Recent results on polyphase sequences

Abstract

A polyphase sequence of length n+1, A={a/sub j/}/sub j=0//sup n/, is a sequence of complex numbers, each of unit magnitude. The (unnormalized) aperiodic autocorrelation function of a sequence is denoted by C(/spl tau/). Associated with the sequence A, the sequence polynomial f/sub A/(z) of degree n and the correlation polynomial g/sub A/(z) of degree 2n are defined. For each root /spl alpha/ of f/sub A/(z), 1//spl alpha/* is a corresponding root of f*/sub A/(z/sup -1/). Transformations on the sequence A which leave |C(/spl tau/)| invariant are exhibited, and the effects of these transformations on the roots of f/sub A/(z) are described. An investigation of the set of roots A of the polynomial f/sub A/(z) has been undertaken, in an attempt to relate these roots to the behavior of C(/spl tau/). Generalized Barker (1952, 1953) sequences are considered as a special case of polyphase sequences, and examples are given to illustrate the relationship described above.