Detection and localization of active and passive targets using sensor arrays play an important role in the field of array signal processing. In this paper the problem of estimating the direction of arrivals of broad banded linear and quadratic chirp sources using both nested and co-prime arrays is addressed. Traditional uniform arrays can only detect N-1 number of sources with N physical sensors using high resolution beam formers like MUSIC. However the nested and co-prime arrays can detect more number of sources than the number of sensors by exploiting the difference co-array structure based on the correlation of the observations. Difference co-array is the distinct sensor locations obtained by taking all possible pairwise differences of sensor locations in the original array. As the chirp signal, commonly used in both radar and sonar systems is better processed in the fractional Fourier domain, the detection is done using fractional Fourier transform (FrFT). But as the traditional FrFT is limited to the analysis of linear chirps, detection using modified FrFT is found to be the apt choice for quadratic chirps. Subsequently, the direction of arrival estimation is achieved using subspace methods which include MUSIC and minimum-norm in the proposed work. The effectiveness of the algorithm is validated through different signals including real data obtained from a practical sonar array. It is seen from the computer simulations that nested-MUSIC combination has better resolution and accuracy than all other combinations.

Direction of arrival estimation, Fractional Fourier transform, Chirp sources, Nested array, Coprime array

[1] M.A. Richards, “Fundamentals of Radar Signal Processing”, McGraw-Hill Publication, India, 2005.
[2] S. Zhou, P. Willett, “Submarine location estimation via a network of detection only sensors”, IEEE Trans. Signal Process., Vol. 55, Issue 6, pp. 3104 - 3115, 2007.
[3] H. Krim, M. Viberg, “Two decades of array signal processing research: The parametric approach”, IEEE Signal Process. Mag., Vol. 13, pp. 67-94, 1996.
[4] D.H. Johnson, D.E. Dudgeon, “Array Signal Processing Concepts and Techniques”, Prentice-Hall, USA, 1993.
[5] J. Capon, “High-resolution frequency-wavenumber spectrum analysis”, In the Proceedings of the IEEE, USA, pp. 1408-1418, 1969.
[6] R. Schmidt, “Multiple emitter location and signal parameter estimation”, IEEE Trans. on Antennas and Propagation, Vol. 34, Issue 3, pp. 276-280, 1986.
[7] B.D. Rao, K.V.S. Hari, “Performance Analysis of Root-Music”, IEEE Trans. on Acoust., Speech and Signal Process., Vol. 37, Issue 12, pp. 1939-1949, 1989.
[8] H.L. Van Trees, “Detection, Estimation, and Modulation Theory- Part IV, Optimum Array Processing”, Wiley, USA, 2002.
[9] R.Roy, T.Kailath, “Estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Acoust., Speech, Signal Process., Vol. 37, Issue 7, pp. 984-995, 1989.
[10] J.R. Klauder, A.C. Price, S. Darlington, W.J. Albersheim, “The theory and design of chirp radars”, Bell Syst. Tech. Jour., Vol. 39, Issue 4, pp. 745-808, 1960.
[11] P.H. Leong, T.D. Abhayapala, T.A. Lamahewa, “Multiple target localization using wideband echo chirp signals”, IEEE Trans. Signal Process., Vol.61, Issue 16, pp. 4077-4089, 2013.
[12] I.S. Yetik, A. Nehorai, “Beamforming using the fractional Fourier transform”, IEEE Trans. Signal Process., Vol. 51, pp. 1663-1668, 2003.
[13] A.A. Winder, “Sonar system technology”, IEEE Trans. Sonics Ultrason., Vol. 22, Issue 5, pp. 291-332, 1975.
[14] M. Wax, T.J. Shan, T. Kailath, “Spatio temporal spectral analysis by eigenstructure methods”, IEEE Trans. on Acoust., Speech and Signal Process., Vol. 32, Issue 4, pp. 817-827, 1984.
[15] G. Su, M. Morf, “The signal subspace approach for multiple wideband emitter location”, IEEE Trans. on Acoust., Speech and Signal Process., Vol. 31, Issue 6, pp.1502-1522, 1983.
[16] H. Wang, M. Kaveh, “Coherent signal-subspace processing for the detection and estimation of angles of arrival of multiple wide-band sources”, IEEE Trans. Acoust., Speech, Signal Process., Vol. 33, Issue 4, pp. 823-831, 1985.
[17] M. A. Doron, A. J. Weiss, “On focusing matrices for wide-band array processing”, IEEE Trans. on Signal Process., Vol. 40, Issue 6, pp. 1295-1302, 1992.
[18] E.D. Claudio, R. Parisi, “WAVES: weighted average of signal subspaces for robust wideband direction finding”, IEEE Trans. on Signal Process., Vol. 49, Issue 10, pp. 2179-2191, 2001.
[19] Piya Pal, P.P. Vaidyanathan, “A novel autofocusing approach for estimating directions of arrival of wideband signals”, In the Proceedings of the 43rd Asilomar Conference on Signals, Systems and Computers, USA, pp. 1663-1667, 2009.
[20] R. Jacob, T. Thomas, A. Unnikrishnan, “Applications of fractional Fourier transform in sonar signal processing”, IETE Jour. Res. Vol. 55, pp. 16-27, 2009.
[21] S. Sahay, D. Pande, V. Gadre, P. Sohani, “A novel generalized time-frequency transform inspired by the fractional Fourier transform for higher order chirps”, In the Proceedings of the International Conference on Signal Processing (SPCOM), INDIA, pp. 1-5, 2012.
[22] S. Sahay, T. Meghasyam, R.K. Roy, G. Pooniwala, S. Chilamkurthy, V. Gadre, “Parameter estimation of linear and quadratic chirps by employing the fractional Fourier transform and a generalized time frequency transform”, Sadhana, Vol. 40, Issue 4, pp. 1049-1075, 2015.
[23] Z. K. Peng, G. Meng, Z. Q. Lang, F. L. Chu, W. M. Zhang, Y. Yang, “Polynomial chirplet transform with application to instantaneous frequency estimation”, IEEE Trans. Meas. and Instrum., Vol. 60, Issue 9, pp. 3222-3229, 2011.
[24] S. Mann, S. Haykin, “The chirplet transform: Physical considerations”, IEEE Trans. Signal Process., Vol. 43, pp. 2745-2761, 1995.
[25] P. Pal, P. Vaidyanathan, “Nested arrays: A novel approach to array processing with enhanced degrees of freedom”, IEEE Trans. Signal Process., Vol. 58, Issue 8, pp. 4167-4181, 2010.
[26] P. Vaidyanathan, P. Pal, “Sparse sensing with co-prime samplers and arrays”, IEEE Trans. Signal Process., Vol. 59, Issue 2, pp. 573-586, 2011.
[27] Namias, “The fractional order Fourier transform and its application to quantum mechanics”, J. Inst. Math. Appln., Vol. 25, pp. 241-265, 1980.
[28] H.M. Ozaktas, O. Arkan, M.A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform”, IEEE Trans. Signal Process., Vol. 44, Issue 9, pp. 2141-2150, 1996.
[29] H. M. Ozaktas, M. A. Kutay, Z. Zalevsky, “The fractional Fourier transform with applications in optics and signal processing”, Wiley, USA, 2000.
[30] Y. Cui, K. Liu, J. Wang, “Direction of arrival estimation of coherent wideband LFM signals in multipath environment”, In the Proceedings of the International Conf. on Signal Process., USA, pp. 58-61, 2010.
[31] X. Jin, T. Zhang, J. Bai, D. Zhao, “DoA estimation of coherent wideband LFM signals based on fractional Fourier transform and virtual array”, In the Proceedings of the IEEE CISP, USA, pp. 4380-4384, 2010.
[32] C. Capus and K. Brown, “Short-time fractional Fourier methods for the time-frequency representation of chirp signals”, J. Acoust. Soc. Amer., Vol. 113, Issue 6, pp. 3253-3263, 2003.
[33] C. Capus, Y.Rzhanov, L.Linnett, “Analysis of multiple linear chirp signals”, In the Proceedings of the IEE Symposium on Time-Frequency Analysis and Applns., London, pp. 41-47, 2000.
[34] C.L. Liu and P.P. Vaidyanathan, “Remarks on the spatial smoothing step in coarray MUSIC”, IEEE Signal Process. Lett., Vol. 22, Issue 9, pp. 1438–1442, 2015.
[35] G. Sreekumar, Leena Mary and A. Unnikrishnan, “Beam-forming of broad-band QFM sigals using generalised time-frequency transform”, Elseiver Signal Process., submitted.
[36] S. Bardhan, D. Rajapan, S. Zacharia, M. Eldhose, P.M. Rajeshwari, D.S. Sreedev, C. Kannan , S. Jacob and M.A. Atmanand, “Detection of buried objects using active sonar”, IEEE-UT, 2015.
[37] S. Bardhan and S. Jacob, “Experimental observation of DoA estimation algorithms in a tank environment for sonar application”, In the Proceedings of the SYMPOL, INDIA, 2015.