## Abstract

The problem of obtaining long sequences with finite alphabet and peaky aperiodic auto-correlation is important in the context of radar, sonar and system identification and is called the coded waveform design problem, or simply the signal design problem in this limited context. It is good to remember that there are other signal design problems in coding theory and digital communication. It is viewed as a problem of optimization. An algorithm based on two operational ideas is developed. From the earlier experience of using the eugenic algorithm for the problem of waveform design, it was realised that rather than random but multiple mutations, all the first-order mutations should be examined to pick up the best one. This is called Hamming scan, which has the advantage of being locally complete, rather than random. The conventional genetic algorithm for non-local optimization leaves out the anabolic role of chemistry of allowing quick growth of complexity. Here, the Hamming scan is made to operate on the Kronecker or Chinese product of two sequences with best-known discrimination values, so that one can go to large lengths and yet get good results in affordable time. The details of the ternary pulse compression sequences obtained are given. They suggest the superiority of the ternary sequences.

## Keywords

Coded waveform design global optimization bi-parental products Hamming scan## Preview

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## References

- Barker R H 1953 Group synchronization of binary digital systems. In
*Communication theory*(ed.) W Jackson (London: Butterworths)Google Scholar - Baumert L D 1971
*Cyclic difference sets*(Berlin: Springer-Verlag)MATHGoogle Scholar - Beenker G F M, Claasen T A C M, Heime P W C 1985 Binary sequences with a maximally flat amplitude spectrum.
*Phillips J. Res.*40: 289–304MATHGoogle Scholar - Bernasconi J 1987 Low autocorrelation binary sequences: statistical mechanics and configuration space analysis.
*J. Phys.*48: 559–567Google Scholar - Bernasconi J 1988 Optimization problems and statistical mechanics.
*Proc. Workshop on Chaos and Complexity 1987*(Torino: World Scientific)Google Scholar - Boehmer M A 1967 Binary pulse compression codes.
*IEEE Trans. Inf. Theory*IT-13: 156–167CrossRefGoogle Scholar - Brewere J W 1978 Kronecker products and matrix calculus in system theory.
*IEEE Trans. Circuits Syst.*CAS-25: 772–781CrossRefGoogle Scholar - De Groot C, Wurtz D, Hoffman K H 1992 Low autocorrelation binary sequences: exact enumeration and optimization by evolutionary strategies.
*Optimization*23: 369–384MATHCrossRefMathSciNetGoogle Scholar - Golay M J E 1972 A class of finite binary sequences with alternate autocorrelation values equal to zero.
*IEEE Trans. Inf. Theory*IT-18: 449–450CrossRefGoogle Scholar - Golay M J E 1977 Sieves for low autocorrelation binary sequences.
*IEEE Trans. Inf. Theory*IT-23: 43–51CrossRefGoogle Scholar - Golay M J E 1982 The merit factor of long low autocorrelation binary sequences.
*IEEE Trans. Inf. Theory*IT-28: 543–549CrossRefGoogle Scholar - Golay M J E 1983 The merit factor of Legendré sequences.
*IEEE Trans. Inf. Theory*IT-29: 934–936CrossRefGoogle Scholar - Golay M J E, Harris D 1990 A new search for skew-symmetric binary sequences with optimal merit factors.
*IEEE Trans. Inf. Theory*36: 1163–1166CrossRefGoogle Scholar - Hoholdt T, Jensen H E, Justesen J 1985 Aperiodic correlations and the merit factor of a class of binary sequences.
*IEEE Trans. Inf. Theory*IT-31: 549–552CrossRefMathSciNetGoogle Scholar - Hoholdt T, Justesen J 1988 Determination of the merit factor of Legendré sequences.
*IEEE Trans. Inf. Theory*IT-34: 161–164CrossRefMathSciNetGoogle Scholar - Holland J H 1992 Genetic algorithms.
*Sci. Am.*267: 66–72CrossRefGoogle Scholar - Jensen J M, Jensen H E, Hoholdt T 1991 The merit factor of binary sequences related to difference sets.
*IEEE Trans. Inf. Theory*37: 617–626CrossRefMathSciNetGoogle Scholar - Kerdock A M, Meyer R, Bass D 1986 Longest binary pulse compression codes with given peak sidelobe levels.
*Proc. IEEE*74: 366CrossRefGoogle Scholar - Koestler A 1969 Beyond atomism and holism: the concept of the holon. In
*Beyond reductionism*(eds) A Koestler, J R Smythies (London: Hutchinson)Google Scholar - Michalewicz Z 1992
*Genetic algorithms + data structures = evolution programs*(Berlin: Springer-Verlag) p 250MATHGoogle Scholar - Moharir P S 1974 Ternary Barker codes.
*Electron. Lett.*10: 460–461CrossRefGoogle Scholar - Moharir P S 1975 Generation of the approximation to binary white noise.
*J. Inst. Electron. Telecommun. Eng.*21: 5–7Google Scholar - Moharir P S 1976 Signal design.
*Int. J. Electron.*41: 381–398CrossRefGoogle Scholar - Moharir P S 1977 Chinese product theorem for generalized PN sequences.
*Electron. Lett.*13: 121–122CrossRefMathSciNetGoogle Scholar - Moharir P S 1992
*Pattern-recognition transforms*(Taunton: Research Studies Press) p 256MATHGoogle Scholar - Moharir P S, Varma S K, Venkatrao K 1985 Ternary pulse compression sequences.
*J. Inst. Electron Telecommun. Eng.*31: 33–40Google Scholar - Newmann D J, Byrnes J S 1990 The L norm of a polynomial with coefficients + 1.
*Am. Math. Monthly*97: 42–45MATHCrossRefGoogle Scholar - Reddy V U, Rao K V 1986 Biphase sequence generation with low sidelobe autocorrelation function.
*IEEE Trans. Aerosp. Electron. Syst.*AES-22: 128–133CrossRefGoogle Scholar - Simon H A 1981
*The sciences of the artificial*(Cambridge, MA: MIT Press)Google Scholar - Singh R, Moharir P S, Maru V M 1996 Eugenic algorithm-based search for ternary pulse compression sequences.
*J. Inst. Electron. Telecommun. Eng.*42: 11–19Google Scholar - Turyn R 1963 Optimum code study. Sylvania Electric Systems, Rep. F 437-1Google Scholar
- Turyn R 1968 Sequences with small correlation. In
*Error correcting codes*(ed.) H B Mann (New York: Wiley) pp 195–228Google Scholar