symbol i as ( , )x y z x iy . The characte r i satisfies the property 21 i . The modulus of a complex number is calculated as 22||z x y , and the argument is as arctg( ) yx . Physically realizable signals are used in radio engineering, therefore, f or the formation and processing of complex signals, it is necessary to represent them in a real vector or matrix extension . Therefore, we write a complex number in the form of a vector and a matrix, as [4]: x yz и xy yxZ . (1) The real part of a complex number in matrix representation corresponds to a matrix with units on the main diagonal : 10 01E . (2) The imaginary unit i corresponds to a 90° rotation, so in matrix extension it has the form : cos 2 sin 2 0 1 sin 2 cos 2 1 0           I . (3) The matrix imaginary unit I satisfies the properties corresponding to the properties i: TII , 2IE , 3 I EI I , 4 I II E , 5I EI I … . Эти свойства запишем в виде таблицы 1. Table 1. Result of matrix multiplication (2) and (3) .  EI E E I I I E . The matrices E and I will be called the basis matrices of a complex number in algebraic form in the matrix representation . Using the basis matrices (2) and (3), we write the complex number z x iy in matrix form (1) as xyZ E I . Note that the matrix E is orthogonal, since TE E E and the matrix I are orthogonal, since TTI I II E . All operations with complex numbers in the matrix representation correspond in their re- sults to operations of the same name on complex numbers in algebraic form. The square of the modulus of a complex number in the matrix representation is equal to the determinant of the ma- trix: 2 2 2||xyz x yyx   Z . The complex conjugate number z x iy in matrix form is formed by transposing the original matrix: Tx y x yxyy x y x             Z E I . Consider the product of complex numbers in matrix expansion as a multiplication of a transposed matrix by a non -transposed one: 1 1 2 2 1 2 1 2 1 2 2 1 T 12 1 1 2 2 1 2 2 1 1 2 1 2() x y x y x x y y x y x y y x y x x y x y x x y y                  ZZ . (4) The product of complex numbers can be written as well as the product of a vector on a matrix: 1 1 2 1 2 1 2 T 12 1 1 2 1 2 2 1x y x x x y y y x y x y x y             Zz . (5) The result co rresponds to the first column of the matrix (4) obtained by matrix multiplication. Writing a complex number in polar form corresponds to an exponential function of a com- plex number. This function converts the complex number z to a complex number e e e e (cos sin )z x iy x iyw y i y u iv        , where ex , cosuy - real part, sinvy - imaginary part of the exponential function of iy. We write the exponential function in matrix form using the basis matrices (2), (3) in the form:  cos sincos sinsin cosw     EI . (6) Since, according to the physical meaning, the imaginary part y of the complex number z in algebraic form is the angle of rotation of the corresponding vector on the complex plane, we will replace its designation w ith the designation of the angle φ. In the vector representation (1) of the complex number z, the scalar product is calculated as [4] 1 2 2 1 2 1 2 1 1 1 2 1 2 1 2 2| || | cosx x xx y x x y yy y y                    z z z z , (7) where θ is the angle between the vectors . The maximum of the scalar product is achieved when the vectors are collinear, i.e . at 0 . The cross product in the vector representation of a complex number is calculated as 12 1 2 1 2 1 2 2 1 12| || | sinxxx y x yyy               z z z z . (8) The maximum of the vector product is reached at 90 . Because () 1 2 1 2 1 2 1 2 1 2 | | | | | || | | || | | || | (cos sin )i i i iz z z e z e z z e z z e z z i           , where    , then as a result of multiplying the conjugate complex number by the non -con- jugate one, we obtain the sum of scalar and vector multiplication [4]. As can be seen from the results obtained, the scalar product (7) and ve ctor product (8) are the result of the product of matrices (4) and a vector by matrix (5). Thus, in the algebraic representation of a complex number, multiplying a conjugate number by a non -conjugate number or in a matrix representation, multiplying a matr ix by a transposed one, we get the sum of the scalar and vector product. Real numbers give us only the dot product. This is explained by the fact that complex numbers, unlike real ones, combine the different nature of energy, namely, potential and kinetic. With an exponential display, the real number character- izes the degree of stretching of the complex vector, and the imaginary number characterizes the rotation speed at which potential energy transforms into kinetic energy and vice versa. In other words, t he real part of the complex number goes into the imaginary and vice versa, according to the Cauchy -Riemann conditions. At present, the representation of information transmission lines by a state -space model is becoming more widespread, which has significan t advantages over the widely used input -output model [5]. According to this model, the equation of dynamics in continuous time without input action is written as ( ) ( )ttx Ax , (9) where the state transition matrix (STM) A differentiates the s tate vector ()tx . The matrix exponent etA is the solution to the differential equation e e et t t tI I AIA . From where, S TM AI , where ω is the circular frequency, rad/s. The exponential mapping of the STM A when it changes in time is found as the inverse Laplace transform [5]: 11e L ( )tsAEA . As a result of the calculation, we obtain the fundamental matrix : 2 2 2 21 2 2 2 2cos sin( , ) Lsin coss tt ssts t t ss          Φ . (10) Using the basis matrices (2) and (3), we write the fundamental matrix (10) in the form: ( , ) cos( ) sin( )ct c c ct e t t    IΦ E I . (11) where c is the circular frequency of the carrier. Matrix (11) is called the complex carrier in the matrix representation. Fundamental matrix (11) is orthogonal, because TT( , ) ( , ) ( , ) ( , )cctt c c c ct t t t e e     IIΦ Φ Φ Φ    cos( ) sin( ) cos( ) sin( )c c c ct t t t       E I E I E . Thus, from the above information about complex numbers, we can conclude that in order to obtain a greater energy gain when tran smitting information, it is necessary to use complex num- bers, consisting of both real and imaginary parts and combining potential and kinetic energy. At the same time, in order to get rid of imaginary units, it is necessary to represent complex numbers in a matrix expansion. This representation leads us to a state -space MIMO 2×2 scheme described in terms of dynamics and output equations. III. MIMO link model with complex carrier To form a MIMO channel, we use a complex carrier (11). The sequence of informa tion pulses 1 , arriving at the input of the transmitter, is divided into pairs of pulses and written as a sequence of information vectors, as the vectors of the initial states  T 01 (0) (0) (0) xxx of the dynamic system (9), the solution of which is the complex carrier (11). For binary transmission, we will have only 4 combinations of initial vectors: 1 1 1 1(0)1 1 1 1x . (12) We consider the information vector as a complex number in vector representation (1), so the sq uare of the modulus of the complex number or the square of the norm o f each vector will be equal to 22 T 2 2 01 (0) (0) (0) (0) (0) (0) 2 xx      x x x x . Let us perform phase manipulation of the complex carrier with the initial state vector (0)x , for which we multiply (11) by the incoming initial state vector and obtain the modulated com- bined output vector : 0 0 1 1 0 1( , ) (0)cos( ) (0)sin( )( , ) ( , ) (0)( , ) (0)sin( ) (0)cos( )c c c cc c c cy t x t x ttty t x t x t                  y Φx . (13) In accordance with the vector notation (1) of a complex number, element 0( , )c yt repre- sents the real part of the combined vector ( , )cty , and element 1( , )cyt represents the imaginary part. Since the fundamental matrix (11) performs an exponential mapping of a complex number, in the process of changing time, the combined o utput vector rotates counterclockwise with the center at the origin. Due to the orthogonality of ()ctΦ , when multiplying the initial vector (0)x by it, we obtain the output vector (13) with the same norm 2 T 2 2 01 ( , ) ( , ) ( , ) (0) (0)c c ct t t x x     y y y . As can be seen from (13), each element of the output vector ( , )cty contains information about two elements of the input vector (0)x , i.e. about the transmitted information elements. In other word s, we have a MIMO 2×2 scheme. Each element of vector ( , )cty is the difference and the sum of cosines and sines with amplitudes corresponding to the values of the information vector. Therefore, they can be drawn on the complex plane and the initial phase can be calculated as the argument of the complex number at t=0. The calculated initial phases of the modulated elements of the combined vectors (13) for the values of the information vectors (12) will be written as a matrix of initial pha ses: 45 135 45 135 45 45 135 135          Θ . (14) As can be seen from (14), the phase difference between the real part of the combined vector and the imaginary part is 90°, which ensures the maximum value of the vector product (8). Modulated oscillations with initial phases (14) correspond to QPSK. The difference lies in the transmission method. With QPSK, only the real part of the combined vector is transmitted (13) [1]. In our case, we will sequentially transmit first the real part 0( , )c yt , and the n the imaginary 1( , )cyt . Figure 1. The elements of the modulated combined vector for T(0) 1 1x . Figure 1 shows the elements of the modulated combined vector arranged sequentially in time. The amplitude A of each oscillation (13) for information pulses (12) with their quadrature addition will be equal to 2 . Therefore, the power of each emitted element (13) will be 221 PA . By direct integratio n over the pulse duration T, we obtain the same result:   2 22 0 1 0 10011(0)cos( ) (0)sin( ) cos ( ) sin ( ) 1TT c c c c P P P x t x t dt t t dtTT          . The energy of the elements is calculated as 2 01 2 E E E PT A T T     . Knowing the energy or power of the signal, one can calculate the amplitude of oscillations as 22 A E T P . With E=1 we get 2 AT . Since we are transmitting two information symbols, when two elements of the output mod- ulated vector (13) are sequentially transmitted, the information transfer rate remains unchanged. Howev er, once again we note that each element of the output modulated vector contains infor- mation about two information symbols 0(0)x and 1(0)x . Further, the generated signal is propagated through the communication chan nel. We will assume that when propagating through the communication channel, interference in the form of white noise is added to the signal. Since the elements of the combined vector are transmitted se- quentially in time, the noise for each element will be uncorrelated. We write the model of the received signal in the form: ( ) ( ) ( ) ( ) (0) ( )c t t t t t     s y n Φ x n , (15) where T 01 ( ) ( ) ( )t s t s ts - 2D received signal vector ,  T 01 ( ) ( ) ( )t n t n tn - 2D complex white noise vector with circular symmetry, i.e. unco rrelated, with variance 2 and power spectral den- sity 0N . Since the noise is complex, i.e. two -dimensional, then the square of the norm of the vector 2 2 2 2 0 1 00 01 ( ) N 2 N 2t       n , where 2 0 , 2 1 and 00N , 01N are the dispersions and spectral densities of the elements of the nois e vector . In accordance with the maximum likelihood criterion, the optimal receiver includes a cor- relator or a matched filter that calculates the scalar product of the received vector (15) and selects the most likely vector [6]. Since the QPSK signal is transmitted as the real part of the combined vector (13), the re- ceiver has only t wo basic normali zed functions: 1( ) 2 cosc t T t and 2( ) 2 sinc t T t , which form the real plane. The simplest receiver represents two correlators [1]. The signal phase is calculated from the values of the received signal, and then the closest quadrant on the real plane is selected from the phase value. In our case, we transmit sequentially both the real and imaginary parts of the modulated vector (13). Then, the optimal receiver is a correlator that multiplies the received vector (15) by the basis matri ces of the transposed fundamental matrix (11) and compares the result with all pos- sible combinations of information vectors. As bases, we consider two basis matrices transposed and normalized by response after in- tegration over duration T: T cos( ) 0 22( , ) cos( )0 cos( )c cc ctttt TT EE , (16) T 0 sin( ) 22( , ) sin( )sin( ) 0c cc ctttt TT  II . (17) Note that in the case of real signals, we have two basis functions that correspond to the coordinate axes, and in the case of complex signals, we operate with two 2D matrices that corre- spond to two -dimensio nal planes. The matrices are orthogonal when integrated over duration T, since 2 022cos ( ) 12T cTt dtTT , 2 022sin ( ) 12T cTt dtTT . Recall that orthogonal matrices do not change the norm of vectors when they are multiplied by these matrices. IV. Diversity From the elements of the modulated signal received in series, we form a vector (15). We first multiply the vector (15) with amplitude 2 AT by the basis matrix (16) and integrate over the pulse duration T. With equal energies of the elements of the signal vector, we obtain the signal component: 012 00 0 E 2 1 10(0) cos ( )(0) 2 (0)(0) cos ( )Т c T cx t dtx x Tx t dt      S . (18) Interference component covariance matrix for the same interference power spectral densi- ties: 012 0 0 E2 0 1N 2 0 0 0 N 2 0    N . (19) As can be seen from the co variance matrix, the interference is not correlated, as it acts on different elements of the combined modulated vector transmitted sequentially. Index 01E in the notation of the signal and noise components shows that the real part T( , )ctE of the basis matrix ( , )ctΦ was used and the cosine component was extracted from the combined vector with com- plex noise and the sequence of responses corresponds to indices 0 and 1 in the vectors. Note that, due to the orthogonality of the basis matrices (16) and (17), the dispersion of interference elements is equal to the dispersion of interference in the communication channel (15). Figure 2. Scheme of the impact of interference during reception using a rea l matrix (16). The interference is cosine. Figure 3. Scheme of the impact of interference during reception using an imaginary matrix (17). The interference is sinus. Figure 2 schematically shows the order of elements in the signal and interference vector when received using a real basis matrix . It is also symbolically shown that the cosine component of the interference is ex tracted using the basis matrix T( , )ctE and the sequence of responses corresponds to indices 0 , 1 in the signal vec tor and 0 , 1 in the noise vector . The diagram also shows that the estimation of information element ˆ(0)x is fed to the decision device. The estimate of the signal in the form of a sample after integration over time T write as:   TT cos01 cos0 cos1 0 cos0 1 cos1ˆ ˆ ˆ (0) (0) (0) (0) (0) (0) (0) x x x x x x    x , (20) where 0(0)x and 1(0)x - expectation value (18) , cos 0(0)x and cos1(0)x - projections onto the real coordinate axis of additive noise with dispersion (19) . cosˆ()tx Let us now multiply the received signal (15) with amplitude 2 AT by matrix (17) and integrate over the pulse duration T. As a result, we get the signal component: 012 00 0 I 2 1 10(0) sin ( )(0) 2 (0)(0) sin ( )T c T cx t dtx x Tx t dt      S . (21) Let us c alculate the covariance matrix of the interference component: 102 10 I2 0 00 N 2 0 0 N 2 0    N . (22) By analogy with (20), we write the estimate of the signal in the form of a sample after integration over time T in the form:   TT sin10 sin1 sin 0 1 sin 0 0 sin1ˆ ˆ ˆ (0) (0) (0) (0) (0) (0) (0) x x x x x x    x , (23) where 0(0)x and 1(0)x - expectation value (21), sin 0(0)x and sin1(0)x - projections onto the imag- inary coordinate axis of additive noise with dispersion (22). Figure 3 schematically shows the order of elements in the signal and interference vector when received using a n imaginary basis matrix T( , )ctI . The index sin10 in the notation of the signal and noise components shows that the sine component is selected from t he combined vector and the sequence of res ponses corresponds to indices 1, 0 in the signal vector and 0, 1 in the noise vector. As you can see, the order of the elements in the noise vector differs from the order in the signal vector. Figure 3 schematicall y shows how the order of interference effects on signal ele- ments changes. Combining the circuits shown in Figures 1 and 2, we get a MIMO 2×2 scheme . a) The result of signal energy accumulation after multiplication by T( , )ctE b) The result of signal energy accumulation after multiplication by T( , )ctI . Figure 4. Energy accumulation results for a signal with additive noise - solid lines and a signal without noise - dotted lines. Figure 4 shows the results of signal reception simulation using the basis matrices (16), (17). As can be seen from the figures, all 4 interferences differ from each other. Thus, we have 2 different uncorrelated noises that act on elements of the combined modu- lated signal vector that are different in time. Each of these noises is projected onto the real and imaginary axes of the coordinate system. The responses of these noises after integration are also uncorrelated due to the orthogonality of the coordinate axes. As a result, we obtained two estimates (20) and (23) of the vector of information elements under the influence of various uncorrelated noises. Consider diversity of the obtained results for different values of information symbols. As is known, when estimatin g by the maximum likelihood, it is assumed that the parameter is deter- ministic, but unknown [6]. In our case, the value of information symbols in a 2D information vector acts as a parameter. We will estimate according to the mathematical expectation (20) a nd (23). When impact ed the "white noise" interference, the mathematical expectation is equal to the values of the infor- mation vectors (12). Since the information vector is multiplied by the fundamental matrix (11) during modulation, each information vector forms a complex number in the matrix expansion. Let's represent the relationship of the information vector and the corresponding complex number in the form: 1 1 1 1 1 1           , 1 1 1 1 1 1             , 1 1 1 1 1 1          , 1 1 1 1 1 1            . (24) The diversity between information vectors is calculated as the scalar product (7). The larger the scalar product, the closer the vectors are to each other. In our case, we have a matrix represen- tation of complex numbers on complex plane s, therefore, to calculate the diversity of planes, we will use the Frobenius norm for matrices [7]. In other words, we will not calculate the distance between 2D vectors, but the distance between two 2D planes. When calculating the Frobenius norm for comp lex numbers in the form (4), we obtain the sum of the scalar and vector product. Квадрат нормы Фробениуса для комплексных чисел в матричном представлении (24) вычисляется, как:   2 T(0) tr (0) tr 2 (0)F    Z E I E I x Ex . At (0) 1x ,  (0) tr 2 (0) 4F Z Ex . It is known that Binary Phase Shift Keying (BPSK) gives the maximum diversity between real signals equal to 2, since the signals are oppo- site. In our case, the signals (12) are not opposite and the closest signals differ by 1 and their scalar product is 0. However, when calculating the Frobenius norm, we get a diversity equal to 4, which is 2 times greater than the diversity for BPSK . V. CONCLUSION Thus, it is shown that the use of complex numbers in the matrix representatio n makes it possible to increase the diversity of information symbols upon reception by a factor of 2. In this case, unlike the Alamouti scheme, one antenna is used for transmission and one for reception. Moreover, instead of a complex system for estimating complex 2×2 channel gains, it is sufficient to use a conventional phase locking system. The proposed scheme can also be used in wired com- munication channels.