Radio-Engineering Means of Transmission, Reception and Processing of Signals Formation of Minimax Ensembles of Aperiodic Gold Codes

Introduction . Signals constructed on the basis of ensembles of code sequences are widely used in digital communication systems. The development of such systems should be based on the analysis, synthesis and implementation of periodic signal ensembles. Although a number of theoretical methods have been developed for synthesizing periodic signal ensembles, there is a lack of approaches to constructing aperiodic signal ensembles. Aim . To construct aperiodic Gold code ensembles characterized by optimal values in terms of the ratio of their code length to volume among the currently known binary codes. Materials and methods . The methods of directed enumeration and discrete choice of the best ensemble based on unconditional preference criteria were used. Results. Full and truncated aperiodic Gold code ensembles with a given length and ensemble volume were constructed. The parameters and shape of auto- and mutual correlation functions were shown for a number of constructed ensembles. The obtained results were compared with those described in literature for periodic Gold code ensembles regarding an increase in the values of the minimax correlation function depending on the code length and ensemble volume. Conclusion. The developed algorithms can be used to obtain both full ensembles and ensembles truncated in volume. In addition, the developed algorithms can be extended to the tasks of forming ensembles from other families, e.g., assembled from code sequences belonging to different families.

Introduction. Signals constructed on the basis of ensembles of code sequences (CS) are widely used in discrete data transmission radio systems, as well as in radar and radio navigation systems. A prospective research direction in the development of such systems consists in the selection of ensembles exhibiting good correlation properties [1].
Recently, much research effort has been devoted to the creation of ensembles of polyphase and com-plementary CS, as well as those with a zero correlation zone [1]. To this end, analytical synthesis methods [2][3][4], genetic [5] or evolutionary algorithms [6], as well as various modifications of these computational procedures [7] are widely applied. At the same time, complementary CS are characterized by such limitations, as the number of sequences in the set, available sequence lengths and the requirement of power amplifier linearity. The latter also pertains to polyphase CS. Therefore, the interest in binary CS is still growing. However, the problem of designing sequence families of the required size, sequence length and aperiodic properties remains to be solved [1].
In contemporary radio systems, both periodic and aperiodic discrete signals are implemented. During their simultaneous transmission in the common frequency band, minimization of mutual interference is of great importance, which is achieved by CS minimax ensembles, i.e. CS ensembles optimal in terms of the minimax criterion [8].
For periodic discrete signals, there exist methods for synthesizing CS minimax ensembles [8] and for estimating values of periodic autocorrelation functions (PACF) and periodic cross-correlation functions (PCCF) achievable by such ensembles. However, for aperiodic CS ensembles, there is not only a lack of a versatile method for synthesizing binary phase-manipulated signals optimal by the minimax criterion, but also it is unclear whether the known signals with a large number of N positions are close to optimal [9].
Moreover, as indicated in [9], despite intensive studies, all existing methods for synthesizing single aperiodic CS include enumeration as one of the stages. Among them are [9, 10]: 1. Directed enumeration method. All the binary Barker sequences, ternary aperiodic quasi-orthogonal CS, including ternary Barker sequences of 31 N < , were obtained using this method. The minimum peak value of the side lobes for the aperiodic autocorrelation function (AACF) was determined up to [11]. The directed enumeration method involves two stages. The first stage is focused on narrowing the enumerated area, consisting in formulating the necessary conditions and admissible combinatorial parameter relations. The second stage involves the development of efficient enumeration algorithms.
2. Synthesis of aperiodic CS on the basis of periodic CS. This method is based on the relationship of AACF and its ( ) can be obtained. Thus, sequences with "good" AACFs can only be found among sequences with "good" PACF [12]. This approach also comprises two stages. The first implies a search for a CS with a "good" PACF. The second implies a search for initial conditions optimal by the minimax criterion. Following this approach, binary and ternary sequences optimal by the minimax criterion can be obtained.
3. Synthesis of CS signals according to a given AACF. Depending on the criterion used and the method for calculating deviations, the following methods are distinguished [9]: -uniform approximation method; -minimum RMSD method; -coordinate descent method; -minimum generalized mean deviation method; -asymptotic synthesis method. All these methods refer to the group of iterative methods and imply a laborious search process. The best results of binary sequence synthesis are significantly inferior to the minimax AACF side lobes synthesized by one of the previously mentioned methods.
As noted in [9], the problem of improving these methods consist in the synthesis of CS with "good" PACF, as well as in reducing the duration of the enumeration stage.
Problem statement. The method of constructing aperiodic CS on the basis of periodic CS can also be applied for constructing minimax ensembles of aperiodic CS.
For minimax ensembles of binary periodic CSs, estimates are known, depending on the N code length and the K ensemble volume. For a number of popular ensembles, such estimates are provided in [12] (Table 1). As can be seen from Table 1, for large K values close to N, Gold codes are recommended. For example, only Gold codes with a length of and above and Kasami ensembles with a length of and above satisfy the required ensemble volume. This determined the choice of a Gold code ensemble in the present study for subsequent construction of an aperiodic CS minimax ensemble, although, for periodic Gold codes, the estimate is somewhat worse than for other ensembles provided in Table 1. The present article aims to construct minimax ensembles of aperiodic Gold code sequences with a volume close to the code length. It should be noted that Gold ensembles are extremely popular in contemporary code division multiple access (CDMA) systems including, in particular, GPS, UMTS, etc. [13]. Such ensembles are applied for extending sequences converting an information signal into a The normalized AACF of the sequence is defined as [12]: is the Euclidean norm, the same for all code vectors of is the energy of each ki a code sequence.
The normalized ACCF for and two sequences of the same length is 1 , 0.   Algorithm for constructing a minimax ensemble of aperiodic Gold codes. On the basis of the definition provided above, this section describes a procedure for constructing a minimax ensemble, which consists of the following operations: -formation of Gold code ensembles; -choice of the best Gold ensemble from the set on the basis of the two-criteria choice algorithm [17].
According -all the n-degree primitive polynomials forming m-sequences are obtained; -each of the obtained m-sequences is decimated by the q decimation coefficients presented below; -from the m-sequences obtained after decimation, the polynomials generating them are obtained using the Berlekamp-Massey algorithm [16]; -mirror copies of the obtained polynomial pairs are discarded.
The remaining pairs represent the preferred pairs. In order to obtain the decimation coefficient, the following relationship between the roots of some primitive polynomials is used: the roots of the one polynomial are the q-th degree roots of anoth- When determining pairs of m-sequences for constructing a Gold ensemble, ensembles based on cyclic shifts of the original m-sequences were not considered. Enumeration of the shifted CS can serve as an additional reserve for optimization.
Let us consider the problem of compiling preferred m-sequence pairs using the example of a 127 N = length sequence generated by a primitive polynomial of the 7 n = degree. For , 18 primitive polynomials are available, 9 of which are provided in [15]. The other nine represent their mirror polynomials. All 18 polynomials correspond to mutually inverse m-sequences, i.e. m-sequence pairs, connected by a decimation coefficient: Thus, by the n degree of the primitive polynomial, all possible ensembles of Gold codes are determined. This article restricts to 6, 7, 9 and 10. The calculated parameters for these n values are given in Table 2.
Subsequently, for the received Gold code ensembles, the and , , values are determined by enumeration. The choice of the best Gold ensemble in terms of the minimax criterion represents the task of a twocriterion discrete choice. To this end, the resulting set is divided into sets of "worst" and "not worst" cases by applying an unconditional preference criterion. Next, the rectangle method [17] is applied illustrated in Fig. 1 for and 511. The rectangle method consists of the following: 1. The and indicators are plotted along the axes of the coordinate plane in ascending order. Results. In accordance with the described algorithms, full and truncated Gold ensembles were constructed for the parameters indicated in Table 2. The selection of optimal minimax ensembles was performed. In Table 3 Table 3, where, for , two sets of minimax values are given. Fig. 2 demonstrates the superimposed AACF of minimax full and truncated ensembles from Table 3 for 127 N = with analogous AACF for 511 N = represented in Fig. 3. Fig. 4 shows the superimposed ACCF of the Gold code ensemble selected by the algorithm [17].   N   5  31  6  15  12  6  63  6  15  6  7  127  18  153  90  9  511  48  1128  288  10 1023 60 1770 300 Discussion. The application of the widely-used method for constructing Gold ensembles followed by selecting the best ensemble on the basis of the unconditional preference criterion [17] allowed minimax ensembles of aperiodic Gold codes to be constructed. The developed algorithms ensure the construction of both full and limited in volume (truncated) ensembles. In addition, these algorithms can be extended to the problem of constructing ensembles from other families, e.g., assembled from CS belonging to different families.

K N
Let us compare the obtained values characterizing minimax ensembles of aperiodic Gold CS with those published in literature.
1. There exist fundamental restrictions on the side lobes of AACF. For arbitrary single binary CS, the   (Table 4 [ 11]). According to Fig. 5, for small code lengths 63, 127), the minimax values for the AACF lobes of the full Gold ensembles obtained in this article significantly exceed those specified in Table 4. However, as compared to K, a decrease in the volume of ensembles to leads to a sharp decrease in minimax lobes.
Due to the absence of available literature data, it was impossible to compare the obtained minimax values for the side lobes of full ensembles with the absolutely minimal normalized values of for and 1023. However, it should be noted that, for these lengths, the values of the minimax lobes decrease significantly with the limited ensemble length in comparison with 63, 127. In addition, for insignificant lengths, a decrease in the ensemble volume, in comparison with the full one, provides no significant decrease in minimax lobes. At the same time, when obtaining, e.g., a c 100 K = ensemble, then, for 1023, the values of minimax lobes are close to those indicated in Table 4 for 2. In [8], periodic CS ensembles minimax by the criterion are considered: (1) where is the PCCF of the k and l sequences; is the PACF of the k-th sequence.
For volumes and odd n, the Grz lower boundaries for periodic binary Gold CSs satisfying (1) coincide with the values of max p ρ obtained from Table 1, i.e. these ensembles are strictly optimal by the minimax criterion (1). 3. A number of works (e.g., [4]) discuss the possibility of changing the maximum level of side lobes 1 .