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About 10,000 papers for “Complementary sequences”

Aperiodic Complementary Sequences

Zhimin Sun, Xiangyong Zeng, Yang Yang · IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences · 2014

For an integer q≥2, new sets of q-phase aperiodic complementary sequences (ACSs) are constructed by using known sets of q-phase ACSs and certain matrices. Employing the Kronecker product to two known sets of q-phase ACSs, some sets of q-phase aperiodic complementary sequences with a new length are obtained. For an even integer q, some sets of q-phase ACSs with new parameters are generated, and their equivalent matrix representations are also presented.

Cited by 1

Quaternary Golay Complementary Sequences Derived from Binary Golay Complementary Sequences

Fanxin Zeng, Xiaoping Zeng, Xiangyong Zeng et al. · 2012

Golay complementary sequences play quite an important role in communication and radar. In this paper, we present a family of quaternary Golay complementary sequences. The resultant sequences result from conversion of the existing binary Golay sequences, and both have the same length and number of subsequences. Our method can provide rich quaternary Golay complementary sequences due to a large number of binary Golay complementary sequences available.

Cited by 1

Complementary sets of sequences

Chin-Chong Tseng, C. Liu · IEEE Transactions on Information Theory · 1972

A set of equally long finite sequences, the elements of which are either + 1 or - 1, is said to be a complementary set of sequences if the sum of autocorrelation functions of the sequences in that set is zero except for a zero-shift term. A complementary set of sequences is said to be a mate of another set if the sum of the cross-correlation functions of the corresponding sequences in these two sets is zero everywhere. Complementary sets of sequences are said to be mutually orthogonal complementary sets if any two of them are mates to each other. In this paper we discuss the properties of such complementary sets of sequences. Algorithms for synthesizing new sets from a given set are given. Recursive formulas for constructing mutually orthogonal complementary sets are presented. It is shown that matrices consisting of mutually orthogonal complementary sets of sequences can be used as operators so as to per form transformations and inverse transformations on a one- or two-dimensional array of real time or spatial functions. The similarity between such new transformations and the Hadamard transformation suggests applications of such new transformations to signal processing and image coding.

Cited by 624

Algorithms for Complementary Sequences

Chai Wah Wu · arXiv (Cornell University) · 2024

Finding the $n$-th positive square number is easy, as it is simply $n^2$. But how do we find the complementary sequence, i.e., the $n$-th positive non-square number? For this case there is an explicit formula. However, for general constraints on numbers, a formula is harder to find. In this paper, we study how to compute the $n$-th integer that does (or does not) satisfy a certain condition. In particular, we consider it as a fixed point problem, relate it to the iterative method of Lambek and Moser, study a bisection approach to this problem, and provide novel formulas for various complementary sequences including the non-$k$-gonal numbers, non-$k$-gonal-pyramidal numbers, non-$k$-simplex numbers, non-sum-of-$k$-th-powers, and non-$k$-th-powers. For example, we show that the $n$-th non $k$-gonal number is given by $n+\text{round}\left(\sqrt{\frac{2n-2+\left\lfloor\frac{k+1}{4}\right\rfloor}{k-2}}\right)$ and that the $n$-th non-second-hexagonal number is $n+\left\lceil\sqrt{\frac{n}{2}}\right\rceil-1$.

Cited by 0Open access

Derived sequences of complementary symmetric Rote sequences

Kateřina Medková, Edita Pelantová, Laurent Vuillon · Springer Link (Chiba Institute of Technology) · 2019

Complementary symmetric Rote sequences are binary sequences which have factor complexity C(n) = 2n for all integers n ≥ 1 and whose languages are closed under the exchange of letters. These sequences are intimately linked to Sturmian sequences. Using this connection we investigate the return words and the derived sequences to the prefixes of any complementary symmetric Rote sequence v which is associated with a standard Sturmian sequence u. We show that any non-empty prefix of v has three return words. We prove that any derived sequence of v is coding of three interval exchange transformation and we determine the parameters of this transformation. We also prove that v is primitive substitutive if and only if u is primitive substitutive. Moreover, if the sequence u is a fixed point of a primitive morphism, then all derived sequences of v are also fixed by primitive morphisms. In that case we provide an algorithm for finding these fixing morphisms.

Cited by 0

Derivated sequences of complementary symmetric Rote sequences

Kateřina Medková, Edita Pelantová, Laurent Vuillon · arXiv (Cornell University) · 2018

Complementary symmetric Rote sequences are binary sequences which have factor complexity $\mathcal{C}(n) = 2n$ for all integers $n \geq 1$ and whose languages are closed under the exchange of letters. These sequences are intimately linked to Sturmian sequences. Using this connection we investigate the return words and the derivated sequences to the prefixes of any complementary symmetric Rote sequence $\mathbf{v}$ which is associated with a standard Sturmian sequence $\mathbf{u}$. We show that any non-empty prefix of $\mathbf{v}$ has three return words. We prove that any derivated sequence of $\mathbf{v}$ is coding of three interval exchange transformation and we determine the parameters of this transformation. We also prove that $\mathbf{v}$ is primitive substitutive if and only if $\mathbf{u}$ is primitive substitutive. Moreover, if the sequence $\mathbf{u}$ is a fixed point of a primitive morphism, then all derivated sequences of $\mathbf{v}$ are also fixed by primitive morphisms. In that case we provide an algorithm for finding these fixing morphisms.

Cited by 2Open access

Synthesis of multilevel complementary sequences

M. Darnell, Andrew H. Kemp · Electronics Letters · 1988

Methods of synthesising multilevel complementary sequence sets of both odd and even lengths are described, thus extending previous work which was restricted to even-length binary sets. Also, techniques for the synthesis of uncorrelated sets of multilevel complementary sequences are given. Possible application areas are then discussed briefly.

Cited by 21

Binary odd-periodic complementary sequences

Hans Dieter Lüke · IEEE Transactions on Information Theory · 1997

A pair of binary sequences is called complementary if the sum of their aperiodic autocorrelation functions is zero except for zero shift. Such sequences are known for relatively few lengths. This correspondence therefore introduces pairs of binary odd-periodic complementary sequences, which can be constructed using q-ary m-sequences for many more lengths.

Cited by 44

Beyond Beatty sequences: Complementary lattices

Sam Vandervelde · Canadian Mathematical Bulletin · 2020

Abstract By taking square lattices as a two-dimensional analogue to Beatty sequences, we are motivated to define and explore the notion of complementary lattices. In particular, we present a continuous one-parameter family of complementary lattices. This main result then yields several novel examples of complementary sequences, along with a geometric proof of the fundamental property of Beatty sequences.

Cited by 0

Complementary means and double sequences

Gheorghe Toader · SHILAP Revista de lepidopterología · 2007

We look after the complementary means with respect to a weighted geometric mean of Stolarsky means in the family of Gini means and in the family of Stolarsky means.

Cited by 3Open access

Periodic complementary sets of binary sequences

Dragomir Ž. Djoković · ArXiv.org · 2007

Let PCS_p^N denote a set of p binary sequences of length N such that the sum of their periodic auto-correlation functions is a delta-function. In the 1990, Boemer and Antweiler addressed the problem of constructing such sequences. They presented a table covering the range p <= 12, N <= 50 and showing in which cases it was known at that time whether such sequences exist, do not exist, or the question of existence is undecided. The number of undecided cases was rather large. Subsequently the number of undecided cases was reduced to 26 by the author. In the present note, several cyclic difference families are constructed and used to obtain new sets of periodic binary sequences. Thereby the original problem of Boemer and Antweiler is completely solved.

Cited by 5Open access

Complementary classification approaches for protein sequences

Jason T.L. Wang, Thomas G. Marr, Dennis Shasha et al. · Protein Engineering Design and Selection · 1996

We have studied five methods of protein classification and have applied them to the 768 groups of related proteins in the PROSITE catalog. Four of these methods are based on searching a database of blocks, and the other uses the frequently occurring motifs found in the protein families combined with a fingerprint technique. Our experimental results show that the block-based methods perform well when taking into account the probability of amino acids occurring in a block. Furthermore, the five methods give information that is complementary to each other. Thus, using the five methods together, one can obtain high confidence classifications (if the results agree) or suggest alternative hypotheses (if the results disagree). We also list those proteins whose current families documented in the PROSITE catalog differ from those suggested by our results. There are remarkably few of them, which is a testimony to the quality of PROSITE.

Cited by 28Open access