Binary Sequence Pairs Research Articles

On Optimal Construction of Two Classes of ZCZ Codes

This paper presents constructions of two kinds of sets of sequences with a zero correlation zone, called ZCZ code, which can reach the upper bound of the member size of the sequence set. One is a ZCZ code which can be constructed by a unitary matrix and a perfect sequence. Especially, a ternary perfect sequence with elements ±1 and zero can be used to construct the proposed ZCZ code. The other is a ZCZ code of pairs of ternary sequences and binary sequences which can be constructed by an orthogonal matrix that includes a Hadamard matrix and an orthogonal sequence pair. As a special case, an orthogonal sequence pair, which consists of a ternary sequence and a binary sequence, can be used to construct the proposed ZCZ code. These codes can provide CDMA systems without co-channel interference.

The Bank of Matched Filters for an Optical ZCZ Code Using a Sylvester Type Hadamard Matrix

The optical ZCZ code is a set of pairs of binary and bi-phase sequences with zero correlation zone. An optical M-ary direct sequence spread spectrum (M-ary/DS-SS) system using this code can detect a desired sequence without interference of undesired sequences. However, the bank of matched filters in a receiver circuit may fall into large scale. In this paper, we propose the compact construction of a bank of matched filters for an M-ary/DS-SS system using an optical ZCZ code. This filter bank can decrease the number of 2-input adders from O(N2) to O(N) and delay circuits from O(N2) to O(N log2 N), respectively, and is implemented on a field programmable gate array (FPGA) corresponding to 400,000 logic gates.

Research on perfect dyadic binary sequence pair

In this paper, the perfect dyadic binary sequence pair with one-value dyadic correlation functions is presented. That is, the perfect dyadic binary sequence pair is a perfect discrete signal, for its dyadic relative function is δ-function. The transformation features and some existing admissibility conditions of perfect dyadic binary sequence pair are discussed, and the properties for this kind of code in Walsh transformation spectrum and weight spectrum are also analyzed. From above, It is found that the perfect dyadic binary sequence pair can easily differentiate from its dyadic shifting. So these good signals can used in engineering as synchronization code, multi-user code and so on.

A new synchronization scheme using Perfect Punctured Binary Sequence Pairs in DS-UWB system

Ultra WideBand (UWB) radio is a new wireless technology that transmits extremely short duration radio impulses. In this paper, a new synchronization scheme is proposed in Direct Sequence (DS) UWB system using Perfect Punctured Binary Sequence Pairs (PPBSP) as the preamble. It can acquire both Pseudo-Noise (PN) sequence and frame synchronization at the same time. The properties and the combinatorial admissibility conditions of PPBSP are presented. The simulation results show that PPBSP is good for synchronization by their good cross-correlation properties both under Additive White Gaussian Noise (AWGN) channel and modified Saleh-Valenzuela channel.

An Iterative Approach to Determining the Length of the Longest Common Subsequence of Two Strings

This paper concerns the longest common subsequence (LCS) shared by two sequences (or strings) of length N, whose elements are chosen at random from a finite alphabet. The exact distribution and the expected value of the length of the LCS, k say, between two random sequences is still an open problem in applied probability. While the expected value E(N) of the length of the LCS of two random strings is known to lie within certain limits, the exact value of E(N) and the exact distribution are unknown. In this paper, we calculate the length of the LCS for all possible pairs of binary sequences from N=1 to 14. The length of the LCS and the Hamming distance are represented in color on two all-against-all arrays. An iterative approach is then introduced in which we determine the pairs of sequences whose LCS lengths increased by one upon the addition of one letter to each sequence. The pairs whose score did increase are shown in black and white on an array, which has an interesting fractal-like structure. As the sequence length increases, R(N) (the proportion of sequences whose score increased) approaches the Chvatal–Sankoff constant a c (the proportionality constant for the linear growth of the expected length of the LCS with sequence length). We show that R(N) is converging more rapidly to a c than E(N)/N.

Complementary sequence inverse keying for indoor wireless infrared channels

Complementary sequence inverse keying is proposed to ameliorate multipath ISI effects in an indoor wireless infrared communication system. The method involves the transmission of a pair of complementary binary sequences. The system BER performance is presented in comparison with OOK and the results demonstrate that CSIK, unlike OOK, does not incur a power penalty due to ISI.

A complete description of Golay pairs for lengths up to 100

In his 1961 paper, Marcel Golay showed how the search for pairs of binary sequences of length $n$ with complementary autocorrelation is at worst a $2^{\frac {3n}{2}-6}$ problem. Andres, in his 1977 master’s thesis, developed an algorithm which reduced this to a $2^{\frac {n}{2}-1}$ search and investigated lengths up to 58 for existence of pairs. In this paper, we describe refinements to this algorithm, enabling a $2^{\frac {n}{2}-5}$ search at length 82. We find no new pairs at the outstanding lengths 74 and 82. In extending the theory of composition, we are able to obtain a closed formula for the number of pairs of length $2^kn$ generated by a primitive pair of length $n$. Combining this with the results of searches at all allowable lengths up to 100, we identify five primitive pairs. All others pairs of lengths less than 100 may be derived using the methods outlined.

Perfect punctured binary sequence pairs

This paper presents the definition of perfect punctured binary sequence pair and the combinatorial admissible conditions for searching the perfect punctured binary sequence pairs and proves that all of the pseudo-random binary sequences satisfy these conditions. Computer calculation verifies that the pseudo-random binary sequences with length up to N=127 can be easily turned into perfect punctured binary sequence pairs.

Maximizing the Shannon Capacity of Constrained Systems with Two Constraints

In this paper, we consider the problem of finding the set $\{A,B\} \subset {0,1}m that maximizes, among all 2-subsets of ${\{0,1\}}^m$, the Shannon capacity, H(A,B), of a constrained system of binary sequences that do not contain A or B as a contiguous subsequence. This problem is motivated by the problem of finding a pair of length-m binary sequences, called markers, that achieves the maximum rate, R(2,m,n), of a (2,m,n) periodic prefix-synchronized (PPS) code. A (2,m,n) PPS code is a binary block code with two length-m markers, A,B, and codewords of length n that inserts A and B alternately at regular intervals in the encoded bitstream, with the additional constraint that A and B may not appear anywhere in the encoded bitstream other than where inserted. We show that for any $m \geq 2$, $\lim_{n \rightarrow \infty} R(2,m,n) = \max\{H(A,B): \{A,B\} \subset {\{0,1\}}^m\} = \log_2\rho_{m-1}$, where $\rho_{m-1}$ is the largest-magnitude zero of the polynomial $z^{m-1} - z^{m-2} - \cdots - 1$. Moreover, we completely characterize the sequences A and B that achieve $\max H(A,B), as well as those that achieve R(2,m,n) for all sufficiently large n.

Periodic complementary binary sequence pairs

A new set of binary sequences—Periodic Complementary Binary Sequence Pair (PCSP) is proposed. A new class of block design—Difference Family Pair (DFP) is also proposed. The relationship between PCSP and DFP, the properties and existing conditions of PCSP and the recursive constructions for PCSP are given.

A Theory of Ternary Complementary Pairs

Sequences with zero autocorrelation are of interest because of their use in constructing orthogonal matrices and because of applications in signal processing, range finding devices, and spectroscopy. Golay sequences, which are pairs of binary sequences (i.e., all entries are ±1) with zero autocorrelation, have been studied extensively, yet are known only in lengths 2a10b26c. Ternary complementary pairs are pairs of (0, ±1)-sequences with zero autocorrelation (thus, Golay pairs are ternary complementary pairs with no 0's). Other kinds of pairs of sequences with zero autocorrelation, such as those admitting complex units for nonzero entries, are studied in similar contexts. Work on ternary complementary pairs is scattered throughout the combinatorics and engineering literature where the majority approach has been to classify pairs first by length and then by deficiency (the number of 0's in a pair); however, we adopt a more natural classification, first by weight (the number of nonzero entries) and then by length. We use this perspective to redevelop the basic theory of ternary complementary pairs, showing how to construct all known pairs from a handful of initial pairs we call primitive. We display all primitive pairs up to length 14, more than doubling the number that could be inferred from the existing literature.

Complementary sequences for ISI channel estimation

A merit factor based on the sequence autocorrelation function, whose minimization leads to the reduction in the Cramer-Rao lower bound (CRLB) for the variance of two-sided intersymbol interference (ISI) channel estimation is introduced. Pairs of binary pilot symbol sequences (a preamble and a postamble) for channel estimation are jointly designed to minimize this merit factor. Given that the number of channel taps is L and the length of a pilot symbol sequence is (N+L-1), where N/spl ges/L, we distinguish between the case when N is even and the case when it is odd. For even N, we show that complementary sequences not only minimize the merit factor, but also the CRLB. For a subset of odd N we construct almost-complementary periodic sequence pairs that minimize the merit factor. The optimal pilot symbol block signaling requires alternating between two (in most cases) different binary sequences that form the merit-minimizing pair.

The performance of direct-sequence spread spectrum with complex processing and quaternary data modulation

The receiver processing that is often employed for direct-sequence spread spectrum with quaternary data modulation operates separately on in-phase and quadrature components of the spread-spectrum signal. Previous results on complex quaternary sequences, which are also referred to as four-phase sequences, provide a strong motivation to employ alternative processing methods that permit the receiver to benefit from the correlation properties of these sequences. With this goal in mind we present receiver architectures that use complex processing and are capable of demodulating the signals proposed for the third-generation mobile cellular code-division multiple-access systems. The performance of the proposed receiver architecture with complex processing is shown to be determined by the complex aperiodic correlation functions for the spreading sequences. Comparisons are made with systems in which the quaternary sequences are derived from pairs of common binary spreading sequences and the receiver employs separate in-phase and quadrature processing. We find that in spite of their superior periodic crosscorrelation properties, the previously developed quaternary sequences do not provide significantly smaller error probabilities in multiple-access systems.

Binary odd-periodic complementary sequences

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.

Shrinking generators and statistical leakage

Shrinking is a newly proposed technique for combining a pair of pseudo random binary sequences, ( a , s ), to form a new sequence, z , with better randomness, where randomness here stands for difficulty of prediction. The ones in the second sequence s are used to point out the bits in the sequence a to be included in z . The generator that performs this process is known as the shrinking generator (SG). In this paper, it is shown for the existing combining method that deviation from randomness in the statistics of a leads to the leakage of this statistics into z . We also show that it is sufficient for constructing a statistically balanced SG to at least have one statistically balanced generator. A new shrinking rule that yields statistically balanced output, even if a and s are not balanced, is then proposed. Self-shrinking in which a single pseudo random bit generator (PRBG) shrinks itself is also investigated and a modification of the existing shrinking rule is proposed. Simulation results show the robustness of the proposed methods. For self-shrinking, in particular, results show that the proposed shrinking rule yields sequences with balanced statistics even for extremely biased generators. This suggests possible application of the new rule to strengthen running key generators.

New class of pseudorandom ternary sequences

The crosscorrelation function between certain pairs of binary maximum-length sequences of the same period has only three levels. The method of formation of a class of these ternary crosscorrelation sequences is described, and some of the properties of the class are listed.