Paraunitary-Based Boolean Generator for QAM Complementary Sequences of Length $2^{K}$

A Boolean generator for a large number of standard complementary QAM sequences of length 2^K is proposed. This Boolean generator is derived from the authors' earlier paraunitary generator, which is based on matrix multiplications. Both generators are based on unitary matrices. In contrast to previous Boolean QAM algorithms which represent complementary sequences as a weighted sum, our algorithm has a multiplicative form. Any element of a sequence can be generated efficiently by indexing the entries of unitary matrices with the binary representation of the discrete time index (which is easily implemented as a binary counter). Our 1Qum (based on one QAM unitary matrix) and 2Qum (based on two QAM unitary matrices) algorithms generate generalized Case I-III sequences and generalized Case IV and V sequences, respectively, as specified by Liu et al. in 2013, in addition to many new 2Qum sequences. The ratio of the numbers of sequences that are generated by our new construction and the previous construction increases with the constellation size. For example, for a 1024-QAM sequence of length 1024, this ratio is 4.4. However, if we compare only 2Qum sequences to Case IV and V sequences, this ratio is 267.

I. Introduction

Binary complementary sequences were introduced by Golay in the field of multi-slit spectrometry [1]. After their introduction, radar researchers were the first to analyze their use as radar signals [2], [3]. However, the first widespread applications appeared in the area of wireless communications including Wi-Fi [4] and 3G [5], [6]. Since OFDM is accepted as the solution of choice for 4G systems, it is clear that the Peak to Average Power Ratio (PAPR) problem severely limits its practical implementation. Davis and Jedwab [7] proposed complementary sequences whose spectral PAPR is bounded by two, as a possible solution to the PAPR problem.