Utilization of a Unitary Transform for Efficient Computation in the ...

180 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 1, JANUARY 2006
For higher values of the SNR, the bias of the estimator decreases,
as expected.
Fig. 2.
RMSE of the DOA versus the number of antenna elements.
VI. CONCLUSION
In this paper, it has been shown that the computational complexity
can be reduced for the MP method for DOA estimation.
A unitary transformation applied to the MP method has been
successfully formulated and utilized to convert the complex data
matrix to a real matrix, hence reducing the computational complexity
significantly. It is seen that when the SNR of the data is
greater than 10 dB, then both the MP and the new UMP method
can be used to model a given data set by a sum of complex exponentials
and the UMP can be implemented on a DSP chip using
only real arithmetic. The surprising part is that the real computations
come at a cost, particularly for low values of SNR where
the variance of the estimates due to the UMP method is larger
than that for the MP method.
ACKNOWLEDGMENT
Grateful acknowledgment is made to the reviewers for suggesting
ways to improve the readability of the paper.
Fig. 3.
Bias of the estimator versus SNR.
pointed out in [9]. This causes some degradation for low SNR
case. That explains the performance difference between the MP
and the UMP methods for the low SNR case. After 8 dB SNR,
the eigenvalues are all real. Since the results for are similar
to we are giving the results for only . The optimum value
for the pencil length, , is chosen to be 3 for efficient noise
filtering which is explained in [7].
The root mean square error (RMSE) of the DOA versus the
number of antenna elements for the UMP is shown in Fig. 2. As
the number of antenna elements increase, the RMSE decreases
as expected. This simulation is based on the value of an SNR of
20 dB.
The bias for the estimate of the DOA has also been studied.
The bias is computed from
(45)
where denotes the expected value. The bias of the estimator
versus SNR is shown in Fig. 3.
REFERENCES
[1] J. C. Liberti and T. S. Rappaport, Smart Antennas for Wireless Communications.
Upper Saddle River, NJ: Prentice Hall, 1999.
[2] J. Capon, “Maximum likelihood spectral estimation,” Nonlinear
Methods of Spectral Analysis, pp. 155–179, 1979.
[3] R. O. Schmidt, “Multiple emitter location and signal parameter estimation,”
IEEE Trans. Antennas Propag., vol. 34, no. 3, pp. 276–280, Mar.
1986.
[4] A. J. Barabell, “Improving the resolution performance of eigenstructurebased
direction finding algorithm,” in Proc. IEEE Int. Conf. Acoustics,
Speech, and Signal Processing, 1983, pp. 336–339.
[5] K. Takao and N. Jijuma, “An adaptive array utilizing an adaptive spatial
averaging technique for multi-path environments,” IEEE Trans. Antennas
Propag., vol. 35, no. 12, pp. 1389–1396, 1987.
[6] Y. Hua and T. K. Sarkar, “Generalized pencil-of-function method for
extracting poles of an EM system from its transient response,” IEEE
Trans. Antennas Propag., vol. 37, no. 2, pp. 229–234, 1989.
[7] T. K. Sarkar and O. Pereira, “Using the matrix pencil method to estimate
the parameters of a sum of complex exponentials,” IEEE Antennas
Propag. Mag., vol. 37, no. 1, pp. 48–54, 1994.
[8] K. C. Huang and C. C. Yeh, “Unitary transformation method for angle
of arrival estimation,” IEEE Trans. Signal Processing, vol. 39, no. 4, pp.
975–977, 1991.
[9] 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, no. 5, pp. 1232–1242, 1995.
[10] A. Lee, “Centrohermitian and skew-centrohermitian matrices,” Linear
Algebra and Its Applications, vol. 29, pp. 205–210, 1980.
[11] L. Datta and D. M. Salvatore, “Some results on matrix symmetries and
a pattern recognition application,” IEEE Trans. Acoust., Speech, Signal
Processing, vol. ASSP-34, no. 4, pp. 992–993, Aug. 1986.
[12] R. Bachl, “The forward and backward averaging technique applied to
TLS-ESPRIT processing,” IEEE Trans. Signal Processing, vol. 43, no.
11, pp. 2691–2699, Nov. 1995.
[13] G. Xu, R. H. Roy, and T. Kailath, “Detection of number of sources
via exploitation of centro-symmetric property,” IEEE Trans. Signal Processing,
vol. 42, no. 1, pp. 102–111, Jan. 1994.
[14] H. V. Trees, Optimum Array Processing, Part IV of Detection, Estimation,
and Modulation Theory. New York: Wiley, 2002.
[15] Y. Hua, “On Techniques for Estimating the Parameters of Exponentially
Damped/Undamped Sinusoids in Noise,” PhD dissertation, Syracuse
Univ., Syracuse, New York, Aug. 1988.
[16] L. C. Godara, Smart Antennas. Boca Raton, FL: CRC Press, 2004.