A set consisting of K subsets of Msequences of length L is called a complementary sequence set expressed by A(L, K, M), if the sum of the out-of-phase aperiodic autocorrelation functions of the sequences within a subset and the sum of the cross-correlation functions between the corresponding sequences in any two subsets are zero at any phase shift. Suehiro et al. first proposed complementary set A(Nn, N, N) where N and n are positive integers greater than or equal to 2. Recently, several complementary sets related to Suehiro's construction, such as N being a power of a prime number, have been proposed. However, there is no discussion about their inclusion relation and properties of sequences. This paper rigorously formulates and investigates the (generalized) logic functions of the complementary sets by Suehiro et al. in order to understand its construction method and the properties of sequences. As a result, it is shown that there exists a case where the logic function is bent when n is even. This means that each series can be guaranteed to have pseudo-random properties to some extent. In other words, it means that the complementary set can be successfully applied to communication on fluctuating channels. The logic functions also allow simplification of sequence generators and their matched filters.

A polyphase sequence set with orthogonality consisting complex elements with unit magnitude, can be expressed by a unitary matrix corresponding to the complex Hadamard matrix or the discrete Fourier transform (DFT) matrix, whose rows are orthogonal to each other. Its matched filter bank (MFB), which can simultaneously output the correlation between a received symbol and any sequence in the set, is effective for constructing communication systems flexibly. This paper discusses the compact design of the MFB of a polyphase sequence set, which can be applied to any sequence set generated by the given logic function. It is primarily focused on a ZCZ code with q-phase or more elements expressed as A(N=qn+s, M=qn-1, Zcz=qs(q-1)), where q, N, M and Zcz respectively denote, a positive integer, sequence period, family size, and a zero correlation zone, since the compact design of the MFB becomes difficult when Zcz is large. It is shown that the given logic function on the ring of integers modulo q generating the ZCZ code gives the matrix representation of the MFB that M-dimensional output vector can be represented by the product of the unitary matrix of order M and an M-dimensional input vector whose elements are written as the sum of elements of an N-dimensional input vector. Since the unitary matrix (complex Hadamard matrix) can be factorized into n-1 unitary matrices of order M with qM nonzero elements corresponding to fast unitary transform, a compact MFB with a minimum number of circuit elements can be designed. Its hardware complexity is reduced from O(MN) to O(qM log q M+N).