Uniquely generated paraunitary-based complementary QAM sequences

A Boolean generator for a broad set of standard pairs of complex valued complementary sequences of length 2^K is proposed. Binary, M-PSK and QAM sequences can be generated. The Boolean generator is derived from our earlier paraunitary algorithm from 2013 that is based on matrix multiplication. Both algorithms are based on unitary matrices. In contrast to previous Boolean QAM algorithms, which have an additive form, our algorithm has a multiplicative form. Any element of the sequence can be efficiently generated by indexing entries of K unitary matrices using a binary counter. MQum generation algorithm uses exactly M <K QAM unitary matrices each having at least one non-unit-norm entry. Greatest common divisor of Gaussian integers plays a key role in ensuring that the algorithm generates a set of unique sequences and, consequently, in deriving their enumeration formula. Our 1Qum and 2Qum algorithms generate generalized Case I, II and III, as well as, generalized Case IV and V sequences given by Liu et al. in 2013, respectively, in addition to many other sequences. The ratio of the total number to Case I-V sequences grows with the constellation size and the sequence length. As an example, for 1024-QAM and length 1024, our algorithm generates 340% additional sequences.