1.Sixty-four additions for each of 64 snapshots to transform from complex-valued space to real-valued space;

2.Calculation of the three “largest” left singular vectors of a 64 × 128 real-valued matrix;

3.Calculation of the solution to two systems of equations in the form AX = B, where A and B are both 64 × 3 and real-valued;

4.Calculation of the eigenvalues of a 3 × 3 matrix.

In the case of a reduced dimension, the two-dimensional DFT beamspace, the computations required for a single trial run are:

1.Nine sets of 64 multiplications and 63 additions for each of 64 snapshots to transform from element space to beamspace;

2.Calculation of the three “largest” left singular vectors of a 9 × 128 real-valued matrix;

3.Calculation of the eigenvalues of a 3 × 3 matrix.

As can be seen now, the ESPRIT in the DFT beamspace requires less computational expenditure than that in the element space.

6.5 Conclusion

In this chapter, in-depth analysis of ESPRIT-based algorithms for a DOA estimation is performed. The performances of the standard ESPRIT and the unitary ESPRIT are presented. In addition, the beamspace approaches for the unitary ESPRIT in one and two dimensions are also analyzed in detail. Finally, the computational complexity of the algorithms is studied in terms of FLOPS necessary for the algorithm.

It is found that the accuracy in a DOA estimation increases with an increasing number of snapshots, the SNR, and the number of array elements. Also, it is observed that there is a limitation in the number of signals that the DOA algorithms can handle and the resolution that the algorithms can achieve. In a practical noisy environment, the number of the signals that an array can resolve is often less than the theoretically predicted one, the number of elements less than one. Finally, the unitary ESPRIT is shown to have the best performance in general with the least the computational expenditures needed.

References

[1]Recktenwald, G., Numerical Methods with MATLAB Implementation and Application, Upper Saddle River, NJ: Prentice-Hall, 2000.

[2]Hanselman, D., and B. Littlefield, Mastering MATLAB 5: A Comprehensive Tutorial and Reference, Upper Saddle River, NJ: Prentice-Hall, 1998.

[3]vander Veen, A. J., E. F. Deprettere, and A. L. Swindlehurst, “Subspace Based Signal Analysis Using Singular Value Decomposition,” Proc. IEEE, Vol. 81, No. 9, September 1993, pp. 1277–1308.

[4]Pillai, S. U., Array Signal Processing, New York: Springer-Verlag, 1989.

[5]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.

[6]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.

experience on DOA estimation algorithms and lays the theoretical foundation for real-time physical implementation.

7.2 Advanced Topics on DOA Estimations

This book aims to provide an introduction to the fundamentals of the basic DOA estimation methods. The radio signals under study have been assumed to be of narrowband and the antenna arrays have been assumed to be stationary in one dimension or two dimensions with their elements uniformly distributed in space. The sources have also been assumed to be stationary and in the far-field regions of the arrays.

More advanced topics on DOA estimations deal mainly with the removal of one or more of the above assumptions in addition to further improvement of DOA estimation accuracies and computational efficiency. These topics include DOA estimations

1.With nonuniform arrays;

2.With three-dimensional arrays;

3.Of wideband signal sources;

4.Of moving sources;

5.With delay or time-of-arrival estimations;

6.With pulsed systems;

7.In the presence of interferences and mutual coupling among elements;

8.With other techniques such as switched arrays, mechanically rotating arrays, and improving algorithms by signal property exploitation.

These advanced topics can be considered as based on or as extensions of the DOA estimation algorithms described in this book. A large body of literature can be found in public domain on these advanced topics, for instance, in IEEE Xplore. References [1–18] are a subset of this literature. In particular, Chandran [1] covers quite a wide range of topics on modern DOA techniques.

Finally, the physical implementation of these algorithms in a DSP-based smart antenna system with DOA capability forms a very exciting and challenging task for the ultimate proof of the utility of the