Leopold Bömer

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 >

Complex sequences over GF(p/sup M/) with a two-level autocorrelation function and a large linear span

New complex sequences with elements are proposed that have constant absolute values of 1. The periodic autocorrelation functions of these sequences are shown to be two-level. The sequences are generated by three consecutive mapping processes. The number of sequences of fixed length is determined, which is larger than the number of binary sequences. For cryptographical reasons, the large linear span of the sequences is of great interest. The linear span of the new sequences is examined and is proven to be larger than the linear span of complex m-sequences, if the parameters of the mapping processes are appropriately chosen. The formula for generating the sequences is generalized to enlarge the linear span of sequences. >

Perfect N-phase sequences and arrays (spread spectrum communication)

Perfect sequences and arrays have periodic autocorrelation functions whose out-of-phase values are zero. Time-discrete N-phase sequences and arrays have complex elements of magnitude one, and one of (2 pi /N)n, 0 >

Perfect binary arrays

Abstract Perfect arrays have an impulse-like periodic autocorrelation function. Necessary conditions for the existence of perfect binary arrays are derived. With three specified construction procedures and based on few starting arrays, many new perfect binary arrays with increasing dimensions and sizes can be synthesized. The first construction method is based on a periodic multiplication of arrays in different dimensions. The further methods represent a specified Kronecker product and higher-dimensional folding and refolding of arrays. Applications are found for example in synchronization, transform coding, channel coding or synthetic aperture imaging.

Perfect ternary arrays

Signals of two or more dimensions with ideal impulse-like periodic autocorrelation functions are used in higher-dimensional signal processing or in radar systems. Four synthesizing methods for perfect ternary arrays are presented. Based on known perfect binary and ternary sequences and arrays, many new perfect ternary arrays with increasing number of elements are constructed. New strong conditions for the existence of perfect arrays are developed. By combining these conditions with an advanced computer search, new ternary arrays are found. These new ternary arrays can be used as starting arrays to construct additional perfect ternary sequences and arrays. >

Two-dimensional perfect binary arrays with 64 elements

Perfect binary arrays can exist for numbers of elements that are even square numbers. Two-dimensional (2-D) perfect binary arrays are 4, 16, 36, and 144 elements are known. Perfect binary arrays with 64 elements can be constructed for three or more dimensions. Here, two-dimensional binary arrays with 64 elements are examined. The results of computer search for symmetric binary arrays with 64 elements are presented in the form of two 2-D perfect binary arrays with sizes 8*8 and 4*16. Applications of these perfect binary arrays are in 2-D synchronization, time-frequency coding, and coding loudspeaker fields. >

Perfect three-level and three-phase sequences and arrays

Introduces construction methods for synthesizing new classes of perfect sequences and arrays. Time discrete sequences and arrays are perfect, if their periodic autocorrelation function sidelobes are zero. The construction of perfect three-level and three-phase sequences is performed in two steps. In the first step, a maximal length shift-register sequence with elements in GF(q) is built for a length of g/sup m//spl minus/1, where q is a prime power and m is a positive integer. In the second step, a sequence is constructed by mapping the elements of the shift-register sequence to 1, b/sub 1/ or b/sub 2/. Two real numbers b/sub 1/ and b/sub 2/ are determined such that the sequence becomes perfect and has high energy efficiency. The phase values for complex numbers b/sub 1/ and b/sub 2/ with magnitude 1 are given for perfect three-phase sequences. It is shown that the phase values of b/sub 1/ and b/sub 2/ tend for growing lengths to 2/spl pi//3 and 4/spl pi//3. A different similar synthesizing method starts with Legendre sequences. The construction of perfect three-level and three-phase arrays is performed by several methods, which make use of the new respective sequences. >

Binary and biphase sequences and arrays with low periodic autocorrelation sidelobes

Binary sequences and arrays with the highest possible PACF merit factor, which is a measure for a low PACF (periodic autocorrelation function) sidelobe energy, are investigated. It is shown that some of these sequences and arrays can be obtained with known construction schemes; others are found by methods of complete computer search. The resulting PACF merit factors are given in a table up to 39 elements. A transformation from pseudonoise sequences into biphase sequences is introduced for the construction of a new class of biphase sequences and arrays with zero autocorrelation sidelobes.<<ETX>>

Quadratic residue arrays

In this paper, a new construction method for synthesizing twoand higher-dimensional quadratic residue arrays is introduced. The construction is mainly based on addressing the elements of an m-dimensional array by the elements of Gajois Field GF(p), where p denotes an odd prime and m denotes an integer. With a following mapping, the elements of the m-dimensional array, except the leading element, are set to either +1 or — 1. The dimensions of the resulting m-dimensional arrays have the size p. It is shown that the periodic autocorrelation function of these quadratic residue arrays are twoor three-level, depending on whether the number of elements is N = 1 mod 4 or N = 3 mod 4. With these quadratic residue arrays, new classes of twoand higher-dimensional pseudonoise and Legendre arrays are obtained. Übersicht: Es wird ein neues Konstruktionsverfahren zur Erzeugung von zweiund höher-dimensionale quadratische Residuen-Arrays vorgestellt. Die Konstruktion basiert auf der Adressierung der Elemente eines m-dimensionalen Arrays unter Verwendung der Elemente des Galois-Fcldcs GF(p), wobei p eine ungerade Primzahl und m einen Integer darstellt. Durch eine eingeführte Abbildung bekommen bis auf das erste Element die Elemente des m-dimensionalen Arrays den Wert +1 oder — l zugeordnet. Die Abmessung jeder Dimension der erzeugten Arrays beträgt den Primzahlwcrt p. Es wird gezeigt, daß quadratische Residuen-Arrays eine zweioder dreiwertige periodische Autokorrelationsfunktion annehmen, je nachdem, ob die Anzahl der Elemente N die Kongruenz N = l mod 4 oder N = 3 mod 4 erfüllt Die Quadratic Residue Arrays stellen eine neue Klasse von zweiund höherdimensionalen Pseudonoiseund Legendre-Arrays dar. Für die Dokumentation: Zweiund höherdimensionale Arrays / Pseudonoise-Arrays / Legendre-Arrays / binäre Arrays / periodische Autokorrelationsfunktion

Long energy efficient Huffman sequences

An iterative method for increasing the energy efficiencies of given Huffman sequences is introduced. The iteration scheme works by varying the roots and the radius of the Z-transformation of Huffman sequences. The algorithm allows the intensity of the variations of the possible root positions to be changed. In this way the energy efficiencies of long real and complex valued Huffman sequences can be optimized. The algorithm has been applied to known Huffman sequences. The resulting efficiencies of real and complex valued Huffman sequences are given up to 100 elements.<<ETX>>