Periodic Binary Sequences Research Articles

Oversampling architecture for analog harmonic modulation

We propose a technique for sinusoidal modulation of analog signals, which requires neither multiplication nor analog sine wave generation. The modulating signal is an oversampled representation of a sine in the form of a periodic binary sequence, optimized to minimize intermodulation distortion. A 2-to-1 multiplexer performs the modulation multiplication. With third-order low-pass filtering of the input signal, the technique achieves 60 dB of linear dynamic range for a sequence of length 256, conveniently generated with an 11-address decoder and an 8-b counter.

Storage efficient decoding for a class of binary be Bruijn sequences

A binary span k de Bruijn sequence is a binary sequence of period 2 k such that each k-tuple of bits occurs exactly once as a subsequence in a period of the sequence. The de Bruijn sequences have applications in position sensing and range finding, as well as in other areas, by virtue of this subsequence property. To make use of a particular de Bruijn sequence, we need to be able to determine the position of an arbitrary k-tuple in the sequence. We refer to this as ‘decoding’ the sequence. In this paper we use the structure of a class of de Bruijn sequences, originally constructed by Lempel, to give an algorithm for decoding that is fast, yet more space efficient than a complete look-up table of subsequences and their positions in a sequence. We also consider the case where the displacement between consecutive subsequences whose positions are decoded is restricted and obtain a further marked improvement in storage.

On evaluating the linear complexity of a sequence of least period 2 n

The linear complexity of a periodic binary sequence is the length of the shortest linear feedback shift register that can be used to generate that sequence. When the sequence has least period 2 n ,n≥0, there is a fast algorithm due to Games and Chan that evaluates this linear complexity. In this paper a related algorithm is presented that obtains the linear complexity of the sequence requiring, on average for sequences of period 2 n ,n≥0, no more than 2 parity checks sums.

On the existence of periodic complementary binary sequences

Using the connection between periodic complementary binary sequences and difference families given by Bomer and Antweiler [2], we use number theoretic techniques to obtain necessary conditions for their existence when two or three sequences are involved. Our results provide theoretical proof for the nonexistence of PCS320(ai) and settle the open case PCS236(ai) of [2], negatively.

On the quadratic spans of DeBruijn sequences

The quadratic span of a periodic binary sequence is defined to be the length of the shortest quadratic feedback shift register (FSR) that generates it. An algorithm for computing the quadratic span of a binary sequence is described. The required increase in quadratic span is determined for the special case of when a discrepancy occurs in a linear FSR that generates an initial portion of a sequence. The quadratic spans of binary DeBruijn sequences are investigated. An upper bound for the quadratic span of a DeBruijn sequence of span n is given; this bound is attained by the class of DeBruijn sequences obtained from m-sequences. It is easy to see that a lower bound is n+1, but a lower bound of n+2 is conjectured. The distributions of quadratic spans of DeBruijn sequences of span 3, 4, 5 and 6 are presented. >

On the linear span of binary sequences obtained from q-ary m-sequences, q odd

A class of periodic binary sequences that are obtained from q-ary m-sequences is defined, and a general method to determine their linear spans (the length of the shortest linear recursion over the Galois field GF(2) satisfied by the sequence) is described. The results imply that the binary sequences under consideration have linear spans that are comparable with their periods, which can be made very long. One application of the results shows that the projective and affine hyperplane sequences of odd order both have full linear span. Another application involves the parity sequence of order n, which has period p/sup m/-1, where p is an odd prime. The linear span of a parity sequence of order n is determined in terms of the linear span of a parity sequence of order 1, and this leads to an interesting open problem involving primes. >

Periodic complementary binary sequences

Existence conditions and recursive construction procedures for sets of periodic complementary binary sequences are given. Relationships to sets of aperiodic complementary binary sequences and to perfect binary arrays, whose two-dimensional periodic autocorrelation function is a delta function, are noted. The connections of periodic complementary binary sequences and difference families are given. Sets of periodic complementary binary sequences, which result from computer search, are presented. A diagram showing what is currently known about the existence of periodic complementary binary sequences with P<or=12 sequences of N<or=50 elements is given.<<ETX>>

False Lock and Bifurcation in Costas Loops

New results are given on the phenomenon of data-related false lock in a Costas loop that contains a perfect integrator in its loop filter. The input carrier to the loop is assumed to be bi-phase modulated with a $T'$ periodic binary sequence constructed from Manchester symbols; there are P symbols per period. As a result, the false locked loop is described by a non-linear system with periodic coefficients. A loop gain parameter $\delta $ appears in this system, and it is used as a perturbation parameter. A constant $\omega _f $ also appears; this constant represents the closed loop frequency error in the false locked loop. Under some general conditions on modulation m it is shown that bifurcation occurs at $\delta = 0$ in this equation for each value$\omega _f = k[ \pi/T'],\,k = 1,2, \cdots $. For each of these values the nonlinear system is shown to have four distinct periodic solutions. The loop has a false locked state that corresponds to each solution. A simple test is derived to analyze the stability of these false locked states. Finally, the theory is applied to a simple loop with proportional plus integral loop filter, and the results are displayed graphically.

Sequential binary arrays and circulant matrices

AbstractA periodic binary array on the sequare grid is said to be sequential if and only if each row and each column of the array contains a given periodic binary sequence or some cyclic shift or reversal of this sequence. Such arrays are of interest in connection with experimental layouts. This paper extends previous results by characterizing sequential arrays on sequences of the type (1,…,1,0,…,0) and solving the problem of equivalence of such arrays (including a computation of the number of equivalence classes).

Characterisation of some sparse binary sequential arrays

A periodic binary array is said to besequential if and only if every line of the array is occupied by a given periodic binary sequence or by some cyclic shift or reversal of this sequence. This paper extends earlier results for arrays built on the square grid, characterising further sequential arrays with two ones per period, and those with three consecutive ones per period.

A generalized recursive construction for de Bruijn sequences

This paper is concerned with the construction of de Bruijn sequences of span <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> --binary sequences of period <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^{n}</tex> in which every binary <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -tuple appears as some <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> consecutive terms in one period of the sequence. Constructions in the literature are based on maximum length linear sequences, algorithms which start from scratch, and recursive methods which start with a single de Bruijn sequence of span <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> and produce one of span <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n+1</tex> . We give a more general recursive construction which takes two de Brnijn sequences of span <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> and produces a de Bruijn sequence of span <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n+1</tex> . In addition, for a special case of the construction, the complexity (shortest linear recursion that generates the sequence) of the resulting sequence is determined in terms of the complexities of the ingredient sequences. In particular, de Bruijn sequences of span <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n+1</tex> with maximum complexity <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^{(n+1)}-1</tex> are obtained from maximum complexity sequences of span <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> . Reverse-complementary de Bruijn sequences are also considered.

On the characteristics of PN sequences (Corresp.)

Balanced binary sequences of period <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^{n}-1</tex> with the run property and the two-level autocorrelation property are not necessarily PN sequences.

Measurement of Residence Time Distribution with Radiotracers Using Periodic Pseudo-Random Binary Signal Sequences

A review about the properties of several types of tracerinput signals is given and in particular the stochastic signals are discussed. An estimation of the necessary measuring time shows, that true stochastic signals are unimportant for tracer experiments. The use of periodic pseudo-random binary signals (PRBS) permits a strong reduction of the measuring time. Therefore their practical application for tracer experiments is possible. The PRBS-method has special advantages in comparison with impulse methods, if fluctuations and disturbances exist within the investigated systems. Correlations for the estimation of parameters, needed for the realization of PRBS-measurements, are given and several measuring arrangements are discussed. The application of the PRBS-method for measurements of residence time distributions (RDC) of a mixer-settler extractor is described.

Sequential binary arrays. I. The square grid

A periodic binary array is said to be sequential if and only if every line of the array is occupied by a given periodic binary sequence or by some cyclic shift or reversal of this sequence. Such arrays are of interest in connection with experimental layouts. This paper, which deals only with arrays built on the square grid, gives some constructions for sequential arrays and some results on the types of arrays possible for certain sequences. The x-step circulant arays are characterized. A technique for exhaustive search to find all possible arrays with a given sequence is outlined and a listing is given for square sequential arrays of period not exceeding seven.

On the classification of balanced binary sequences of period&amp;lt;tex&amp;gt;2^n-1&amp;lt;/tex&amp;gt;(Corresp.)

Let U be the set of all binary sequences of period p=2^{n}-1 containing (p+1)/2 ones and (p-1)/2 zeros per period. There is a lattice of interesting subsets of U , the smallest of which is the set PN (the maximum-length linear shift register sequences of period p ). In between are sets with the run statistics of PN , with the correlation properties of PN , with the n (that every nonzero subseqnonce of length n occurs in each period), and others. Results concerning the interrelationships of these subsets are obtained, examples are given to show that certain intersections of subsets are nonempty, and conjectures are formulated regarding other intersections of subsets. For example, it is conjectured that all span- n sequences with the two-level autocorrelation property are in class PN . Some relationships between run properties and correlation properties of binary sequences are also obtained.

Bounds on aperiodic cross-correlation for binary sequences

Results are presented which relate the mean-squared value of the aperiodic crosscorrelation function for periodic binary sequences to their aperiodic autocorrelation function. Bounds on the mean-squared aperiodic crosscorrelation are given in terms of the mean-squared aperiodic autocorrelation.

On the modulo-two-sum decomposition of binary sequences of finite periods

A method of decomposing finite periodic binary sequences into modulo-2-sum of several finite periodic binary sequences is presented. The method is extended to the linear feedback shift register realization of the given sequences which produces the minimum polynomials of the given sequences and the fractional polynomial representation of the sequences. The sequence decomposition problem is then handled algebraically. A necessary and sufficient condition for the decomposability of a sequence of finite period is obtained. A modulo-2-sum sequence decomposition procedure is given.

Estimating frequency-response function using periodic signals and the f.f.t.

A technique for estimating a frequency-response function using a periodic pseudorandom binary sequence which is compatible with the fast-Fourier-transform algorithm is presented. A method of synchronising the p.r.b.s. and the sampling for the f.f.t. using a single clock source and two binary counters is described. A simple check on the synchronisations and the identification of an 8th-order lowpass Butterworth filter is provided.

The design of feedback shift registers and other synchronous counters

This paper is concerned with digital sub-systems comprising bistable flip-flops connected to a common source of clock-pulses. This configuration, of which the feedback shift register is the most important special case, is useful for the generation of periodic binary codes and sequences. It is assumed that the building-blocks for these circuits are integrated J-K flip-flops and NAND gates, and design techniques appropriate for this situation are described.

A class of nonlinear error correcting codes based upon interleaved two-level sequences (Corresp.)

In this correspondence, we consider nonlinear binary error-correcting codes based upon the n -fold interleaving of a periodic binary sequence of length L with out-of-phase (normalized) autocorrelation -1/L . Examples of binary sequences with such out-of-phase autocorrelation (termed two-level sequences) are: 1) maximal-length linear shift-register \footnote[1]{sequences}, 2) twin-prime \footnote[2]{sequences}, 3) quadratic residue \footnote[3]{sequences}, and 4) Hall \foonote[4]{sequences}.