Optimal Families of Perfect Polyphase Sequences from Cubic Polynomials

Abstract

For an odd prime p and a positive integer k ≥ 2, we propose and analyze construction of perfect p k -ary sequences of period p k based on cubic polynomials over the integers modulo p k . The constructed perfect polyphase sequences from cubic polynomials is a subclass of the perfect polyphase sequences from the Mow's unified construction. And then, we give a general approach for constructing optimal families of perfect polyphase sequences with some properties of perfect polyphase sequences and their optimal families. By using this, we construct new optimal families of p k -ary perfect polyphase sequences of period p k . The constructed optimal families of perfect polyphase sequences are of size p − 1. key words: perfect polyphase sequences, cubic polynomials, optimal families of perfect polyphase sequences 1. Introduction Various sequences have been widely used in modern digital communication systems [8], [10], [11] and radar systems [14]. Recent applications include such commercial mobile communication systems as CDMA, WCDMA, and 3GPP LTE [1] and global navigation satellite systems as GPS [2] and GALILEO [7]. These applications require sequences with good correlation, or the minimum possible correlation magnitude for all the non-trivial phase shifts. Thus, it is best that sequences for these applications have zero autocorrelation at all out-of-phases, and such sequences are called perfect sequences. A sequence is called a polyphase sequence if all the symbols are on the complex unit circle [8], [16], [18]. For many decades, perfect polyphase sequences (PPSs) have attracted engineers and researchers [5], [6], [9], [12], [15], [16], [18]-[20], [24]. For an integer N ≥ 2, Frank and Zadoff constructed N-ary sequences of period N 2 in 1962 [9]. There is another class of PPSs, called the Zadoff-Chu sequence, whose phase sequence are generated by a quadratic polynomials. This class was proposed by Chu in 1972 [5], and generalized by Popovic [20]. A generalized version of Chu's sequences was named generalized chirp-like sequences by Popovic. In [16], [18], Mow classified all the known PPSs into four classes: the generalized Frank