The authors construct new and improved sonar sequences by applying rotation, multiplication, and shearing transformations to Costas sequence constructions. A catalog of the best known sonar sequences with up to 100 symbols is given.>

An efficient and flexible multiple mainbeam architecture for performing adaptive sidelobe cancellation using a single set of n auxiliary antennas is described. The design can easily incorporate constrained and/or excess degrees-of-freedom processing if this proves necessary in a given application. The approach is to compute the QR decomposition of the auxiliary data matrix and then send this information to mainbeam processors, where the residual for each mainbeam is computed without explicitly calculating and applying auxiliary weights. A new fast algorithm and architecture for implementing constraints are presented. The proposed approach reduces the number of operations in the constrained mainbeam processor from order n/sup 3/ to order n/sup 2/.>

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.>

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.>

The algebraic-integer number representation, in which the signal sample is represented by a set of (typically four to eight) small integers, combines with residue number system (RNS) processing to produce processors composed of simple parallel channels. The analog samples must first be quantized into the algebraic-integer representation, and the final algebraic-integer result converted back to an analog or digital form. In between these two conversions, the algebraic-integer representation must be converted into and out of two levels of RNS parallelism. The authors address these quantization and conversion problems and demonstrate their solution by implementing a 128-tap algebraic-integer filter using the moduli 17 and 31, with four parallel channels per modulus. This processor performs equivalently to an integer processor with 19 bits of dynamic range. It is concluded that the algebraic-integer representation is best suited to control the dynamic range requirements of integer processors in situations where there is a high sensitivity to quantization and roundoff errors, especially when there is a matching nonuniform input distribution.>

Nondegenerate quadrics of PG(2l, 2^{s})have been used to construct ternary sequences of length(2^{2sl+1} - 1)/(2^{s} - 1)with perfect autocorrelation function. The same construction can be used for degenerate quadrics for this case as well as quadrics of PG(N,q), withNarbitrary andq = p^{s}, for any primep. This is possible because it is shown that ifQ \subseteq {\rm PG} (N, q)is a quadric, possibly degenerate, that has the same size as a hyperplane, then, providedQitself is not a hyperplane, the hyperplanes of PG(N,q)intersectQin three sizes. These sizes depend on whetherNis even or odd and the degeneracy ofQ. Finally, a connection to maximum period linear recursive sequences is made.

An algorithm is described that approximates complex numbers by elements of the algebraic integers ofZ[e^{2 \pi i / 8}]with integer coordinates of at most a prescribed size. The motivating application is to reduce the dynamic range requirements of residue number system implementations of the discrete Fourier transform. The closest points to zero ofZ[e^{2 \pi i / 8}]_{M}gor any integerMare determined. A particular sequence of such points forms the basis of the algorithm. An example of8-bitZ[\omega]_{M}- approximations of the 128th roots of unity is considered. The algorithm yieldsM = 186;with scalingMis reduced to18.

The problem of approximating complex numbers by elements ofZ[\omega], the algebraic integers ofQ(\omega), where\omegais a primitiventh root of unity, is considered. The motivating application is to reduce the dynamic range requirements of residue number system implementations of the discrete Fourier transform. Smallest error tolerances for the case of eighth roots of unity are derived using a geometric argument. Scale factors involved are reduced from\alphato\sqrt{\alpha}for this case with roughly the same percentage errors. The case of sixteenth roots of unity gives even better range reductions and is considered only briefly.

This paper is concerned with the construction of de Bruijn sequences of spann--binary sequences of period2^{n}in which every binaryn-tuple appears as somenconsecutive 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 spannand produce one of spann+1. We give a more general recursive construction which takes two de Brnijn sequences of spannand produces a de Bruijn sequence of spann+1. 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 spann+1with maximum complexity2^{(n+1)}-1are obtained from maximum complexity sequences of spann. Reverse-complementary de Bruijn sequences are also considered.

The complexity of a periodic sequence(s)is defined to be the least number of stages in a linear feedback shift register that generates(s).

The technique of error-trapping decoding for algebraic codes is studied in combinatorial terms of covering systems. Letn, k, andtbe positive integers such thatn \geq k \geq t > 0. An(n, k,t)-covering system is a pair(X, \beta), whereXis a set of sizenand\betais a collection of subsets ofX, each of sizek, such that for allT \subseteq Xof sizet, there exists at least oneB \in \betawithT\subseteq B. Letb(n, k, t)denote the smallest size of\beta, such that(X, \beta)is an(n, k, t)-covering system. It is shown that the complexity of an error-trapping decoding technique is bounded byb(n, k, t)from below. Two new methods for constructing small(n, k, t)-covering systems, the algorithmic method and the difference family method, are given.