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Method of synthesis of ensembles of wide-band signals on the basis of orthogonal unitary matrices forms of the Fourier transform

DOI 10.18127/j00338486-201906(7)-12

Keywords:

A.Yu. Grishentsev – Dr.Sc.(Eng.), Associate Professor, Saint Petersburg National Research University of Information Technologies, Mechanics and Optics (ITMO University)
E-mail: grishentcev@yandex.ru


A new class of matrices with a special form of autocorrelation function having a significantly dominant Central splash over the side petals is proposed and investigated in this paper. Matrices from the proposed class may be obtained by producing orthogonal matrices on a unitary Fourier transform matrix. The proposed matrix class allows to synthesize alphabets of mutually orthogonal code wideband signals, including for multi-component systems (CDMA).
The object of the study is autocorrelation and cross-orthogonal matrices. The subject of the study is the properties and methods of synthesis of autocorrelation and cross-orthogonal matrices.
The paper proposes a method for the synthesis of square matrices of size N×N having high autocorrelation properties, in terms of the ratio of the Central splash to the side petals, with the possibility of the synthesis of orthogonal sets of power equal to N2 that the As-sociation of such matrices with space frequency-time.
The synthesis method is based on the element product of unitary Fourier transform matrices of a given size N×N on matrices obtained on the basis of Walsh orthogonal functions. Thus, it is generated a new class of matrices forming an orthogonal form of the unitary matrices Fourier transform.
In this paper, two theorems are formulated and proved: the first on the number of orthogonal matrices, and the second on the formation of a commutative Abelian group by orthogonal matrices.
It is shown that when comparing the sizes Of N×N orthogonal matrices with the dimensions of the space, the frequency-time W×T can obtain a set of mutually orthogonal codes contributing to the increase in the speed of the transmitted information.
The cumulative results of the studies show that the proposed matrix class can be effectively used for the synthesis of orthogonal sets of signal alphabets of code division signals, including multi-component systems (CDMA).
The work consists of the following sections: «Summary»; «Introduction»; «Ensembles of orthogonal matrices with a special form of autocorrelation»; «Properties and finding orthogonal matrices»; «An example of the synthesis of wideband signals on the basis of or-thogonal unitary matrices forms the Fourier transform of»; «Conclusion of conclusions».
The section «Summary» contains a brief description of the research and the results contained in the publication.
Section «Introduction» contains the justification of the relevance of the research topic, formulation of research objectives.
In the section «Ensembles of orthogonal matrices with a special form of autocorrelation», the theorem on the power of a set of mutually pairwise orthogonal matrices obtained on the basis of unitary Fourier transform matrices is formulated and proved.
In the section «Properties and finding orthogonal matrices», a theorem is formulated and proved that every set of orthogonal matrices generated by Walsh vectors forms a finite commutative Abelian group. The Euclidean distance between orthogonal matrices of the same size is estimated.
In the section «Example of synthesis of wideband signals based on orthogonal forms of unitary Fourier transform matrices», examples of signal synthesis are given, as an example, graphs of auto - and cross-correlation functions are constructed.
In the section «Conclusion of conclusions», conclusions on the work.
The work is performed on the basis of the Federal state Autonomous educational institution of higher education «St. Petersburg national research University of information technologies, mechanics and optics» (ITMO University) within the research methods of analysis and synthesis of broadband signals.

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