Construction of Zero Autocorrelation Stochastic Waveforms

ABSTRACT

Stochastic waveforms are constructed whose expected autocorrelation can be made arbitrarily small outside the origin. These waveforms are unimodular and complex-valued. Waveforms with such spike like autocorrelation are desirable in waveform design and are particularly useful in areas of radar and communications. Both discrete and continuous waveforms with low expected autocorrelation are constructed. Further, in the discrete case, frames for C^{d} are constructed from these waveforms and the frame properties of such frames are studied.

Stochastic waveforms are constructed whose expected autocorrelation can be made arbitrarily small outside the origin. These waveforms are unimodular and complex-valued. Waveforms with such spike like autocorrelation are desirable in waveform design and are particularly useful in areas of radar and communications. Both discrete and continuous waveforms with low expected autocorrelation are constructed. Further, in the discrete case, frames for C

Cite this paper

S. Datta, "Construction of Zero Autocorrelation Stochastic Waveforms,"*Advances in Pure Mathematics*, Vol. 2 No. 6, 2012, pp. 428-440. doi: 10.4236/apm.2012.26065.

S. Datta, "Construction of Zero Autocorrelation Stochastic Waveforms,"

References

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[25] A. E. Litvak, A. Pajor, M. Rudelson and N. Tomczak-Jaegermann, “Smallest Singular Value of Random Matrices and Geometry of Random Polytopes,” Advances in Mathematics, Vol. 195, No. 2, 2005, pp. 491-523. doi:10.1016/j.aim.2004.08.004

[26] E. Candès and T. Tao, “Near Optimal Signal Recovery from Random Projections: Universal Encoding Strategies?” IEEE Transactions on Information Theory, Vol. 52, No. 12, 2006, pp. 5406-5425. doi:10.1109/TIT.2006.885507

[27] E. Candes, “Compressive Sampling,” Proceedings of the International Congress of Mathematicians, Madrid, 22-30 August 2006, pp. 1433-1452.

[28] E. Candès, J. Romberg and T. Tao, “Robust Uncertainty Principles: Exact Signal Reconstruction from Highly In- complete Frequency Information,” IEEE Transactions on Information Theory, Vol. 52, No. 2, 2006, pp. 489-509. doi:10.1109/TIT.2005.862083

[29] D. L. Donoho, “Compressed Sensing,” IEEE Transactions on Information Theory, Vol. 52, No. 4, 2006, pp. 1289-1306. doi:10.1109/TIT.2006.871582

[1] L. Auslander and P. E. Barbano, “Communication Codes and Bernoulli Transformations,” Applied and Computational Harmonic Analysis, Vol. 5, No. 2, 1998, pp. 109-128. doi:10.1006/acha.1997.0222

[2] J. J. Benedetto and J. J. Donatelli, “Ambiguity Function and Frame Theoretic Properties of Periodic Zero Autocorrelation Waveforms,” IEEE Journal of Selected Topics in Signal Processing, Vol. 1, 2007, pp. 6-20.

[3] T. Helleseth and P. V. Kumar, “Sequences with Low Correlation,” Handbook of Coding Theory, North-Holland, Amsterdam, 1998, pp. 1765-1853.

[4] N. Levanon and E. Mozeson, “Radar Signals,” Wiley Interscience, New York, 2004. doi:10.1002/0471663085

[5] M. L. Long, “Radar Reflectivity of Land and Sea,” Artech House, 2001.

[6] W. H. Mow, “A New Unified Construction of Perfect Root-of-Unity Sequences,” Proceedings of IEEE 4th International Symposium on Spread Spectrum Techniques and Applications, Sun City, 22-25 September 1996, pp. 955-959. doi:10.1109/ISSSTA.1996.563445

[7] F. E. Nathanson, “Radar Design Principles: Signal Processing and the Environment,” SciTech Publishing Inc., Mendham, 1999.

[8] G. W. Stimson, “Introduction to Airborne Radar,” SciTech Publishing Inc., Mendham, 1998.

[9] S. Ulukus and R. D. Yates, “Iterative Construction of Optimum Signature Sequence Sets in Synchronous CDMA Systems,” IEEE Transactions on Information Theory, Vol. 47, No. 5, 2001, pp. 1989-1998. doi:10.1109/18.930932

[10] S. Verdú, “Multiuser Detection,” Cambridge University Press, Cambridge, 1998.

[11] J. J. Benedetto and S. Datta, “Construction of Infinite Unimodular Sequences with Zero Autocorrelation,” Advances in Computational Mathematics, Vol. 32, No. 2, 2010, pp. 191-207. doi:10.1007/s10444-008-9100-9

[12] D. Cochran, “Waveform-Agile Sensing: Opportunities and Challenges,” IEEE International Conference on Acoustics, Speech, and Signal Processing, Pennsylvania, 18-23 March 2005, pp. 877-880.

[13] M. R. Bell, “Information Theory and Radar Waveform Design,” IEEE Transactions on Information Theory, Vol. 39, No. 5, 1993, pp. 1578-1597. doi:10.1109/18.259642

[14] S. P. Sira, Y. Li, A. Papandreou-Suppappola, D. Morrell, D. Cochran and M. Rangaswamy, “Waveform-Agile Sensing for Tracking,” IEEE of Signal Processing Magazine, Vol. 26, No. 1, 2009, pp. 53-64.

[15] H. Boche and S. Stanczak, “Estimation of Deviations between the Aperiodic and Periodic Correlation Functions of Polyphase Sequences in Vicinity of the Zero Shift,” IEEE 6th International Symposium on Spread Spectrum Techniques and Applications, Parsippany, 6-8 September 2000, pp. 283-287.

[16] R. Narayanan, “Through Wall Radar Imaging Using UWB Noise Waveforms,” IEEE International Conference on Acoustics, Speech and Signal Processing, Beijing, 31 March - 4 April 2008, pp. 5185-5188.

[17] R. M. Narayanan, Y. Xu, P. D. Hoffmeyer and J. O. Curtis, “Design, Performance, and Applications of a Coherent Ultra-Wideband Random Noise Radar,” Optical Engineering, Vol. 37, No. 6, 1998, pp. 1855-1869.

[18] D. Bell and R. Narayanan, “Theoretical Aspects of Radar Imaging Using Stochastic Waveforms,” IEEE Transactions on Signal Processing, Vol. 49, No. 2, 2001, pp. 394-400. doi:10.1109/78.902122

[19] A. F. Karr, “Probability of Springer Texts in Statistics,” Springer-Verlag, New York, 1993.

[20] O. Christensen, “An Introduction to Frames and Riesz Bases,” Birkhauser, 2003.

[21] I. Daubechies, “Ten Lectures on Wavelets,” CBMS-NSF Regional Conference Series in Applied Mathematics, 1992. doi:10.1137/1.9781611970104

[22] W. Hoeffding, “Probability Inequalities for Sums of Bounded Random Variables,” Journal of American Statistical Association, Vol. 58, No. 301, 1963, pp. 13-30,. doi:10.1080/01621459.1963.10500830

[23] Z. D. Bai, “Methodologies in Spectral Analysis of Large-Dimensional Random Matrices: A Review,” Statistica Sinica, Vol. 9, No. 3, 1999, pp. 611-677.

[24] M. Ledoux, “The Concentration of Measure Phenomenon,” Mathematical Surveys and Monographs, American Mathematical Society, 2001.

[25] A. E. Litvak, A. Pajor, M. Rudelson and N. Tomczak-Jaegermann, “Smallest Singular Value of Random Matrices and Geometry of Random Polytopes,” Advances in Mathematics, Vol. 195, No. 2, 2005, pp. 491-523. doi:10.1016/j.aim.2004.08.004

[26] E. Candès and T. Tao, “Near Optimal Signal Recovery from Random Projections: Universal Encoding Strategies?” IEEE Transactions on Information Theory, Vol. 52, No. 12, 2006, pp. 5406-5425. doi:10.1109/TIT.2006.885507

[27] E. Candes, “Compressive Sampling,” Proceedings of the International Congress of Mathematicians, Madrid, 22-30 August 2006, pp. 1433-1452.

[28] E. Candès, J. Romberg and T. Tao, “Robust Uncertainty Principles: Exact Signal Reconstruction from Highly In- complete Frequency Information,” IEEE Transactions on Information Theory, Vol. 52, No. 2, 2006, pp. 489-509. doi:10.1109/TIT.2005.862083

[29] D. L. Donoho, “Compressed Sensing,” IEEE Transactions on Information Theory, Vol. 52, No. 4, 2006, pp. 1289-1306. doi:10.1109/TIT.2006.871582