Costas Arrays: Survey, Standardization, and MATLAB Toolbox

A Costas array is an arrangement of N dots on an N-by-N grid, one per row, one per column, such that no two dots share the same displacement vector with any other pair. Costas arrays have applications in SONAR/RADAR systems, communication systems, cryptography, and other areas. We present a standardization of notation and language which can be used to discuss Costas array generation techniques and array manipulations. Using this standardization we can concisely and clearly state various theorems about Costas arrays, including several new theorems about the symmetries of Costas arrays. We also define labels for each array (generated, emergent, and sporadic), which describe whether the array is generated using a known technique, generated using a semiempirical variation of a known technique, or of unexplained origin. A new method for obtaining emergent Costas arrays, the DRT expansion, is also given here for the first time. A MATLAB Costas array toolbox has also been developed which implements the proposed standardization. The toolbox contains a comprehensive set of functions covering Costas array generation, manipulation and classification.

[1]  S.V. Marie,et al.  Using Costas Arrays to Construct Frequency Hop Patterns for OFDM Wireless Systems , 2006, 2006 40th Annual Conference on Information Sciences and Systems.

[2]  J.P. Costas,et al.  A study of a class of detection waveforms having nearly ideal range—Doppler ambiguity properties , 1983, Proceedings of the IEEE.

[3]  Gary L. Mullen,et al.  Primitive elements in finite fields and costas arrays , 2005, Applicable Algebra in Engineering, Communication and Computing.

[4]  James K. Beard.  Costas array generator polynomials in finite fields , 2008, 2008 42nd Annual Conference on Information Sciences and Systems.

[5]  K. Drakakis,et al.  A review of Costas arrays , 2006 .

[6]  K. Erickson,et al.  Combinatoric collaboration on Costas arrays and radar applications , 2004, Proceedings of the 2004 IEEE Radar Conference (IEEE Cat. No.04CH37509).

[7]  S. Rickard,et al.  The Enumeration of Costas Arrays of Size 26 , 2006, 2006 40th Annual Conference on Information Sciences and Systems.

[8]  L. D. Westbrook,et al.  Carrier frequency allocation in FDMA military SATCOMs , 2001, 2001 MILCOM Proceedings Communications for Network-Centric Operations: Creating the Information Force (Cat. No.01CH37277).

[9]  Francesco Iorio,et al.  The enumeration of Costas arrays of order 28 , 2010, ITW.

[10]  Weita Chang.  A remark on the definition of Costas arrays , 1987 .

[11]  Francesco Iorio,et al.  Results of the Enumeration of Costas Arrays of Order $27$ , 2008, IEEE Transactions on Information Theory.

[12]  Edward L. Titlebaum,et al.  Optimum Costas-like decompositions of Costas arrays for channel characterization and communications , 1994, Proceedings of ICASSP '94. IEEE International Conference on Acoustics, Speech and Signal Processing.

[13]  John J. Healy,et al.  A stochastic analysis approach in the search for Costas arrays , 2008 .

[14]  O. Moreno,et al.  A class of frequency hop codes with nearly ideal characteristics for multiple-target recognition , 1997, Proceedings of IEEE International Symposium on Information Theory.

[15]  Francesco Iorio,et al.  The enumeration of Costas arrays of order 28 , 2010, 2010 IEEE Information Theory Workshop.

[16]  Jiann-Ching Guey,et al.  Hopping pilot pattern for interference mitigation in OFDM , 2008, 2008 IEEE 19th International Symposium on Personal, Indoor and Mobile Radio Communications.

[17]  O. Moreno,et al.  New Enumeration Results for Costas Arrays , 1993, Proceedings. IEEE International Symposium on Information Theory.

[18]  Konstantinos Drakakis,et al.  On the symmetry of Welch- and Golomb-constructed Costas arrays , 2009, Discret. Math..

[19]  Oscar Moreno,et al.  On periodicity properties of Costas arrays and a conjecture on permutation polynomials , 1996, IEEE Trans. Inf. Theory.

[20]  Liu Tao,et al.  On the Existence and Counting Problems of Costas Arrays and its Signature Application , 2006, 2006 40th Annual Conference on Information Sciences and Systems.

[21]  Francesco Iorio,et al.  Results of the Enumeration of Costas Arrays of Order 27 , 2008, IEEE Trans. Inf. Theory.

[22]  Oscar Moreno,et al.  Exhaustive search for Costas-type sequences for multi-target recognition , 2003, The Ninth IEEE Workshop on Future Trends of Distributed Computing Systems, 2003. FTDCS 2003. Proceedings..

[23]  Scott Rickard.  Searching for Costas arrays using periodicity properties , .

[24]  Scott T. Rickard,et al.  On the Complexity of the Verification of the Costas Property , 2009, Proceedings of the IEEE.

[25]  Oscar Moreno,et al.  Costas sequences for multiple targets , 2002, Proceedings IEEE International Symposium on Information Theory,.

[26]  Scott Rickard,et al.  Large sets of frequency hopped codes with nearly ideal orthogonality properties , 1993 .

[27]  Solomon W. Golomb,et al.  Algebraic Constructions for Costas Arrays , 1984, J. Comb. Theory, Ser. A.

[28]  Weita Chang.  A remark on the definition of Costas arrays , 1987, Proceedings of the IEEE.

[29]  Oscar Moreno,et al.  A Parallel Algorithm for the Enumeration of Costas Sequences , 1995, PPSC.

[30]  J.K. Beard.  Generating Costas Arrays to Order 200 , 2006, 2006 40th Annual Conference on Information Sciences and Systems.

[31]  S. Golomb,et al.  Constructions and properties of Costas arrays , 1984, Proceedings of the IEEE.

[32]  Keith M. Martin,et al.  On the Applicability of Combinatorial Designs to Key Predistribution for Wireless Sensor Networks , 2009, IWCC.