5.8 Conclusion

Table 5.5

Summary of Two-Dimensional Unitary ESPRIT in DFT Beamspace

1. Transformation to Beamspace: Compute the two-dimensional DFT of array outputs at each snapshot and apply the vec(•) operator and place the result as column

Υ = (WBby WBbx )X, B = B x B y

2. Signal Subspace Estimation: Compute Es

as the d dominant left singular vectors of [Re{Y} Im{Y}] (square-root approach)

or as the d dominant eigenvectors of Re{Y} Re{Y}H + Im{Y}Im{Y}H (covariance approach) .

3. Solution of the Invariance Equation: Solve the following equations for

Γ μ1Es ϒ μ ≈ Γ μ2Es and Γν1Es ϒ ν ≈ Γν 2Es

by means of LS or TLS

4. Spatial Frequency Estimation: Calculate the eigenvalues of the complex-valued d × d matrix Γ μ + jϒ ν = TΛT−1 with Λ = diag{λi}δi =1

Then extract the angular DOA information via

between them. ESPRIT exploits such a displacement property which translates into an underlying rotational invariance of signal subspaces spanned by two data vectors received by two subarrays.

Unitary ESPRIT retains the simplicity and high resolution capability of a standard ESPRIT scheme and achieves a superior performance with a reduced computational complexity. It is used with centro-symmet- ric array configurations and includes forward-backward averaging. In contrast to standard ESPRIT, unitary ESPRIT is formulated in terms of real-valued computations. In the two-dimensional case, unitary ESPRIT can obtain automatically paired azimuth and elevation angle estimates.

Finally, DFT beamspace versions of oneand two-dimensional unitary ESPRIT are also described. DFT beamspace unitary ESPRIT enables the DOA processing on a particular directional sector of interest and thereby reduces the computational complexity significantly. However, this is under the assumption that a priori knowledge of where the signal sources of interest is required.

References

[1]Haardt, M., Efficient One-, Two-, and Multidimensional High-Resolution Array Signal Processing, New York: Shaker Verlag, 1997.

[2]Roy, R., and T. Kailath, “ESPRIT-Estimation of Signal Parameters Via Rotational Invariance Techniques,” IEEE Trans. on Acoust., Speech, Signal Processing, Vol. 37, No. 7, July 1989, pp. 984–995.

[3]Rao, B. D., and K. S. Arun, “Model Based Processing of Signals: A State Space Approach,” Proc. IEEE, Vol. 80, No. 2, February 1992, pp. 283–309.

[4]Kariya, T., and H. Kurata, Generalized Least Squares, New York: John Wiley & Sons, 2004.

[5]Golub, G. H., and C. F. van Loan, Matrix Computations, 3rd ed., Baltimore, MD: John Hopkins University Press, 1996.

[6]Haardt, M., and J. A. Nossek, “Unitary ESPRIT: How to Obtain Increased Estimation Accuracy with a Reduced Computational Burden,” IEEE Trans. on Signal Processing, Vol. 43, No. 5, May 1995, pp. 1232–1242.

[7]Lee, A., “Centrohermitian and Skew-Centrohermitian Matrices,” Linear Algebra and Its applications, Vol. 29, 1980, pp. 205–210.

[8]Haardt, M., “ 2D Unitary ESPRIT For Efficient 2D Parameter Estimation,” Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, Vol. 3, May 9–12, 1995, pp. 2096–2099.

[9]Zoltowski, M. D., M. Haardt, and C. P. Mathews, “Closed-Form 2D Angle Estimation with Rectangular Arrays in Element Space or Beamspace Via Unitary ESPRIT,” IEEE Trans. on Signal Processing, Vol. 44, No. 2, February 1996, pp. 316–328

[10]Xu, G., et al., “Beamspace ESPRIT,” IEEE Trans. on Signal Processing, Vol. 42, No. 2, February 1994, pp. 349–355.

[11]Gansman, J. A., M. D. Zoltowski, and J. V. Krogmeier, “Multidimensional Multirate DOA Estimation in Beamspace,” IEEE Trans. on Signal Processing, Vol. 44, No. 11, November 1996, pp. 2780–2792.

•Array signal-to-noise ratio (SNR): The SNR observed at the antenna array plays a vital role as that in all communication problems. This experiment demonstrates the variations in performance with a variation of SNR. The higher the SNR is, the better the performance. This simulation studies the behaviors versus SNR.

•Number of signals: The number of signals impinging on the array (or the model order) casts a basic limitation on performance of an algorithm because of the fundamental fact that the number of signals should be less than the number of elements minus one for

an antenna array. Theoretically, an M element antenna can detect M − 1 number of signals at maximum. However, in a realistic environment with noises, the performance deteriorates and an array does not behave as theoretically expected. This simulation attempts to study this by checking the performance against the number of signals.

•Number of elements: This simulation studies the DOA estimation accuracy versus the number of antenna elements. Intuitively, the accuracy is high with more numbers of antenna elements and especially when the number of signals is less than the number of elements. However, the computational expenditure and hardware implementation complexity are directly proportional to the number of antenna elements. As a result, it is necessary and desirable to use the number of elements just sufficient to obtain a reliable DOA estimate.

•Variation of power levels: This simulation is of importance due to the fact that signal sources are located at various distances from the receiving antenna array. Hence, the signals from different sources arrive at an array at different power levels. This experiment studies how the power level differences can affect the performance of a DOA estimation algorithm, in particular, in the presence of the weak signals and strong signals.

•Angle of separation: This simulation directly measures the resolution of a DOA estimation algorithm. The resolution of an algorithm is counted on its capability of resolving two closely spaced signals. This experiment studies variations in error of DOA estimates when the impinging signals are closely placed.

•Computational complexity: This simulation attempts to give an idea about the implementation of a DOA finding algorithm in terms of the FLOPS algorithm demands. The algorithms are programmed and the simulations are conducted on the MATLAB platform. Therefore, the MATLAB FLOP counter has been used for this study [1, 2].

In all the simulations, the performance of the algorithms in coherent scenarios is also studied. In these scenarios, spatial smoothing with forward-backward averaging is used to decorrelate the signals. The main objective is to show that the algorithms perform well in coherent environments provided the data is preprocessed. However, only simulations with respect to the key parameters such as snapshots and SNR are presented since the performance with other parameters is similar to that of uncorrelated signals.

In all the simulations to follow, RMS (root mean square) error is used as a metric to measure the accuracy of the DOA estimations. For a single trial of a DOA estimation, the RMS error of the estimated DOAs is defined as

where θi is the DOA estimate obtained by a DOA estimation algorithm

and θi is the actual angle of arrival.

In our study, the DOA estimations are run with several trials (say, K independent trials) in order to assess the overall errors of a DOA algorithm. As a result, an estimate of the RMS error should involve the results from the K independent trials; the corresponding RMS error is then redefined as

where θik is the DOA estimate obtained by a DOA estimation algorithm

at the kth trial based on (6.1).

Finally all the simulations abide to the data model assumptions presented in Chapter 3.