Investigation Of Multilevel Hadamard Matrices, Aperiodic Sequences And Mismatched Sequences

Hadamard matrix named after the French mathematician Jacques Hadamard is originally defined as a square matrix whose entries are either +1 or -1 and whose rows are mutually orthogonal. Such orthogonal matrices can almost be directly used as an error-correcting code, orthogonal spreading sequences, and so on. On the other hand, various sequences have been widely used in such diverse applications as CDMA systems, radar, sonar, pulse compression, channel estimation, etc. There are different forms of Hadamard matrices and sequences, such as binary, complex and multilevel Hadamard matrices, periodic and aperiodic sequences, matched filtering sequences and mismatched filtering sequences, one-dimensional sequences and two-dimensional arrays, etc. In practice, different applications may require different Hadamard matrices and sequences with specific parameters and properties. This dissertation investigates mainly on the design of multilevel Hadamard matrices, maximum imbalance of binary sequences with low aperiodic autocorrelation, and mismatched filtering sequences with zero correlation zone (ZCZ).The dissertation begins with the design of multilevel Hadamard matrices which are closely related to binary Hadamard matrices and orthogonal design, together with its application in the construction of multilevel ZCZ sequences. All elements of these matrices are integer numbers and thus binary Hadamard matrices are special cases of multilevel Hadamard matrices. Multilevel Hadamard matrices with arbitrary order can be constructed if such matrices having only two different elements in their entries. Based on the circulant matrices or cyclic difference sets, multilevel Hadamard matrices having three different elements of order m~2 or An are constructed, respectively. Multilevel Hadamard matrices with higher order can be easily obtained by Kronecker product based on two existing multilevel Hadamard matrices. In the case of elements which are successive integers, we confidently believe that such matrices with odd order do not exist except an unknown case. In the case of even order, the multilevel Hadamard matrices with order 2~r can be derived. Further, based on multilevel Hadamard matrices, multilevel zero correlation zone sequences which may find applications in quasi-synchronous -CDMA system are obtained.Then the maximum imbalance of binary sequences with low aperiodic autocorrelations is discussed and some related theoretical bounds are derived. Specifically, Barker sequences, quasi-Barker sequences and minimum peak sidelobe sequences are investigated in detail. It is assumed that all sequences achieve the minimum peak sidelobe level P within a certain window W centered at the mainlobe. Based on the link between sequence imbalance and aperiodic autocorrelation functions, the maximum imbalance of such sequences are bounded by the length of sequences N, minimum peak sidelobe P, and window size W. Our analysis and numerical results show that the imbalance can achieve the theoretical bounds for the above mentioned sequences with some given lengths. It is very special for these lengths that inequivalent sequences are very rare. With these bounds, instead of using exhaustive search for all imbalances of a given length to search the best possible binary sequences, one only needs to search sequences with imbalances having the same parity properties and imbalances below the bounded value.Finally real-valued mismatched sequence sets with zero correlation zone are investigated. A theoretical bound for such sequence set on the family size, length and zero correlation zone (ZCZ) is derived. Two methods for constructing set of sequences with periodic ZCZ property using mismatched filtering are presented from a pair of sequences with ideal impulse-like crosscorrelation. The idea of the first method is based on the Hadamard matrices and circulant matrices which are constructed from a pair of sequences with ideal impulse-like crosscorrelation. The second method is also based on Hadamard matrices and a pair of sequences with ideal impulse-like crosscorrelation, but not through the circulant matrices. It is shown that the second method results in an optimal periodic mismatched ZCZ sequence set. Besides, the energy efficiency of the obtained ZCZ sequence set is the same as that of the pair of sequences with ideal impulse-like crosscorrelation. These mismatched filtering ZCZ sequences as well as matched filtering ZCZ sequences presented in the dissertation are suitable for applications in quasi-synchronous CDMA system to reduce or eliminate MAI and, therefore, increasing the system capacity significantly.