A unique decomposition of arbitrary pairs of complementary sequences (including binary, polyphase and QAM) based on paraunitary matrices is presented. This decomposition allows us to describe the internal structure of any sequence pair of length N using basic paraunitary matrices defined by N complex coefficients named omega vector. When the omega vector is sparse a particularly compact canonic form exists and leads to an efficient implementation of a generator-correlator. The equivalence of paraunitary matrices and Z transforms of complementary sequences allows us to apply the rich results from the theory of perfect reconstruction filter-banks to the field of sequence design. Based on this equivalence a new algorithm for generating/correlating 16-, 64-, and 256-QAM sequences is introduced.