RU2809479C1 - Method of information transmission using mimo scheme in hypercomplex space - Google Patents

Abstract

FIELD: radio engineering.

SUBSTANCE: in the method of transmitting information using the MIMO scheme in N=2ⁿ, n=2, 3, 4 hypercomplex space, sequences of binary data pulses consisting of zeros and ones arriving at the input of the transmitting device are converted into bipolar pulses, which are converted into N -dimensional information vector consisting of N information pulses and forming a hypercomplex number in a vector representation, while modulation on the transmitting side is carried out by multiplying the information vector by a hypercomplex carrier, which is a hypercomplex number in polar form with different N-1 reference frequencies in a matrix representation and acting as a channel matrix MIMO channel, while the multi-frequency channel matrix MIMO is decomposed into the sum of single-frequency matrices, which are modulated separately and their results are added up, forming multi-frequency elements of an N-dimensional vector, while the phase-modulated elements of the output multi-frequency modulated vector are transmitted sequentially in time either with code division or with frequency division, while the signal vectors are coherently demodulated by multiplying the received vector by transposed basis matrices of single-frequency matrices representing an orthogonal basis, and the reception of the signal vector arriving at the receiver input is carried out using the integration of the results of the multiplication followed by addition for each single-frequency matrix of the orthogonal basis and making a decision in the decision device about the value of the vector elements, while the hypercomplex carrier is a hypercomplex number in polar form and in matrix representation in the form of a square NxN matrix with N-1 reference frequencies, combinations of which form 2^N-2 combinational frequencies, and the same number of single-frequency matrices.

EFFECT: increased noise immunity and information transmission rate in wireless and wired communication systems.

2 cl, 33 dwg

Description

The invention relates to information transmission systems, in particular methods for implementing the MIMO scheme [H04B7/0413].

Known from the prior art is a MIMO WIRELESS COMMUNICATION METHOD AND SYSTEM USING JOINT HADAMARD CONVERSION AND ALAMUTI SCHEME (KR20160010799(A), published 01/28/2016) including: modulating data into digital symbols, performing a Hadamard transform of modulated symbols to generate a new constellation; and encoding the generated constellation using the Alamouti method as input. In this modulation step, the QPSK scheme is modulated by a combination of the Hadamard transform and the Alamouti scheme. In this transformation step, two points of the modulated symbol are combined into one symbol using a combination of the Hadamard transform and the Alamouti scheme. In this transformation step, a new population is created by the sum or difference of two points using a combination of the Hadamard transform and the Alamouti method. In this case, the MIMO wireless communication system contains: a modulator for modulating data into digital symbols on the transmission side of the MIMO wireless communication system; a converter for Hadamard transform of the modulated symbol to create a new constellation; and an encoder for encoding the generated constellation as input using the Alamouti method. Here, the modulator is a MIMO wireless communication system with QPSK modulation using a combination of Hadamard transform and Alamouti transform, and the converter combines two modulated symbol points into one symbol by combining the Hadamard transform and Alamouti scheme. In this case, the converter generates a new constellation by summing or subtracting two points.

The disadvantage of analogue is the low speed of information transfer.

There is also a known METHOD FOR DATA TRANSMISSION IN THE "MIMO" COMMUNICATION SYSTEM (RU2553679C2, publ. 06/20/2015) using space-time coding, which uses a mapping table that maps a set of symbols to antennas and to transmission resources, which can be time slots or sections of the OFDM frequency band. The mapping table contains nested primary Alamouti code segments, namely the character-level Alamouti encoding, within secondary segments, which may contain the Alamouti encoding of the primary segments.

The disadvantage of analogue is the low speed of information transfer.

The closest in technical essence is the DATA TRANSMISSION METHOD USING SPATIO-TEMPORAL BLOCK CODES (US2008063110 A1, publ., 03/13/2008) which is characterized by the fact that for data transmission a set of at least 2 ^m nxn matrices is provided, which represent an extension of the group no fixed points. Each of the 2 ^m matrices is assigned a unique binary number ranging from 0 to 2 ^m -1, thereby allowing data to be identified and displayed according to a specific distribution. For optimal transmission, matched matrices are recommended to be sent using n antennas, with each antenna corresponding to one row of each matrix, which provides multi-dimensional space-time coding with maximum data transmission bandwidth and reliability.

The main technical problem of the prototype is limited throughput, since the use of the Space-Temporal Block Codes (STBC) method in a multiple-input multiple-output (MIMO) system requires the transmission of additional information to encode data blocks, which leads to an increase in transmission duration. This limits system throughput and may result in poor performance in time-constrained environments. As a consequence, low information transfer speed. Another technical problem of the prototype is low noise immunity, since the Space Time Block Code (STBC) method can be sensitive to interference and distortion, especially in negatively correlated MIMO antenna channels. This can lead to deterioration in the quality of received data and increase the likelihood of errors during decoding, resulting in low noise immunity.

The objective of the invention is to eliminate the disadvantages of analogues and the prototype.

The technical result of the invention is to increase noise immunity and information transmission speed in wireless and wired communication systems.

The specified technical result is achieved due to the fact that the method of transmitting information using the MIMO scheme in N=2 ⁿ , n= 2, 3, 4 hypercomplex space, characterized by the fact that the sequences of binary data pulses consisting of zeros and units, are converted into bipolar pulses, which are converted into an N-dimensional information vector, consisting of N information pulses and forming a hypercomplex number in a vector representation, while modulation on the transmitting side is carried out by multiplying the information vector by a hypercomplex carrier, which is a hypercomplex number in polar form with different N-1 reference frequencies in matrix representation and acting as a MIMO channel matrix of the channel, while the multi-frequency MIMO channel matrix is decomposed into the sum of single-frequency matrices, which are modulated separately and their results are added up, forming multi-frequency elements of the N-dimensional vector, while modulated in phase, the elements of the output multi-frequency modulated vector are transmitted sequentially in time or with code division, or with frequency division, while the signal vectors are coherently demodulated by multiplying the received vector by transposed basis matrices of single-frequency matrices, which represent an orthogonal basis, and the reception of the vector arriving at the input of the receiver signal is carried out using the integration of the results of multiplication followed by addition for each single-frequency matrix of the orthogonal basis and making a decision in the solver about the value of the vector elements.

In particular, a hypercomplex carrier is a hypercomplex number in polar form and in matrix representation as an NxN square matrix with N-1 reference frequencies, the combinations of which form 2 ^N-2 combination frequencies, and the same number of single-frequency matrices.

Brief description of the drawings.

In fig. Figure 1 shows vectors of information pulses in the form of a quaternion for a positive value of the first element of the information vector.

In fig. Figure 2 shows vectors of information pulses in the form of a quaternion for the negative value of the first element of the information vector.

In fig. Figure 3 shows a three-frequency phase-modulated vector with information vector X0 [11-11] No. 2.

In fig. Figure 4 shows a three-frequency phase-modulated vector with information vector X0=[1-1-11] No. 6.

In fig. Figure 5 shows single-frequency phase-modulated vectors with information vector X0=[11-11] No. 2.

In fig. Figure 6 shows single-frequency phase-modulated vectors with information vector X0=[1-1-11] No. 6.

In fig. Figure 7 shows the rotation orbit of the scalar part of the three-frequency quaternion for the initial vector X0=[11-11].

In fig. Figure 8 shows the rotation orbit of the scalar part of the three-frequency quaternion for the initial vector X0=[1-1-11].

In fig. Figure 9 shows the rotation orbit of the scalar part of the three-frequency quaternion for the initial vector X0=[-11-11].

In fig. Figure 10 shows the rotation orbit of the scalar part of the three-frequency quaternion for the initial vector X0=[-1-1-11].

In fig. Figure 11 shows the result of demodulation using the basis matrix $${\text{E}}_1\left({\Omega }_1,t\right)$$ with frequency $${\Omega }_1={\omega }_i+{\omega }_j+{\omega }_k$$ .

In fig. Figure 12 shows the result of demodulation using the basis matrix $$I_1\left({\Omega }_1,t\right)$$ with frequency $${\Omega }_1={\omega }_i+{\omega }_j+{\omega }_k$$ .

In fig. Figure 13 shows the result of demodulation using the basis matrix $${\text{J}}_1\left({\Omega }_1,t\right)$$ with frequency $${\Omega }_1={\omega }_i+{\omega }_j+{\omega }_k$$ .

In fig. Figure 14 shows the result of demodulation using the basis matrix $$K_1\left({\Omega }_1,t\right)$$ with frequency $${\Omega }_1={\omega }_i+{\omega }_j+{\omega }_k$$ .

In fig. Figure 15 shows the sum of values for the basis matrices with frequency $${\Omega }_1={\omega }_i+{\omega }_j+{\omega }_k$$ .

In fig. Figure 16 shows the result of demodulation using the basis matrix $${\text{E}}_2\left({\Omega }_2,t\right)$$ with frequency $${\Omega }_2={\omega }_i+{\omega }_j-{\omega }_k$$ .

In fig. Figure 17 shows the result of demodulation using the basis matrix $$I_2\left({\Omega }_2,t\right)$$ with frequency $${\Omega }_2={\omega }_i+{\omega }_j-{\omega }_k$$ .

In fig. Figure 18 shows the result of demodulation using the basis matrix $$J_2\left({\Omega }_2,t\right)$$ with frequency $${\Omega }_2={\omega }_i+{\omega }_j-{\omega }_k$$ .

In fig. Figure 19 shows the result of demodulation using the basis matrix $$K_2\left({\Omega }_2,t\right)$$ with frequency $${\Omega }_2={\omega }_i+{\omega }_j-{\omega }_k$$ .

In fig. Figure 20 shows the sum of values for the basis matrices with frequency $${\Omega }_2={\omega }_i+{\omega }_j-{\omega }_k$$ .

In fig. Figure 21 shows the result of demodulation using the basis matrix $${\text{E}}_3\left({\Omega }_3,t\right)$$ with frequency $${\Omega }_3={\omega }_i-{\omega }_j+{\omega }_k$$ .

In fig. Figure 22 shows the result of demodulation using the basis matrix $$I_3\left({\Omega }_3,t\right)$$ with frequency $${\Omega }_3={\omega }_i-{\omega }_j+{\omega }_k$$ .

In fig. Figure 23 shows the result of demodulation using the basis matrix $$J_3\left({\Omega }_3,t\right)$$ with frequency $${\Omega }_3={\omega }_i-{\omega }_j+{\omega }_k$$ .

In fig. Figure 24 shows the result of demodulation using the basis matrix $$K_3\left({\Omega }_3,t\right)$$ with frequency $${\Omega }_3={\omega }_i-{\omega }_j+{\omega }_k$$ .

In fig. Figure 25 shows the sum of values for the basis matrices with frequency $${\Omega }_3={\omega }_i-{\omega }_j+{\omega }_k$$ .

In fig. Figure 26 shows the result of demodulation using the basis matrix $$E_4\left({\Omega }_4,t\right)$$ with frequency $${\Omega }_4={\omega }_i-{\omega }_j-{\omega }_k$$ .

In fig. Figure 27 shows the result of demodulation using the basis matrix $$I_4\left({\Omega }_4,t\right)$$ with frequency $${\Omega }_4={\omega }_i-{\omega }_j-{\omega }_k$$ .

In fig. Figure 28 shows the result of demodulation using the basis matrix $$J_4\left({\Omega }_4,t\right)$$ with frequency $${\Omega }_4={\omega }_i-{\omega }_j-{\omega }_k$$ .

In fig. Figure 29 shows the result of demodulation using the basis matrix $$K_4\left({\Omega }_4,t\right)$$ with frequency $${\Omega }_4={\omega }_i-{\omega }_j-{\omega }_k$$ .

In fig. Figure 30 shows the sum of values for the basis matrices with frequency $${\Omega }_4={\omega }_i-{\omega }_j-{\omega }_k$$ .

In fig. Figure 31 shows the sum of values for all frequencies.

Fig. 32 is a functional block diagram showing the structure of an exemplary wireless transmitter.

Fig. 33 is a functional block diagram showing the structure of an example wireless receiver.

The figures indicate: 1 – source of information, 2 – hypercomplex number encoder, 3 – hypercomplex number modulator, 4 – transmitting antenna, 5 – receiving antenna, 6 – hypercomplex number demodulator, 7 – hypercomplex number decoder, 8 – information receiver.

Implementation of the invention.

Let's consider a method for transmitting information using a MIMO scheme in a hypercomplex space using a quaternion.

A quaternion represents a hypercomplex number having three imaginary units i, j, k. In algebraic form we write the quaternion as

$$q=s+xi+yj+zk$$ , (1)

where s , x , y , z are real numbers.

Let's present operations with imaginary units in the form of a table.

Table 1. Operation of multiplication of quaternion imaginary units.

$$\begin{array}{ccccc}\times & 1& i& j& k\\ 1& 1& i& j& k\\ i& i& -1& k& -j\\ j& j& -k& -1& i\\ k& k& j& -i& -1\end{array}$$ .

As is known, in problems of radio engineering and information transmission, real signals are used, therefore, in order to get rid of imaginary units, we present the quaternion in algebraic form (1) in matrix form:

$$Q=\left[\begin{array}{cccc}s& x& y& z\\ -x& s& -z& y\\ -y& z& s& -x\\ -z& -y& x& s\end{array}\right]$$ . (2)

Matrix determinant $$\left|Q\right|={\left(s^2+x^2+y^2+z^2\right)}^2$$ . Matrix Q is orthogonal because

$$Q^{\mathrm{T}}Q/\left|Q\right|=Q{Q}^{\mathrm{T}}/\left|Q\right|=E$$ .

We write the imaginary units in the matrix representation for matrix (2) as

$$I=\left[\begin{array}{cccc}0& 1& 0& 0\\ -1& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 0& 1& 0\end{array}\right]$$ , $$J=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ -1& 0& 0& 0\\ 0& -1& 0& 0\end{array}\right]$$ , $$K=\left[\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& -1& 0\\ 0& 1& 0& 0\\ -1& 0& 0& 0\end{array}\right]$$ . (3)

Matrices of imaginary units together with an identity matrix with units on the main diagonal $$E=\mathrm{diag}\left[\begin{array}{cccc}1& 1& 1& 1\end{array}\right]$$ , are called quaternion basis matrices. The quaternion matrix (2) is written using the basis matrices (3) and the identity matrix E as

$$Q=Es+Ix+Jy+Kz$$ . (4)

For the basis matrices (3), a multiplication table similar to Table 1 is valid:

Table 2. The operation of multiplying quaternion imaginary units in matrix representation.

$$\begin{array}{ccccc}\times & E& I& J& K\\ E& E& I& J& K\\ I& I& -E& K& -J\\ J& J& -K& -E& I\\ K& K& J& -I& -E\end{array}$$ .

For the quaternion (1) we write the exponential function as

$${\mathrm{e}}^q={\mathrm{e}}^{s+xi+yj+zk}={\mathrm{e}}^s\left(\cos x+i\sin x\right)\left(\cos y+j\sin y\right)\left(\cos z+k\sin z\right)$$ . (5)

Note that when $$x=y=z$$ expression (5) will take the form:

$${\mathrm{e}}^q={\mathrm{e}}^{s+x(i+j+k)}={\mathrm{e}}^s{\mathrm{e}}^{x(i+j+k)}={\mathrm{e}}^s{\mathrm{e}}^{x\widehat{i}}={\mathrm{e}}^s\left(\cos x+\widehat{i}\sin x\right)$$ ,

Where $$\widehat{i}=(i+j+k)/\sqrt{3}$$ .

Just as for a complex number in exponential display, the coefficients for the imaginary units of the quaternion in the polar form of notation have a physical meaning of the rotation angle. Therefore, we write the angles in (5) as functions of time: $$x(t)={\omega }_{i}t$$ , $$y(t)={\omega }_{j}t$$ , $$z(t)={\omega }_{k}t$$ , Where $${\omega }_i$$ , $${\omega }_j$$ , $${\omega }_k$$ - angular frequencies on axes i , j , k . Let's assume that $${\mathrm{e}}^s=r=\sqrt{s^2+x^2+y^2+z^2}$$ - radius of rotation in 4D space.

After multiplying the expressions in parentheses in formula (5) and grouping by the real and imaginary parts, we obtain an exponential function in the form of a three-frequency quaternion, which we denote as

$$f\left(q(t)\right)=p(t)+iu(t)+jv(t)+kw(t)$$ . (6)

After bringing similar terms, the components in expression (6) of the exponential function of the three-frequency quaternion will take the form:

We write the exponential function of the three-frequency quaternion (5) in matrix form using the obtained expressions (7), as

With matrix representation (8) of expression (6), we obtain a three-frequency fundamental matrix $$\Phi ({\omega }_i,{\omega }_j,{\omega }_k,t)$$ :

We represent the basis matrices (9) in the form (4):

Let us write the decomposition of the fundamental matrix (9) into basis matrices as

Expression (5) for a unit radius can also be represented in matrix form, using single-frequency matrices:

Multiplying the single-frequency basis matrices (12), we obtain matrix (9).

Thus, the fundamental matrix of a 3-frequency quaternion (9) can be represented as a sum of basic 3-frequency matrices (11) or as a product of single-frequency matrices (12).

Let us consider the main properties of the fundamental matrix of a three-frequency quaternion in the form of a sum (11) and a product (12). Fundamental matrix (9) is orthogonal, since

$${\Phi }^{\mathrm{T}}({\omega }_i,{\omega }_j,{\omega }_k,t)\Phi ({\omega }_i,{\omega }_j,{\omega }_k,t)=\Phi ({\omega }_i,{\omega }_j,{\omega }_k,t){\Phi }^{\mathrm{T}}({\omega }_i,{\omega }_j,{\omega }_k,t)=E$$ .

The determinant of matrix (9) is calculated using (7) as

$$\left|\Phi ({\omega }_i,{\omega }_j,{\omega }_k,t)\right|={\left(p^2({\omega }_i,{\omega }_j,{\omega }_k,t)+u^2({\omega }_i,{\omega }_j,{\omega }_k,t)+v^2({\omega }_i,{\omega }_j,{\omega }_k,t)+w^2({\omega }_i,{\omega }_j,{\omega }_k,t)\right)}^2$$ .

Substituting expression (7) into the determinant formula, we obtain

$$\left|\Phi ({\omega }_i,{\omega }_j,{\omega }_k,t)\right|=E$$ .

The determinant of the product of basic single-frequency matrices (12) is equal to the determinant of the 3-frequency matrix (9):

$$\left|{\Phi }_I({\omega }_i,t){\Phi }_J({\omega }_j,t){\Phi }_K({\omega }_k,t)\right|=\left|\Phi ({\omega }_i,{\omega }_j,{\omega }_k,t)\right|$$ .

According to the well-known property of the product of determinants, the determinant of the product of single-frequency fundamental matrices for different frequencies is equal to the product of the determinants of these matrices:

$$\left|{\Phi }_I({\omega }_i,t){\Phi }_J({\omega }_j,t){\Phi }_K({\omega }_k,t)\right|=\left|{\Phi }_I({\omega }_i,t)\right|\left|{\Phi }_J({\omega }_j,t)\right|\left|{\Phi }_K({\omega }_k,t)\right|$$ .

From here,

$$\left|\Phi ({\omega }_i,{\omega }_j,{\omega }_k,t)\right|=\left|{\Phi }_I({\omega }_i,t)\right|\left|{\Phi }_J({\omega }_j,t)\right|\left|{\Phi }_K({\omega }_k,t)\right|=$$

$$={\left({\cos }^2({\omega }_{i}t)+{\sin }^2({\omega }_{i}t)\right)}^2{\left({\cos }^2({\omega }_{j}t)+{\sin }^2({\omega }_{j}t)\right)}^2{\left({\cos }^2({\omega }_{k}t)+{\sin }^2({\omega }_{k}t)\right)}^2=1$$ .

As can be seen from (10), we received 8 combinations of products of cosines and sines of different frequencies. To transform expressions (7), we use well-known trigonometry formulas for products of three combinations with sines and cosines. According to these formulas, the products of three sines and cosines in various combinations are converted into the sum of sines and cosines from the sum of frequencies in various combinations. Let us denote the combinations of angular frequencies obtained as a result of the expansion as

The corresponding matrix for converting reference frequencies into combination frequencies has the form:

$$\Omega =\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& -1\\ 1& -1& 1\\ 1& -1& -1\end{array}\right]$$ . (14)

We obtain the corresponding combination frequencies in the form of a vector $$F={\left[\begin{array}{cccc}{F}_1& F_2& F_3& F_4\end{array}\right]}^{\mathrm{T}}$$ by multiplying the reference frequency vector $$f={\left[\begin{array}{ccc}{f}_1& f_2& f_3\end{array}\right]}^{\mathrm{T}}$$ to the matrix $$\Omega$$ : $$F=\Omega f$$ .

The transformation matrix of combination frequencies into reference frequencies is calculated from (14), as

$$\frac{1}{4}{\Omega }^{\mathrm{T}}=\frac{1}{4}\left[\begin{array}{cccc}1& 1& 1& 1\\ 1& 1& -1& -1\\ 1& -1& 1& -1\end{array}\right]$$ .

Let's look at an example:

$$f=\left[\begin{array}{c}6\\ -2\\ -1\end{array}\right]$$ , $$F=\Omega f=\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& -1\\ 1& -1& 1\\ 1& -1& -1\end{array}\right]\left[\begin{array}{c}6\\ -2\\ -1\end{array}\right]=\left[\begin{array}{c}3\\ 5\\ 7\\ 9\end{array}\right]$$ .

$$f=\frac{1}{4}{\Omega }^{\mathrm{T}}F=\frac{1}{4}\left[\begin{array}{cccc}1& 1& 1& 1\\ 1& 1& -1& -1\\ 1& -1& 1& -1\end{array}\right]\left[\begin{array}{c}3\\ 5\\ 7\\ 9\end{array}\right]=\left[\begin{array}{c}6\\ -2\\ -1\end{array}\right]$$ .

After calculating the products of sines and cosines in (10) using the indicated trigonometry formulas, bringing similar terms and grouping by frequencies, we obtain the following expressions for exponential functions (10) for combination frequencies (13):

Substituting expressions (15) for combination frequencies into the matrix of products (12) and grouping the terms by frequencies, we obtain a representation of the fundamental matrix as a sum of single-frequency matrices:

Where

We denote the corresponding normalized basis matrices for combination frequencies as

In the process of correlation technique, when integrating the products of the basis matrices by the corresponding transposed matrices (17) for duration T = 1, we obtain that

$$E_m{}^{\mathrm{T}}\left({\Omega }_m,t\right)E_m\left({\Omega }_m,t\right)=E$$ , $$I_m{}^{\mathrm{T}}\left({\Omega }_m,t\right)I_m\left({\Omega }_m,t\right)=E$$ , $$J_m{}^{\mathrm{T}}\left({\Omega }_m,t\right)J_m\left({\Omega }_m,t\right)=E$$ ,

$$K_m{}^{\mathrm{T}}\left({\Omega }_m,t\right)K_m\left({\Omega }_m,t\right)=E$$ ,

where m =1, 2, 3, 4.

Thus, we have obtained expressions for the exponential function (5) of a three-frequency quaternion in the form of a sum of functions (6), which are represented by 8 combinations of products of cosines and sines of reference frequencies (7). Also, using trigonometry formulas for the products of three combinations with sines and cosines, we obtained the exponential function (5) in the form of sums of cosines and sines (15) of various combinations of frequencies (13).

Multi-frequency matrix manipulation.

We represent possible combinations of binary symbols in the form of vectors of initial states written in a matrix:

The amplitude of information vectors is equal to

$$A=\sqrt{x_0^2+x_1^2+x_2^2+x_3^2}=\sqrt{\pm 1^2\pm 1^2\pm 1^2\pm 1^2}=\sqrt{4}=2$$ .

The power of information vectors is equal to

$$P=x_0^2+x_1^2+x_2^2+x_3^2=\pm 1^2\pm 1^2\pm 1^2\pm 1^2=4$$ .

We will use the three-frequency fundamental matrix (9) as the channel matrix of the MIMO channel in the mathematical model of the information transmission system. When using a carrier frequency, the reference frequencies will be shifted by the amount of the carrier, producing spectra at the positive and negative reference frequencies. Since when receiving signals at a carrier frequency, inverse conversion occurs, we will consider the channel matrix at reference and combination frequencies.

We write the manipulation in the form of multiplying the vector of initial states by a three-frequency fundamental matrix of reference frequencies, which we define by analogy with modulation methods with signals considered on the real axis or plane, as a “quaternion carrier”, in the general case – a “hypercomplex carrier”:

$$y({\omega }_i,{\omega }_j,{\omega }_k,t)=\Phi ({\omega }_i,{\omega }_j,{\omega }_k,t)x(0)$$ . (19)

As a result of multiplication, we obtain a modulated output vector $$y({\omega }_i,{\omega }_j,{\omega }_k,t)$$ .

Since the multi-frequency matrix is equal to the sum of single-frequency matrices (16), the manipulation of single-frequency matrices will, accordingly, have the form:

$$y({\omega }_i,{\omega }_j,{\omega }_k,t)=\left[{\Phi }_1({\Omega }_1,t)+{\Phi }_2({\Omega }_2,t)+{\Phi }_3({\Omega }_3,t)+{\Phi }_4({\Omega }_4,t)\right]x(0)$$ . (20)

Let us write (20) in the form of separate output vectors:

$$y_1({\Omega }_1,t)={\Phi }_1({\Omega }_1,t)x(0)$$ , $$y_2({\Omega }_2,t)={\Phi }_2({\Omega }_2,t)x(0)$$ , $$y_3({\Omega }_3,t)={\Phi }_3({\Omega }_3,t)x(0)$$ , (21)

$$y_4({\Omega }_4,t)={\Phi }_4({\Omega }_4,t)x(0)$$ .

Possible combinations of information vectors (18) will be depicted as states in 3D space, as shown in Figures 1 and 2. In Figure 1, the positive values of the most significant bit of the information vector are shown at the nodal points; in Figure 2, the negative values of the most significant bit are shown at the nodal points information vector.

Vector elements (19) can be transmitted using different types of division: code, frequency or time.

Figures 3 and 4 show graphs of the output vector (19) with sequential transmission over time for reference frequencies $$f={\left[\begin{array}{ccc}6& -2& -1\end{array}\right]}^{\mathrm{T}}$$ for two possible combinations of initial state vectors. A negative frequency for a sine simply means a change in the sign of the amplitude.

Since the fundamental matrix (9) is orthogonal and does not change the norm of the information vector, the powers of the elements of the output vector are equal to 1.

Figures 5 and 6 show graphs of elements of output vectors (21) for four single-frequency matrices with two combinations of information vectors.

With 4 non-zero values of the elements, the amplitude will be $$\sqrt{2}/2=\sqrt{1/2}$$ , power taking into account the coefficient $$1/2$$ for the harmonic we get $$1/4$$ .

The amplitude of the elements of the output vectors of single-frequency matrices can be A = 1 for 2 elements and 0 for the other 2 elements. Element power taking into account the coefficient $$1/2$$ for harmonica it will be $$1/2$$ , i.e. the power of 1 element will be 2 times less.

In sum over the elements, the power of the vector will be 1 for the elements of the vectors in Figure 5 and in Figure 6. The power of one vector of a single-frequency matrix will be 1. For 4 vectors in single-frequency matrices, the power will be equal to 4.

Thus, the power of the output vector will be equal to the power of the input vector over the multi-frequency matrix and in total over the single-frequency matrices.

Figures 7 and 8 show graphs of the rotation orbits of the scalar part of the quaternion for various initial vectors normalized to 1 with a positive scalar part representing the most significant bit of the binary vector.

Figures 9 and 10 show graphs of the rotation orbits of the scalar part of the quaternion for the same initial vectors normalized to 1 with a negative scalar part representing the most significant bit of the binary vector.

In total we get 8 orbits for the positive value of the scalar part and 8 orbits for the negative scalar part. We group these orbits along identical trajectories and write them as numbers of information vectors (18):

1, 6 and 11.16 2.3 and 14.15 4, 7 and 10, 13 5, 8 and 9.12.

As you can see, in total we have 4 different configurations of trajectories for four different information vectors. When exposed to interference in the form of a 4D white noise vector with central symmetry, the interference will not be correlated even if the trajectory is the same, since the initial states are located in different places of the trajectory.

It is known that for a single-frequency quaternion there are also 4 trajectories of rotation of the scalar part. However, these trajectories are located on planes in 3D. For a three-frequency quaternion, since the rotation for the 4 combination frequencies will be at different speeds, the trajectory will be more complex and in 3D trajectories cover a much larger 3D space and, as a result, the signals have a significantly greater variety. As is known, the greater the diversity of the signal, the less susceptible it is to interference and the higher the information transmission speed can be obtained. In the limiting case, when the signal is similar to white noise and occupies the entire volume, we obtain the throughput of the MIMO channel. Note also that the power, and therefore the length of the rotation vector, remains constant.

Passing through the communication channel, interference is added to the modulated oscillations from the combination frequencies. As a noise, consider 4D white noise with circular symmetry. As a result, get the signal vector with noise:

$$s(t)=\left[\begin{array}{c}{y}_1({\Omega }_1,{\Omega }_2,{\Omega }_3,{\Omega }_4,t)\\ y_2({\Omega }_1,{\Omega }_2,{\Omega }_3,{\Omega }_4,t)\\ y_3({\Omega }_1,{\Omega }_2,{\Omega }_3,{\Omega }_4,t)\\ y_4({\Omega }_1,{\Omega }_2,{\Omega }_3,{\Omega }_4,t)\end{array}\right]+\left[\begin{array}{c}{n}_1(t)\\ n_2(t)\\ n_3(t)\\ n_5(t)\end{array}\right]$$ .

Since white noise is considered as interference, the optimal receiver for each element is a correlator.

We will demodulate the received vector using normalized transposed basis matrices (17) for each single-frequency matrix separately, in which we will replace the combination frequencies with reference ones in accordance with (13).

Multiplication of the received sequence of vector elements $$s(t)$$ on the elements of the basis matrices and their integration with subsequent summation of the results is carried out in real time without pulse delay. At the end of the accumulation of signal energy, a count is made and stored. Since each element of the sequence $$s(t)$$ contains all the elements of the information vector, then when multiplied by the elements of the columns of the basis matrices, we obtain estimates of all four information elements. Subsequently, all samples of each element from all basis matrices of each single-frequency matrix are added and fed to the solver.

Figures 11-30 show the results of multiplying the received vector by the basis matrices of single-frequency matrices after integration by pulse duration.

Figure 31 shows the sum of the results from the 4-frequency matrices.

When receiving elements of the output vector (19) by multiplying by basic orthogonal matrices, the norm of the vector does not change. However, in this case, the elements of the interference vector are rearranged in accordance with the structure of the matrices E , I , J , K. Consequently, when adding vector elements, they are added in energy, and interference vector elements are added in power. Therefore, for each matrix we get a gain in signal-to-noise ratio, compared to receiving real signals with BPSK, by a factor of 4. Since the reception of the same values is carried out at different frequencies, the total gain in the signal-to-noise ratio with uncorrelated interference and with sufficient frequency separation to eliminate their influence on each other will be equal to 16, not counting the increase in frequency band. Typically, when operating in high-frequency wave bands, increasing the bandwidth does not entail any large costs.

The technical result of the invention is to increase the noise immunity and speed of information transmission in wireless and wired communication systems, which is achieved due to the fact that the sequences of binary data pulses consisting of zeros and ones arriving at the input of the transmitting device are converted into bipolar pulses, which are converted into N- dimensional information vector consisting of N information pulses and forming a hypercomplex number in a vector representation, which provides a greater energy distance (compared to physical space, space-time code), as well as effective use of the N-dimensional hypercomplex space of a hypercomplex number in time and frequency, increasing the noise immunity of transmitted information, as a result, increasing the speed of information transmission is achieved.

In this case, modulation on the transmitting side is carried out by multiplying the information vector by a hypercomplex carrier, which is a hypercomplex number in polar form with different N-1 reference frequencies in a matrix representation and plays the role of the MIMO channel matrix of the channel, while the multi-frequency MIMO channel matrix is decomposed into the sum of single-frequency matrices that are modulated separately and their results are added to form multi-frequency elements of an N-dimensional vector, while the phase-modulated elements of the output multi-frequency modulated vector are transmitted sequentially in time either with code division or with frequency division, thereby achieving a reduction in the number of transmitting and receiving antennas due to the use of one antenna for transmission and one antenna for reception, while the use of N-dimensional hypercomplex space and rotation of information elements when modulating a quaternion carrier makes it possible to maximize the energy distance between information elements, which leads to improved noise immunity, higher speed of transmitted information, efficient the use of frequency resources and increased secrecy of information transmission, as a result, an increase in noise immunity and information transmission speed is achieved.

In this case, since the channel matrix of the MIMO channel is orthogonal, the norm of the information vector before data transmission through the communication channel will be equal to the norm of the information vector after transmission, if the channel does not change it, as a result, an increase in noise immunity and information transmission speed is achieved.

At the same time, the gain in signal-to-noise ratio when using a single-frequency quaternion channel matrix, compared to receiving real signals with BPSK, is 4 times and 16 times for a 4-frequency channel matrix of combination frequencies.

In this case, the power of the output vector will be equal to the power of the input vector in multi-frequency matrices and in total in single-frequency matrices, thus, the information transmission system maintains the signal power at each frequency or in each individual channel, as a result:

1. Resources are used efficiently: there is no excess energy loss or sudden changes in power (peak factor) at different frequencies. This allows optimal use of the power of the transmitter amplifiers and the available bandwidth of the communication channel;

2. Interference Suppression: Since each information pulse is transmitted at each time position and at each frequency of the modulated signal, and the signal strength is maintained at all frequencies and all time positions, the system can be more resistant to external interference. Interference at one frequency will not affect the signal strength at other frequencies and positions in time. This is especially important in the case of multipathfading or the presence of other sources of interference, as well as in frequency- and time-selective fading;

3. Support high-quality information transmission: Since the signal strength does not change, the quality of the transmitted data is maintained. This is especially important for transmitting analog or digital signals, where changes in power can cause data corruption;

4. Ease of Processing: When the power of the input vector is equal to the power of the output vector, signal processing becomes more predictable and convenient. There is no need to correct or compensate for differences in signal strength at different frequencies.

As a result, an increase in noise immunity and information transmission speed is achieved.

In this case, the total gain in signal-to-noise ratio with uncorrelated interference will be equal to 16, not counting the increase in frequency bandwidth.

In this case, interference on the orthogonal basis is not correlated, as a consequence:

1. Improved signal separability: interference can be more effectively separated from the desired signals;

2. Improved reception quality: Interference does not add up or increase with each other. This means that signals can be received with less distortion and loss of quality. Uncorrelated noise does not create interference at the fundamental frequencies of information transmission;

3. Increased Security: Interference that is not correlated makes it difficult or more difficult for unauthorized access or hacking of a communications system. Uncorrelated interference may make it difficult or impossible to extract sensitive information from the transmitted signal;

4. More efficient use of radio frequency spectrum: Uncorrelated interference can be better isolated and suppressed when using radio frequency spectrum. This allows you to effectively use the available frequency bands and minimize the mutual influence of signals and interference.

As a result, an increase in noise immunity and information transmission speed is achieved.

In this case, the signal vectors are coherently demodulated by multiplying the received vector by transposed basis matrices of single-frequency matrices, which represent an orthogonal basis, and the reception of the signal vector arriving at the input of the receiver is carried out using the integration of the results of multiplication followed by addition for each single-frequency matrix of the orthogonal basis and making a decision in the decisive device about the meaning of vector elements. In this case, it is sufficient to calculate the phase of the carrier wave using conventional synchronization methods and does not require knowledge of the phase from each transmitting antenna to each receiving antenna to determine the parameters of the space-time code. Typically, feedback, a pilot signal, and a complex computing device are used to obtain such data. Thus, technologies for separately receiving information elements based on the basis matrices of the quaternion are implemented, as a consequence:

1. Improved spectral efficiency: increased signal-to-noise ratio; ability to transmit information at higher speeds;

2. Increased reception noise immunity;

3. Increased secrecy of data transmission, since each information symbol is contained in each symbol of the transmitted vector and in one way or another changes its modulation parameters. Therefore, without knowing the channel matrix and its frequencies, it is impossible to select information symbols.

In this case, it is possible to evaluate each element of the information vector when exposed to various interference, since the interference is decomposed into components on an orthogonal basis, and with a narrow-band spectrum does not affect information elements transmitted at other frequencies.

As a result, an increase in noise immunity and information transmission speed is achieved.

In this case, the hypercomplex carrier is a hypercomplex number in polar form and in matrix representation in the form of a square NxN matrix with N-1 reference frequencies, combinations of which form 2 ^N-2 combination frequencies, and the same number of single-frequency matrices.

When using multi-frequency hypercomplex carriers, in the general case, or, in the particular case, three-frequency quaternion carriers, the modulated signals have a greater variety, i.e. occupy a larger volume of hypercomplex space, as shown in Figures 7-10. As a result, they are less susceptible to interference that differs from white noise and is concentrated in time (pulse interference) or frequency (narrowband interference). Since when exposed to white noise interference, the throughput is determined by the difference between the diversity of the signal and the entropy of the interference , then when using a hypercomplex multi-frequency space it is possible to achieve higher signal diversity than with conventional methods of modulating real signals on a plane, even when increasing the number of signal amplitude levels or frequencies. As a result, the throughput of such channels increases significantly.

According to FIG. 32 and 33 Information source 1 converts the binary pulses 0, 1 arriving at the input into bipolar 1, -1 and from them forms, in the general case, an N-dimensional information vector by appropriate delay of the pulses. In this case, the last pulse arrives without delay.

The hypercomplex number encoder 2 generates, in the general case, a hypercomplex multi-frequency carrier in the form of an NxN matrix, in a particular case, a quaternion carrier in the form of a 4x4 matrix with three frequencies. The hypercomplex carrier is an exponential representation of a hypercomplex number in a matrix representation, which, in turn, is decomposed into a sum of single-frequency matrices.

The hypercomplex number modulator 3 modulates the hypercomplex carrier by multiplying single-frequency matrices by an information vector and then adding the results to obtain a modulated multi-frequency oscillation, in the particular case, a four-frequency oscillation.

The transmitting antenna 4 receives modulated signals at a carrier frequency and amplified by a power amplifier. Antenna 4 emits them in a given direction.

The receiving antenna 5 amplifies the received signal, which is then fed to a low-noise amplifier and transferred to the frequencies of the hypercomplex carrier.

The demodulator of hypercomplex numbers 6 performs correlation reception in real time of the received modulated elements of the vector by multiplying it by elements in the columns of transposed basis matrices of single-frequency matrices of combination frequencies, followed by integration and addition of the results.

The hypercomplex number decoder 7 summarizes the results of correlation reception over all multi-frequency matrices, in a particular case, over all 4-frequency matrices of a three-frequency quaternion and feeds them to the decision device. The decision can be made for each information element of the vector separately or with the calculation of the Frobenius norm.

The information recipient 8 converts bipolar pulses 1, -1 into 0, 1 and supplies them to the consumer in the required representation.

Example implementation.

The inventive method was implemented for a quaternion using a 4x4 channel matrix. An M-sequence generator based on JK flip-flops with feedback is used as a data source. To form a 4-dimensional data vector from incoming pulses, the first pulse was delayed using D-flip-flops by 3 clock cycles, the 2nd by 2, the third by 1, and the 4th was received without delay in real time.

The three-frequency channel matrix of reference frequencies was represented as the sum of 4 single-frequency matrices of combination frequencies. To form single-frequency channel matrices, a high-frequency generator was used, which outputted two harmonic oscillations of combination frequencies shifted by π/2, which formed the sum of a cosine and a sine with different signs in accordance with the structure of the channel matrix. Modulation was carried out by multiplying the data vector by the column elements of four single-frequency channel matrices using 4 multipliers for each matrix. A signal from a carrier generator of a given frequency was supplied to one input of the multipliers, and the corresponding element of the pulse vector was supplied to the other. The results obtained at the outputs of the multipliers for matrices of different combination frequencies were summed up in adders, resulting in a 4-dimensional vector of a multi-frequency signal. The described procedure corresponds to the multiplication of a 4-dimensional information vector by a three-frequency matrix of reference frequencies of a quaternion carrier.

The multiplication of the information vector by each column was carried out in one clock cycle of the information pulse duration and the resulting pulse in the form of a sum of 4 combination frequencies was transferred to the carrier frequency and transmitted sequentially in time through a power amplifier to the antenna. In this case, each element was transmitted using the entire power of the power amplifier.

While passing through the communication channel, noise was added to the signal. To obtain noise immunity characteristics, noise from a noise generator was additionally supplied to the receiver input.

The received signal was transferred to the original combination frequencies and received using correlators. As reference frequencies, elements of 4 single-frequency basis matrices were used, into which single-frequency matrices of combination frequencies were decomposed and represented an orthogonal basis.

All 16 multipliers with integrators for each single-frequency matrix were combined into a bus, 4 in each wire. Demodulation was carried out in real time. When the 1st element arrived, it was fed to the 1st wire of the bus and multiplied by the elements of the first columns of the basic single-frequency matrices. At the end of accumulation, the results were counted and summed over all elements of the column for all 4 single-frequency matrices. The result of the sum of 16 samples was fed to the decision device, which made a decision on the value of the information impulse. The probability of error was determined by comparing the supplied pulse with the received one. Similar operations were performed upon receipt of the 2nd received pulse, which was applied to the 2nd wire of the bus, etc.

The signal-to-noise ratio was measured using a selective millivoltmeter. The result was an increase in signal-to-noise ratio by 16 times compared to BPSK and 8 times compared to the Alamouti scheme with 2x2 MIMO.

Claims (2)

1. A method for transmitting information using a MIMO scheme in N=2 ⁿ , n = 2, 3, 4, hypercomplex space, characterized by the fact that sequences of binary data pulses consisting of zeros and ones arriving at the input of the transmitting device are converted into bipolar pulses , which are converted into an N-dimensional information vector consisting of N information pulses and forming a hypercomplex number in vector representation, while modulation on the transmitting side is carried out by multiplying the information vector by a hypercomplex carrier, which is a hypercomplex number in polar form with different N-1 reference frequencies in the matrix representation and acting as the channel matrix of the MIMO channel, while the multi-frequency channel matrix MIMO is decomposed into the sum of single-frequency matrices, which are modulated separately and their results are added up, forming multi-frequency elements of the N-dimensional vector, while the phase-modulated elements of the output multi-frequency modulated vectors are transmitted sequentially in time or with code division, or with frequency division, while the signal vectors are coherently demodulated by multiplying the received vector by transposed basis matrices of single-frequency matrices, which represent an orthogonal basis, and the reception of the signal vector arriving at the receiver input is carried out using integration of the results multiplication followed by addition for each single-frequency matrix of the orthogonal basis and making a decision in the decision device about the value of the vector elements. 2. A method for transmitting information using a MIMO scheme in hypercomplex space according to claim 1, characterized in that the hypercomplex carrier is a hypercomplex number in polar form and in matrix representation in the form of a square NxN matrix with N-1 reference frequencies, combinations of which form 2 ^N-2 combination frequencies and the same number of single-frequency matrices.