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).

To construct good DNA codes based on biologically motivated constraints, it is important that they have a large minimum Hamming distance and the number of GC-content is kept constant. Also, maximizing the number of codewords in a DNA code is required for given code length, minimum Hamming distance, and number of GC-content. In most previous works on the construction of DNA codes, quaternary constant weight codes were directly used because the alphabet of DNA strands is quaternary. In this paper, we propose new coding theoretic constructions of DNA codes based on the binary Hadamard matrix from a binary sequence with ideal autocorrelation. The proposed DNA codes have a greater number of codewords than or the equal number to existing DNA codes constructed from quaternary constant weight codes. In addition, it is numerically shown that for the case of codes with length 8 or 16, the number of codewords in the proposed DNA code sets is the largest with respect to the minimum reverse complementary Hamming distances, compared to all previously known results.

Biphase periodic sequences having elements +1 or -1 with the two-level autocorrelation function are desirable in communications and radars. However, in case of the biphase orthogonal periodic sequences, Turyn has conjectured that there exist only sequences with period 4, i.e., there exist the circulant Hadamard matrices for order 4 only. In this paper, it is described that the conjecture is proved to be true by means of the isomorphic mapping, the Chinese remainder theorem, the linear algebra, etc.

We propose a practical method that acquires dense light transports from unknown 3D objects by employing orthogonal illumination based on a Walsh-Hadamard matrix for relighting computation. We assume the presence of color crosstalk, which represents color mixing between projector pixels and camera pixels, and then describe the light transport matrix by using sets of the orthogonal illumination and the corresponding camera response. Our method handles not only direct reflection light but also global light radiated from the entire environment. Tests of the proposed method using real images show that orthogonal illumination is an effective way of acquiring accurate light transports from various 3D objects. We demonstrate a relighting test based on acquired light transports and confirm that our method outputs excellent relighting images that compare favorably with the actual images observed by the system.

We apply the Hadamard equivalence to all the binary matrices of the size mn and study various properties of this equivalence relation and its classes. We propose to use HR-minimal as a representative of each equivalence class, and count and/or estimate the number of HR-minimals of size mn. Some properties and constructions of HR-minimals are investigated. Especially, we figure that the weight on an HR-minimal's second row plays an important role, and introduce the concept of Quasi-Hadamard matrices (QH matrices). We show that the row vectors of mn QH matrices form a set of m binary vectors of length n whose maximum pairwise absolute correlation is minimized over all such sets. Some properties, existence, and constructions of Quasi-orthogonal sequences are also discussed. We also give a relation of these with cyclic difference sets. We report lots of exhaustive search results and open problems, one of which is equivalent to the Hadamard conjecture.

The present paper introduces a novel construction of ternary sequences having a zero-correlation zone. The cross-correlation function and the side-lobe of the auto-correlation function of the proposed sequence set is zero for the phase shifts within the zero-correlation zone. The proposed sequence set consists of more than one subset having the same member size. The correlation function of the sequences of a pair of different subsets, referred to as the inter-subset correlation function, has a wider zero-correlation zone than that of the correlation function of sequences of the same subset (intra-subset correlation function). The wide inter-subset zero-correlation enables performance improvement during application of the proposed sequence set. The proposed sequence set has a zero-correlation zone for periodic, aperiodic, and odd correlation functions.

We investigate binary sequence pairs with two-level correlation in terms of their corresponding cyclic difference pairs (CDPs). We define multipliers of a cyclic difference pair and present an existence theorem for multipliers, which could be applied to check the existence/nonexistence of certain hypothetical cyclic difference pairs. Then, we focus on the ideal case where all the out-of-phase correlation coefficients are zero. It is known that such an ideal binary sequence pair exists for length υ = 4u for every u ≥ 1. Using the techniques developed here on the theory of multipliers of a CDP and some exhaustive search, we are able to determine that, for lengths υ ≤ 30, (1) there does not exist "any other" ideal/ binary sequence pair and (2) every example in this range is equivalent to the one of length υ = 4u above. We conjecture that if there is a binary sequence pair with an ideal two-level correlation then its in-phase correlation must be 4. This implies so called the circulant Hadamard matrix conjecture.

A novel method is proposed that can estimate the tag population in Radio Frequency Identification (RFID) systems by using a Hadamard code for the tag response. We formulate the maximum likelihood estimator for the tag population using the number of observed footprints. The lookup table of the estimation algorithm has low complexity. Simulation results show that the proposed estimator performs considerably better than the conventional schemes.

Factorization of Hadamard matrices can provide fast algorithm and facilitate efficient hardware realization. In this letter, constructions of factorizable multilevel Hadamard matrices, which can be considered as special case of unitary matrices, are inverstigated. In particular, a class of ternary Hadamard matrices, together with its application, is presented.

The present paper introduces an integrated construction of binary sequences having a zero-correlation zone. The cross-correlation function and the side-lobe of the auto-correlation function of the proposed sequence set is zero for the phase shifts within the zero-correlation zone. The proposed method enables more flexible design of the binary zero-correlation zone sequence set with respect to its member size, length, and width of zero-correlation zone. Several previously reported sequence construction methods of binary zero-correlation zone sequence sets can be explained as special cases of the proposed method.

In this letter, a method to construct good binary and quaternary error correcting codes, called complex Hadamard codes, based on a complex Hadamard matrix is presented. The related properties of the codes are analyzed. In addition, through the operation in Z4 domain, a new simplex soft-decision decoding algorithm for the complex Hadamard codes is also proposed.

In this paper, the previous definition of the Reverse Jacket matrix (RJM) is revised and generalized. In particular, it is shown that the inverse of the RJM can be obtained easily by a constructive approach similar to that used for the RJM itself. As new results, some useful properties of RJMs, such as commutativity and the Hamiltonian symmetry appearing in half the blocks of a RJM, are shown, and also 1-D fast Reverse Jacket transform (FRJT) is presented. The algorithm of the FRJT is remarkably efficient than that of the center-weighted Hadamard transform (CWHT). The FRJT is extended in terms of the Kronecker products of the Hadamard matrix. The 1-D FRJT is applied to the discrete Fourier transform (DFT) with order 4, and the N-point DFT can be expressed in terms of matrix decomposition by using 4 4 FRJT.

This paper discusses factorization of bent function type complex Hadamard matrices of order pn with a prime p. It is shown that any bent function type complex Hadamard matrix has symmetrical factorization, which can be expressed by the product of n matrices of order pn with pn+1 non-zero elements, a matrix of order pn with pn non-zero ones, and the n matrices, at most. As its application, a correlator for M-ary spread spectrum communications is successfully given, which can be simply constructed by the same circuits with reduced multiplicators, before and behind.

It is well known that M-ary/spread spectrum (M-ary/SS) system is superior to direct-sequence spread spectrum system under AWGN, and can achieve high spectral efficiency. On the other hand, however, the main drawback of this system is that the synchronization acquisition is difficult. In this paper, we propose a new synchronization acquisition method of M-ary/SS system. This method acquires the code synchronization by introducing a symmetrical property in spreading sequences, and detecting this property with the differential decoding technique. As spreading sequences, a set of orthogonal sequences and a set of non-orthogonal sequences are considered. The strong features of proposed systems are that the systems can acquire the code synchronization in carrier band and can reduce the complexity of calculation greatly. Among the comparison results of the systems with newly proposed orthogonal and some specific non-orthogonal spreading sequences, it is especially noted that the latter can reduce the mean acquisition time and calculation complexity much greater than the former.

M-ary orthogonal keying (MOK) systems under carrier frequency offset (CFO) are investigated. It is shown that spurious signals are introduced by the offset frequency components of spectrum after multiplication in correlation detection process, and some conditions on robust orthogonal signal sets are derived. Walsh function sets are found to be very weak against CFO, since they produce large spurious signals. As robust orthogonal signal sets against CFO, the rows of circulant Hadamard matrices are proposed and their error performanses are evaluated. The results show that they are good M-ary orthogonal signal sets in the presence of CFO.

This paper discusses the performance of non-coherent reception for M-ary spread-spectrum (M-ary/SS) signals in the presence of carrier frequency offset. In general, the M-ary/SS scheme is expected to be of higher spectral efficiency than the conventional DS/SS schemes, but its performance may be degraded by the carrier frequency offset. We, therefore, analyze the effect of carrier frequency offset on the performance of the non-coherent M-ary/SS system with orthogonal modulation using a set of sequences generated by the Hadamard matrix. As a result of the analysis, it has been found that the carrier frequency offset may cause a great deal of degradation in the performance, and that its effect has a distinctive property which is due to the characteristic of Hadamard matrix, at the same time. Making use of this property, we propose two schemes that can mitigate the effect of carrier frequency offset: one is based on choise of the code sequences, the other is on the error correcting code. The effectiveness of the schemes is evaluated in the terms of symbol-error-rates through analysis and computer simulation.

Synchronization has been one of the problems in M–ary spread spectrum communication systems. In this letter, we propose the frame synchronization method using the Hadamard matrix and a frame synchronization method of PCM communication systems. Moreover, we analyze the probabilities of keeping synchronous state and frame renewal rates, and we evaluate the relationship between these probabilities and the number of stages of counters.

In this letter we present an electronic circuit based on a neural net to compute the discrete Walsh transform. We show both analytically and by simulation that the circuit is guaranteed to settle into the correct values.