A parameterization of perfect sequences over composition algebras over the real number field is presented. According to the proposed parameterization theorem, a perfect sequence can be represented as a sum of trigonometric functions and points on a unit sphere of the algebra. Because of the non-commutativity of the multiplication, there are two definitions of perfect sequences, but the equivalence of the definitions is easily shown using the theorem. A composition sequence of sequences is introduced. Despite the non-associativity, the proposed theorem reveals that the composition sequence from perfect sequences is perfect.

The present paper introduces a novel method for the construction of a class of sequences that have a zero-correlation zone. For the proposed sequence set, both the cross-correlation function and the side lobe of the auto-correlation function are zero for phase shifts within the zero-correlation zone. The proposed scheme can generate a set of sequences of length 8n2 from an arbitrary Hadamard matrix of order n and a set of 2n trigonometric-like function sequences of length 4n. The proposed sequence construction can generate an optimal zero-correlation zone sequence set that satisfies the theoretical bound on the number of members for the given zero-correlation zone and sequence period. The auto-correlation function of the proposed sequence is equal to zero for all nonzero phase shifts. The peak factor of the proposed sequence set is √2, and the peak factor of a single trigonometric function is equal to √2. Assigning the sequences of the proposed set to a synthetic aperture ultrasonic imaging system would improve the S/N of the obtained image. The proposed sequence set can also improve the performance of radar systems. The performance of the applications of the proposed sequence sets are evaluated.

The magnetic properties, abrasivity, corrosion resistance and electromagnetic characteristics of metal tapes containing acicular CrO2 particles were studied. The following results were obtained; The Bs of tapes decreased with CrO2 addition. The playback output level and noise level decreased, but the C/N level increased, at 4 MHz. The abrasivity of the metal tape was improved by up to 96 % that of commercial oxide tape by adding 50 wt% CrO2. The corrosion resistance of the tape dropped significantly.

The present paper introduces a new approach to the construction of a sequence set with a zero-correlation zone for both periodic and aperiodic correlation functions. The proposed sequences can be constructed from a pair of Hadamard matrices of orders n0 and n1. The constructed sequence set consists of n0 n1 ternary sequences, each of length n0(m+2)(n1+Δ), for a non-negative integer m and Δ ≥ 2. The zero-correlation zone of the proposed sequences is |τ| ≤ n0m+1-1, where τ is the phase shift. The proposed sequence set consists of n0 subsets, each with a member size n1. The correlation function of the sequences of a pair of different subsets, referred to as the inter-subset correlation function, has a zero-correlation zone with a width that is approximately Δ times that of the correlation function of sequences of the same subset (intra-subset correlation function). The inter-subset zero-correlation zone of the proposed sequences is |τ| ≤ Δn0m+1, where τ is the phase shift. The wide inter-subset zero-correlation enables performance improvement during application of the proposed sequence set.

The present paper introduces the construction of quadriphase sequences having a zero-correlation zone. For a zero-correlation zone sequence set of N sequences, each of length l, the cross-correlation function and the side lobe of the autocorrelation function of the proposed sequence set are zero for the phase shifts τ within the zero-correlation zone z, such that |τ|≤z (τ ≠ 0 for the autocorrelation function). The ratio $rac{N(z+1)}{ell}$ is theoretically limited to one. When l=N(z+1), the sequence set is called an optimal zero-correlation sequence set. The proposed zero-correlation zone sequence set can be generated from an arbitrary Hadamard matrix of order n. The length of the proposed sequence set can be extended by sequence interleaving, where m times interleaving can generate 4n sequences, each of length 2m+3n. The proposed sequence set is optimal for m=0,1 and almost optimal for m>1.

The present paper introduces a novel construction of structured ternary sequences having a zero-correlation zone (ZCZ) for both periodic and aperiodic correlation functions. The cross-correlation function and the side lobe of the auto-correlation function of the proposed sequence set are zero for phase shifts within the ZCZ. The proposed ZCZ sequence set can be generated from an arbitrary Hadamard matrix of order n. The sequence set of order 0 is identical to the r-th row of the Hadamard matrix. For m≥0, the sequence set of order (m+1) is constructed from the sequence set of order m by sequence concatenation and interleaving. The sequence set of order m has 2m subsets of size n. The length of the sequence is equal to n4m+2m+1(2m-1); The phase shift of the ZCZ for the whole sequence set is from -(2m-1) to (2m-1). The sequence set of order 0 is coincident with the rows of the given Hadamard sequence with no ZCZ. The subsets can be associated with a perfect binary tree of height m with 2m leaves. The r-th sequence subset consists of from the nr-th sequence to the ((n+1)r-1)-th sequence. The r-th subset is assigned to the r-th leaf of the perfect binary tree. For a longer distance between the corresponding leaves to the r-th and s-th sequences, the ZCZ of the r-th and s-th sequences is wider. This tree-structured width of ZCZ of a pair of the proposed sequences enables flexible design in applications of the proposed sequence set. The proposed sequence is suitable for a heterogeneous wireless network, which is one of the candidates for the fifth generation of radio access networks.

To analyze the structure of a set of perfect sequences over a composition algebra of the real number field, transforms of a set of sequences similar to the discrete Fourier transform (DFT) are introduced. The discrete cosine transform, discrete sine transform, and generalized discrete Fourier transform (GDFT) of the sequences are defined and the fundamental properties of these transforms are proved. We show that GDFT is bijective and that there exists a relationship between these transforms and a convolution of sequences. Applying these properties to the set of perfect sequences, a parameterization theorem of such sequences is obtained.

To analyze the structure of a set of high-dimensional perfect sequences over a composition algebra over R, we developed the theory of Fourier transforms of the set of such sequences. We define the discrete cosine transform and the discrete sine transform, and we show that there exists a relationship between these transforms and a convolution of sequences. By applying this property to a set of perfect sequences, we obtain a parameterization theorem. Using this theorem, we show the equivalence between the left perfectness and right perfectness of sequences. For sequences of real numbers, we obtain the parameterization without restrictions on the parameters.

A perfect sequence is a sequence having an impulsive autocorrelation function. Perfect sequences have several applications, such as CDMA, ultrasonic imaging, and position control. A parameterization of a perfect sequence is presented in the present paper. We treat a set of perfect sequences as a zero set of quadratic equations and prove a decomposition law of perfect sequences. The decomposition law reduces the problem of the parameterization of perfect sequences to the problem of the parameterization of quasi-perfect sequences and the parameterization of perfect sequences of short length. The parameterization of perfect sequences for simple cases and quasi-perfect sequences should be helpful in obtaining a parameterization of perfect sequences of arbitrary length. According to our theorem, perfect sequences can be represented by a sum of trigonometric functions.

The present paper introduces a novel type of structured ternary sequences having a zero-correlation zone (zcz) for both periodic and aperiodic correlation functions. The cross-correlation function and the side lobe of the auto-correlation function of the proposed sequence set are zero for phase shifts within the zcz. The proposed zcz sequence set can be generated from an arbitrary pair of an Hadamard matrix of order lh and a binary/ternary perfect sequence of length lp. The sequence set of order 0 is identical to the r-th row of the Hadamard matrix. For m ≥ 0, the sequence set of order (m+1) is constructed from the sequence set of order m by sequence concatenation and interleaving. The sequence set has lp subsets of size 2lh. The periodic correlation function and the aperiodic correlation function of the proposed sequence set have a zcz from -(2m+1-1) to 2m+1-1. The periodic correlation function and the aperiodic correlation function of the sequences of the i-th subset and k-th subset have a zcz from -2m+2-(lh+1)((j-k) mod lp) to -2m+2-(lh+1)((j-k) mod lp). The proposed sequence is suitable for a heterogeneous wireless network, which is one of the candidates for the fifth-generation mobile networks.

The present paper introduces a novel method for the construction of sequences that have a zero-correlation zone. For the proposed sequence set, both the cross-correlation function and the side lobe of the autocorrelation function are zero for phase shifts within the zero-correlation zone. The proposed scheme can generate a set of sequences, each of length 16n2, from an arbitrary Hadamard matrix of order n and a set of 4n trigonometric function sequences of length 2n. The proposed construction can generate an optimal sequence set that satisfies, for a given zero-correlation zone and sequence period, the theoretical bound on the number of members. The peak factor of the proposed sequence set is equal to √2.

The present paper introduces a new approach to the construction of a sequence set with a zero-correlation zone (ZCZ), which is referred to as a ZCZ sequence set. The proposed sequence construction generates a ZCZ sequence set from a ZCZ sequence set. The proposed method can generate an almost optimal ZCZ sequence set, the member size of which approaches the theoretical bound, when an almost optimal ZCZ sequence is used for the sequence construction. The proposed sequence set consists of NO subsets, where a ZCZ sequence set Z(LO, NO, ZO is used in sequence construction. The correlation function of the sequences of a pair of different subsets, referred to as the inter-subset correlation function, has a ZCZ with a width that is about times that of the correlation function of sequences of the same subset (intra-subset correlation function) for integers Λ ≥ 1, T, and m ≥ 0. Wide inter-subset zero-correlation enables improved performance during application of the proposed sequence set.

The present letter describes the estimation of the upper bounds of the correlation functions of a class of zero-correlation-zone sequences constructed from an arbitrary Hadamard matrix.

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.

A perfect array is an array for which the autocorrelation function is impulsive. A parameterization of perfect arrays of real numbers is presented. Perfect arrays are represented by trigonometric functions. Three formulae are obtained according to the parities of the size of the array. Examples corresponding to each formula are shown. In the case of 66 arrays, the existence of a set of perfect arrays having integer components is shown.