The peak sidelobe level of families of binary sequences

A numerical investigation is presented for the peak sidelobe level (PSL) of Legendre sequences, maximal length shift register sequences (m-sequences), and Rudin-Shapiro sequences. The PSL gives an alternative to the merit factor for measuring the collective smallness of the aperiodic autocorrelations of a binary sequence. The growth of the PSL of these infinite families of binary sequences is tested against the desired growth rate o(/spl radic/nlnn) for sequence length n. The claim that the PSL of m-sequences grows like O(/spl radic/n), which appears frequently in the radar literature, is concluded to be unproven and not currently supported by data. Notable similarities are uncovered between the PSL and merit factor behavior under cyclic rotations of the sequences.

[1]  V. Reddy,et al.  Biphase Sequence Generation with Low Sidelobe Autocorrelation Function , 1986, IEEE Transactions on Aerospace and Electronic Systems.

[2]  Siu Lun Ma,et al.  A survey of partial difference sets , 1994, Des. Codes Cryptogr..

[3]  Jonathan Jedwab,et al.  A Survey of the Merit Factor Problem for Binary Sequences , 2004, SETA.

[4]  F. Lemmermeyer.  Error-correcting Codes , 2005 .

[5]  Dilip V. Sarwate.  An upper bound on the aperiodic autocorrelation function for a maximal-length sequence , 1984, IEEE Trans. Inf. Theory.

[6]  Ann M. Boehmer,et al.  Binary pulse compression codes , 1967, IEEE Trans. Inf. Theory.

[7]  J. Lindner,et al.  Binary sequences up to length 40 with best possible autocorrelation function , 1975 .

[8]  D. E. Vakman,et al.  Sophisticated Signals and the Uncertainty Principle in Radar , 1968 .

[9]  Hu Chuan-Gan,et al.  On The Shift Register Sequences , 2004 .

[10]  J. M. Baden,et al.  Minimum peak sidelobe pulse compression codes , 1990, IEEE International Conference on Radar.

[11]  Marvin N. Cohen.  Pulse Compression in Radar Systems , 1987 .

[12]  David Joyner,et al.  Applied Abstract Algebra , 2004 .

[13]  Marcel J. E. Golay.  The merit factor of Legendre sequences , 1983, IEEE Trans. Inf. Theory.

[14]  Peter B. Borwein,et al.  Binary sequences with merit factor greater than 6.34 , 2004, IEEE Transactions on Information Theory.

[15]  Hanfried Lenz,et al.  Design theory , 1985 .

[16]  R. McEliece.  Finite Fields for Computer Scientists and Engineers , 1986 .

[17]  Tom Høholdt,et al.  Determination of the merit factor of Legendre sequences , 1988, IEEE Trans. Inf. Theory.

[18]  Hans D. Schotten,et al.  On the search for low correlated binary sequences , 2005 .

[19]  G. E. Coxson,et al.  New results on minimum-PSL binary codes , 2001, Proceedings of the 2001 IEEE Radar Conference (Cat. No.01CH37200).

[20]  J. Littlewood.  Some problems in real and complex analysis , 1968 .

[21]  W. W. Peterson,et al.  Error-Correcting Codes. , 1962 .

[22]  Robert J. McEliece.  Correlation Properties of Sets of Sequences Derived from Irreducible Cyclic Codes , 1980, Inf. Control..

[23]  Idris David Mercer,et al.  Autocorrelations of Random Binary Sequences , 2006, Combinatorics, Probability and Computing.

[24]  Tom Høholdt,et al.  Aperiodic correlations and the merit factor of a class of binary sequences , 1985, IEEE Trans. Inf. Theory.

[25]  Alexander Pott,et al.  Finite Geometry and Character Theory , 1995 .

[26]  John Michael Baden,et al.  Biphase codes with minimum peak sidelobes , 1989, Proceedings of the IEEE National Radar Conference.

[27]  L. Moser,et al.  On the Correlation Function of Random Binary Sequences , 1968 .

[28]  Marcel J. E. Golay.  A class of finite binary sequences with alternate auto-correlation values equal to zero (Corresp.) , 1972, IEEE Trans. Inf. Theory.

[29]  Tom Høholdt,et al.  Binary Sequences with Good Correlation Properties , 1987, AAECC.