MANY-PARAMETER M-COMPLEMENTARY GOLAY SEQUENCES AND TRANSFORMS

1074 Computer Optics, 2018, Vol. 42(6) MANY-PARAMETER M-COMPLEMENTARY GOLAY SEQUENCES AND TRANSFORMS V.G. Labunets 1, V.P. Chasovskikh 1, Ju.G. Smetanin 2, E. Ostheimer 3 1 Ural State Forest Engineering University, Sibirskiy trakt, 37, Ekaterinburg, Russia, 620100, 2 Federal Research Center “Information and Control” of the RAS, Vavilov street 44 |(2), Moscow, Russia, 119333, 3 Capricat LLC, Pompano Beach, Florida, USA Abstract In this paper, we develop the family of Golay–Rudin–Shapiro (GRS) m-complementary manyparameter sequences and many-parameter Golay transforms. The approach is based on a new generalized iteration generating construction, associated with n unitary many-parameter transforms and n arbitrary groups of given fixed order. We are going to use multi-parameter Golay transform in Intelligent-OFDM-TCS instead of discrete Fourier transform in order to find out optimal values of parameters optimized PARP, BER, SER, anti-eavesdropping and anti-jamming effects.

In previous papers [7,8], we have shown a new unified approach to the GF(p) -, or Clifford-valued complementary sequences and Golay transforms. It was associated not with the triple (Z2, 2, C), but with triples   is a single transform, lg  is an algebra (for example, Clifford algebra).
In this work, we develop a new unified approach to the so-called generalized multi-parameter mcomplementary sequences. This construction has a rich algebraic structure. It is associated not with the triple   The rest of the paper is organized as follows: in Section 2, the object of the study (Golay -Rudin -Shapiro m-ary sequences) is described. In Section 3 we propose method based on new generalized iteration rule with n unitary (m×m)-transforms

The object of the study. New iteration construction for original Golay sequences
We begin by describing the original Golay mcomplementary sequences. [11]

Definition 1. A generalization of the Golay complementary pair, known as the Golay m-Complementary melement Set (m-GCS) of complex-valued sequences
be m n+1 -element set of m complementary sequences (of length m n+1 ), where n+1, tn+1 = 0, 1,.., m n+1 -1 They form rows of a (m n+1 ×m n+1 ) -matrix Let us to select the more fine structure of the m-Golay matrix: Example 1. For n = 1 and n = 2 we have, respectively, is constructed by an iteration construction. The initial matrix 1 [1] m G is formed by starting with an arbitrary unitary (m×m)-matrix (in manyparameter form or not) ...
It is easy to check that is true for an arbitrary unitary (orthogonal) matrix. Hence, and initial sequences in the form of rows of an unitary matrix (in particular case, in the form of characters comk(t1) = (1,  k1 ,  k2 ,…,  k(m-1) ) of cyclic group Zm) are the Golay m-complementary sequences.

Methods
The matrix is constructed by an iteration construction 2 3 1 and, consequently, Where 0, tn+2  0. New sequences in (9) are orthogonal and m-complementary sequences.

Generalizations
In this section, we introduce generalized mcomplementary sequences. It is based on using new permutation matrices n m  P in (7). The mappings g: XX of a set X into (or onto) itself are of particular importance. They form the following set X X : = {g|g: XX }.
Definition 2. One-to-one map from a set X to itself g: XX, x = g(x) = gx is called a transformation of the set X.
If X is finite and consists of m elements (for example, X = {0, 1, 2,…, m}) then a transformation of the set X is called a permutation. As is well known, the set of all permutations of X forms a group Sm = Sum{X} in which the product  of a pair of permutations   is defined by ()x: = (x).
If X contains more than two elements, Sm is not commutative. Any subgroup of Sm is called a permutation group on X, or a group of permutations of X. We shall say that the permutations in Sym(X) act or operate on the elements of X.

Definition 3. A homomorphism of a group on a set h: GrSym{X} is called a permutation representation (or realization) of .
The image h (Gr)  Sym{X} is a permutation group and the elements of are represented as permutations of . A permutation representation is equivalent to an action of on the set : To specify an action, we need to define for element gGr the corresponding permutation h(g) of , that is, h(g)x for any xX. We are going to write h(g)x in the short form gx and to call the group of transformations of . The pair  is called a space with transformation group the elements xX are called points of the space . Definition 4. If is a permutation group of degree , then the permutation representation of is the linear permutation representation of : P: GrGLm(lg) which maps to the corresponding permutation matrix P(g), .
That is, acts on by permuting the standard basis vectors {en}nXlg m such that ,..., ( 1) . m In particular, for m = 2 and m = 3we have In expression (7) It is associated with triple       (8) and (9) for m = 2 (i.e., expressions (6) and (7)      In this paper, we propose a simple and effective antieavesdropping and anti-jamming Intelligent OFDM system, based on MPTs. In our Intelligent-OFDM-TCS we are going to use multi-parameter Golay transform G 2 n(1, 2,…, q) at the place of DFT N. We are going to study of Intell-G 2 n(1, 2,…, q)-OFDM-TCS to find out optimal values of parameters optimized PARP, BER, SER, anti-eavesdropping and anti-jamming effects.