Proposed Complex and Binary Sequences which Achieve Welch Bound.
Technical memo. Jul 75-May 76,
NAVAL WEAPONS CENTER CHINA LAKE CA
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Families of periodic sequences with small side-lobes and low cross-talk cross-correlations are needed in many digital communication systems. L. R. Welch has recently established a lower bound for the largest of the side-lobes and cross-correlation coefficient associated with a family of M distinct sequences of period L. In particular, for two sequences there must occur a coefficient greater than sq. rt. 12L. As the number M of sequences increases to approximately sq. rt. lL, this lower bound increases to sq. rt. 1L. In this report several sets of periodic sequences utilizing complex, ternary, and binary coded signal bits are constructed which nearly meet Welchs bound for M approximately equal to sq. rt. 1L. In this papers context, ternary coding refers to three distinct phase signals, while complex coding refers to phase levels relating to the nth complex roots of unity. Most of these sequences contain only sq. rt. L non-zero entries. However, one type of family consists of M 2 to the m power binary or - 1 sequences of period L 2 to the m power - 1. The maximum coefficient for this family is approximately one-half as large as the corresponding maximum for the Gold sequences of the same period. This represents a considerable improvement over the Gold codes for such codes requiring an even number of shift register stages for generation. Author
- Electricity and Magnetism