we assumed two uncorrelated sources
dB. Fig. 2 shows the results for the
with the fixed SNR
conventional and Unitary root-MUSIC algorithms versus the
number of snapshots . Again (as in Fig. 1), the theoretical and
CRB curves merge here into one curve.
has been taken, and two corIn the third example,
related sources have been assumed with the absolute value of
the correlation coefficient equal to 0.99 and zero intersource
phase in the first sensor. The results for conventional and Unitary root-MUSIC versus the SNR are plotted in Fig. 3.
In the fourth example, we assumed two correlated sources
dB has
with the same parameters again, and the fixed SNR
been taken. Fig. 4 displays the results for the conventional and
Unitary root-MUSIC techniques versus the number of snapshots
.
In the last example, we assumed the fixed SNR
dB and
and considered two correlated sources. Fig. 5 shows
the results for conventional and Unitary root-MUSIC versus the
absolute value of the correlation coefficient.
From Figs. 1–5, we observe that there is an excellent correspondence of the theoretical curves and experimental points at a
high SNR and a large . In the first two examples with the uncorrelated sources (Figs. 1 and 2), the asymptotic performances
of both algorithms are identical. However, from the last three examples with the correlated sources (Figs. 3–5), we see that the
Unitary root-MUSIC algorithm has a superior asymptotic performance. In particular, Fig. 5 demonstrates the difference in the
performances of the Unitary and conventional root-MUSIC algorithms in the situation where the sources become more and
more correlated. Moreover, from our figures, it clearly follows
that both for correlated and uncorrelated source scenarios, Unitary root-MUSIC has a lower SNR threshold and better finitesample performance than conventional root-MUSIC.
VII. REAL DATA EXAMPLES
To compare the conventional and Unitary root-MUSIC algorithms further, experimental sonar and ultrasonic ULA data have
been used. In this section, we describe the data sets and present
the results of real data processing.
A. Sonar Data
The experimental site was located in the Bornholm Deep,
which is to the east from Bornholm island in the Baltic Sea.
The experiments were conducted by STN Atlas Elektronik,
Bremen, Germany, in October 1983. A horizontal ULA of
15 hydrophones was towed by a surface ship. The strongest
moving broadband source was given by another (cooperating)
ship. The sources originating from at least two other (noncooperating) moving ships have been also detected [21]. The
Fig. 1. Comparison of the performances of the conventional and Unitary
root-MUSIC algorithms versus the SNR in the first example. Note that the
theoretical and CRB curves merge.
Fig. 2. Comparison of the performances of the conventional and Unitary
root-MUSIC algorithms versus
in the second example. Note that the
theoretical and CRB curves merge.
N
sea depth at the experimental site was about 60–65 m. The
towed receiving array had the interelement spacing
m, and the sampling frequency was
Hz after
lowpass filtering with a cut-off of 256 Hz. The narrowband
snapshots with the 4-Hz bandwidth have been formed from this
Hz.
broadband data after a DFT at the frequency bin
The sequence of covariance matrices has been estimated using
nonoverlapping sliding windows with 4 s duration each.
A record of 10-min duration (i.e., with 150 sliding windows)
has been chosen. More details on this experiment and experimental data parameters may be found in [21]. In particular, certain array calibration errors may be expected (which occur because of underwater perturbations of the towed receiving array).
Fig. 6(a) and (b) shows the results of real sonar data processing using conventional and Unitary root-MUSIC, respectively. From this figure, we observe that both algorithms have
PESAVENTO et al.: UNITARY ROOT-MUSIC WITH A REAL-VALUED EIGENDECOMPOSITION
Fig. 3. Comparison of the performances of the conventional and Unitary
root-MUSIC algorithms versus the SNR in the third example.
1311
Fig. 5. Comparison of the performances of the conventional and Unitary
root-MUSIC algorithms versus the absolute value of correlation coefficient in
the fifth example.
Fig. 4. Comparison of the performances of the conventional and Unitary
in the fourth example.
root-MUSIC algorithms versus
N
nearly identical performance. In particular, both of them are able
to resolve closely spaced sources. Although some performance
degradation of the unitary algorithm relative to the conventional
one may be expected in the presence of array calibration errors,
our experimental results demonstrate to us that this degradation
is negligible.
B. Ultrasonic Data
The employed ultrasonic data were recorded at University
of Wyoming Source Tracking Array Testbed (UW STAT) [22].
These narrowband six-element array data are available on the
World Wide Web at http://www.eng.uwyo.edu/electrical/array.html and have been used as benchmark
data in [7], [22], and [23]. They have the carrier frequency of
40 kHz and the signal bandwidth of 200 Hz. The receiving
has been used.
ULA with the interelement spacing
Fig. 6. Comparison of the conventional and Unitary root-MUSIC algorithms
using real sonar data.
The data set no. 2 with one stationary source and one moving
(constant velocity) source has been used. A rectangular (maxsnapshots has been
imal-overlap) sliding window with
employed to estimate the source trajectories. The results for conventional and Unitary root-MUSIC are displayed in Fig. 7(a)
and (b), respectively. The true source trajectories are shown as
well. From these plots, we see that both algorithms have serious
problems when the sources become closely spaced. However, it
can be observed that Unitary root-MUSIC has better threshold
performance than conventional root-MUSIC. In particular, the
conventional root-MUSIC algorithm has the breakdown problems approximately between 570–735 snapshots, whereas for
the Unitary root-MUSIC technique this “breakdown interval”
is concentrated approximately between 570–705 snapshots. In
other words, for Unitary root-MUSIC, this interval is narrower
1312
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 5, MAY 2000
(50)
(51)
Inserting (37) into (50) and exploiting the property
we obtain that
,
Fig. 7. Comparison of the conventional and Unitary root-MUSIC algorithms
using real ultrasonic data.
than for the conventional one. This fact demonstrates the superior threshold performance of Unitary root-MUSIC.
(52)
VIII. CONCLUSION
The real-valued (unitary) formulation of the root-MUSIC algorithm has been addressed. Unitary and FB root-MUSIC polynomials have been shown to be exactly identical, but the Unitary
root-MUSIC technique has been proven to enjoy simpler implementation than the conventional and FB root-MUSIC techniques. Closed-form expressions for the large sample meansquare estimation errors of Unitary root-MUSIC have been derived. These expressions have been compared with the wellknown results for conventional root-MUSIC, and it has been
shown that Unitary root-MUSIC performs asymptotically similarly to or better than conventional root-MUSIC. Additionally,
our simulations and real sonar and ultrasonic data examples
have verified the better threshold performance of Unitary rootMUSIC. As both the Unitary and conventional root-MUSIC algorithms are applicable to the same array configuration (ULA),
it can be concluded that, as a rule, Unitary root-MUSIC should
be preferred by the user to conventional root-MUSIC.
APPENDIX A
PROOF OF (45)
For the sake of notational simplicity, denote
. Then
Re
where
It is straightforward to show that the function (38) has the following invariance properties:
(53)
Taking into account (53), rearranging the elements in (52), and
exchanging the indexes in this equation, we obtain that
(54)
and therefore, (52) can be simplified to
(55)
After similar manipulations, (51) can be simplified to
and
(49)
(56)
Inserting (55) and (56) into (49), we prove (45).
APPENDIX B
PROOF OF (46)
From (26), (32), and (41), it follows that
(57)
PESAVENTO et al.: UNITARY ROOT-MUSIC WITH A REAL-VALUED EIGENDECOMPOSITION
where
diag
(58)
From (26), we have
(59)
In addition, according to (12) and (22)
(60)
Using (12), (22), (32), (57), (59), and (60), and denoting
and
, we can transform the left side of (46) in
the following manner:
(61)
. Now, to complete the proof of (46), it remains
where
to be shown that
(62)
To prove (62), let us use (12) and (57) and the following identities:
(63)
(64)
Then, we have
1313
[3] D. A. Linebarger, R. D. DeGroat, and E. M. Dowling, “Efficient direction-finding methods employing forward–backward averaging,” IEEE
Trans. Signal Processing, vol. 42, pp. 2136–2145, Aug. 1994.
[4] M. Haardt and J. A. Nossek, “Unitary ESPRIT: How to obtain increased
estimation accuracy with a reduced computational burden,” IEEE Trans.
Signal Processing, vol. 43, pp. 1232–1242, May 1995.
[5] M. Haardt, “Efficient one-, two-, and multidimensional high-resolution
array signal processing,” Ph.D. dissertation, Tech. Univ. Munich, Munich, Germany, 1997.
[6] A. B. Gershman and M. Haardt, “Improving the performance of Unitary
ESPRIT via pseudo-noise resampling,” IEEE Trans. Signal Processing,
vol. 47, pp. 2305–2308, Aug. 1999.
[7] A. B. Gershman and P. Stoica, “On unitary and forward–backward
MODE,” Digital Signal Process., vol. 9, no. 2, pp. 67–75, Apr. 1999.
[8] R. Roy and T. Kailath, “ESPRIT—Estimation of signal parameters via
rotational invariance techniques,” IEEE Trans. Acoust., Speech, Signal
Processing, vol. 37, pp. 984–995, July 1989.
[9] A. B. Gershman, “Pseudo-randomly generated estimator banks: A new
tool for improving the threshold performance of direction finding,” IEEE
Trans. Signal Processing, vol. 46, pp. 1351–1364, May 1998.
[10] A. J. Barabell, “Improving the resolution performance of eigenstructure-based direction-finding algorithms,” in Proc. ICASSP, Boston, MA,
May 1983, pp. 336–339.
[11] P. Stoica and K. C. Sharman, “Novel eigenanalysis method for direction
estimation,” Proc. Inst. Elect. Eng. F, vol. 137, pp. 19–26, Feb. 1990.
, “Maximum likelihood methods for direction-of-arrival estima[12]
tion,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 38, pp.
1132–1143, July 1990.
[13] B. D. Rao and K. V. S. Hari, “Performance analysis of root-MUSIC,”
IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, pp. 1939–1949,
Dec. 1989.
[14] P. Stoica and M. Jansson, “On forward–backward MODE for array
signal processing,” Digital Signal Process., vol. 7, no. 4, pp. 239–252,
Oct. 1997.
[15] M. Kaveh and A. Barabell, “The statistical performance of MUSIC and
the minimum-norm algorithms in resolving plane waves in noise,” IEEE
Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 331–340,
Apr. 1986.
[16] P. Stoica and A. Nehorai, “MUSIC, maximum likelihood and
Cramér–Rao bound,” IEEE Trans. Acoust., Speech, Signal Processing,
vol. 37, pp. 720–741, May 1989.
[17] S. U. Pillai and B. H. Kwon, “Performance analysis of MUSIC-type high
resolution estimators for direction finding in correlated and coherent
scenes,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, pp.
1176–1189, Aug. 1989.
[18] B. D. Rao and K. V. S. Hari, “Weighted subspace methods and spatial smoothing: Analysis and comparison,” IEEE Trans. Acoust., Speech,
Signal Processing, vol. 41, pp. 788–803, Feb. 1993.
[19] G. M. Kautz and M. Zoltowski, “Performance analysis of MUSIC employing conjugate symmetric beamformers,” IEEE Trans. Signal Processing, vol. 43, pp. 737–748, Mar. 1995.
[20] P. Stoica and A. Nehorai, “Performance study of conditional and unconditional direction-of-arrival estimation,” IEEE Trans. Acoust., Speech,
Signal Processing, vol. 38, pp. 1783–1795, Oct. 1990.
[21] C. F. Mecklenbräuker, A. B. Gershman, and J. F. Böhme, “Broadband
ML-approach to environmental parameter estimation in shallow ocean
at low SNR,” Signal Process., to be published.
[22] J. W. Pierre, E. D. Scott, and M. P. Hays, “A sensor array testbed for
source tracking algorithms,” in Proc. ICASSP, Munich, Germany, 1997,
pp. 3769–3772.
[23] A. B. Gershman and J. F. Böhme, “A pseudo-noise approach to direction
finding,” Signal Process., vol. 71, pp. 1–13, Nov. 1998.
(65)
With (65), the proof of (46) is completed.
REFERENCES
[1] K. C. Huarng and C. C. Yeh, “A unitary transformation method for angle
of arrival estimation,” IEEE Trans. Acoust., Speech, Signal Processing,
vol. 39, pp. 975–977, Apr. 1991.
[2] M. D. Zoltowski, G. M. Kautz, and S. D. Silverstein, “Beamspace rootMUSIC,” IEEE Trans. Signal Processing, vol. 41, pp. 344–364, Jan.
1993.
Marius Pesavento was born in Werl, Germany,
in 1973. He received the M.S. (Diploma) degree
in electrical engineering from Ruhr University,
Bochum, Germany, in 1999.
He is currently with the Department of Electrical
and Computer Engineering, McMaster University,
Hamilton, Ont., Canada. His research interests
include statistical signal and array processing and
adaptive beamforming.
1314
Alex B. Gershman (M’97–SM’98) was born in
Nizhny Novgorod, Russia, in 1962. He received the
M.S. (Diploma) and Ph.D. degrees in radiophysics
from the Nizhny Novgorod State University, in 1984
and 1990, respectively.
From 1984 to 1989, he was with the Radiotechnical and Radiophysical Institutes, Nizhny
Novgorod. From 1989 to 1997, he was with the
Institute of Applied Physics of Russian Academy
of Science, Nizhny Novgorod, as a Senior Research
Scientist. From the summer of 1994 until the beginning of 1995, he held a Visiting Research Fellow position at the Swiss Federal
Institute of Technology, Lausanne. From 1995 to 1997, he held an Alexander
von Humboldt Fellow position at the Ruhr University, Bochum, Germany.
From 1997 to 1999, he was a Member of Staff at the Department of Electrical
Engineering at Ruhr University. Since 1999, he has been with the Department
of Electrical and Computer Engineering, McMaster University, Hamilton, Ont.,
Canada, as an Associate Professor. His research interests are in the area of
signal processing and include statistical signal and array processing, adaptive
beamforming and smart antennas, signal parameter estimation and detection,
spectral analysis, high-order statistics, and signal processing applications to
underwater acoustics, wireless communications, seismology, and radar. He has
published more than 100 journal and conference papers on these topics.
Dr. Gershman was a recipient of the 1993 International Union of Radio
Science (URSI) Young Scientist Award, the 1994 Outstanding Young Scientist
Presidential Fellowship (Russia), the 1994 Swiss Academy of Engineering
Science and Branco Weiss Fellowships (Switzerland), and the 1995–1996
Alexander von Humboldt Fellowship (Germany). He is listed in several Who’s
Who in the World and Who’s Who in Science and Engineering editions. He
is an Associate Editor of IEEE TRANSACTIONS ON SIGNAL PROCESSING and
a Member of the Sensor Array and Multichannel Signal Processing Technical
Committee of the IEEE Signal Processing Society.
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 5, MAY 2000
Martin Haardt (S’90–M’98–SM’99) was born
in Germany on October 4, 1967. He received the
Dipl.-Ing. (M.S.) degree from the Ruhr University,
Bochum, Germany, in 1991 and the Dr.-Ing. (Ph.D.)
degree from the Munich University of Technology,
Munich, Germany, in 1996, both in electrical
engineering.
From 1989 to 1990, he was a Visiting Scholar at
Purdue University, West Lafayette, IN, sponsored
by a fellowship of the German Academic Exchange
Service (DAAD). From 1991 to 1993, he worked
for Corporate Research and Development, Siemens AG, Munich, Germany,
conducting research in the areas of image and biomedical signal processing.
From 1993 to 1996, he was a Research Associate with the Institute of Network
Theory and Circuit Design at the Munich University of Technology, where
his research was focused on smart antennas. In 1997, he joined Siemens
Mobile Networks, Munich, where he was responsible for strategic research
for third-generation mobile radio systems. Since 1998, he has been serving as
Director of International and University Projects in the Chief Technical Office
of Siemens Communication on Air. He also teaches at the Munich University
of Technology and is a contributing author to The Digital Signal Processing
Handbook, (Boca Raton, FL: CRC and Piscataway, NJ: IEEE, 1998). His
current research interests include wireless communications, array and digital
signal processing, and numerical linear algebra.
Dr. Haardt received the Rohde and Schwarz Outstanding Dissertation Award
for his Ph.D. dissertation in 1997. In 1998, he was the co-recipient of the Mannesmann Mobilfunk Innovation Award for outstanding research in mobile communications and received the Best Paper Award from the Information Technology Society (ITG) of the Association of Electrical Engineering, Electronics,
and Information Technology (VDE), Germany. He is a Member of VDE.