Efficiency Estimation of MIMO Communication System Channel Coefficients Based on Markov Model

Based on Markov model of MIMO data transmission system channel coefficients from moving object held synthesis and analysis of channel coefficients error estimation. The model takes into account distribution configuration of reflectors in area, reception center and moving object location at a given time, kind of trajectory and speed of the moving object. A theory discrete Markov processes linear filtration is used by the development of estimation algorithm. Posteriori dispersions matrix is calculated. The simulation of filtering algorithm for different signal-interference situations is carried out.


Introduction
Due to the expansion of applications of mobile objects to accumulate various types of information arises the problem of transmitting information from mobile object to receiving station.Transmission of optical images and radar high-resolution images in real-time shows increased requirements to channel capacity.Implementation of required bandwidth causes difficulties due to multipath propagation of radio waves from mobile object to receiving station.In these conditions data transmission systems based on MIMO technology [1] are widely used.To realize MIMO channel potential performance it is necessary to measure channel coefficients from each transmit antenna to each receive antenna with high accuracy.The channel estimation errors has a significant impact on channel capacity [2][3][4].The reduction of channel coefficients estimation errors can be realize by the most complete account of their statistical properties, as well as development of optimal estimation algorithm.The statistical description of channel coefficients most frequently uses Gaussian probability density distribution.However, the usage of correlation properties of channel coefficients and their dependence on speed of object movement in the estimation algorithm design is not enough to allow potential accuracy.Determination of the optimal mode of channel coefficients matrix estimation is considered in [5], but the possibility of accounting channel coefficients correlation in their estimation also not implemented.
The purpose of the article is to increase the MIMO data transmission system capacity by reduction of channel coefficients estimation error, which is achieved by using the optimal algorithm for filtering channel coefficients based on the Markov model.

Statement of the Problem
To describe the signal propagation in MIMO channel a geometric one-ring scattering model (Figure 1) [6]  In accordance with the model, channel matrix coefficient is determined by the following expression: , where ( ) -the coefficient describes the phase shift of the signal from the transmitting antenna to one of scatterers; ( ) -coefficient describes the phase shift of the signal from one of the scatterers to the receiving antenna, , and it is assumed to be constant when the calculation of channel coefficients.Attenuation of the signal amplitude when passing from the transmitter to the scatterer, from the scatterer to the receiver, and the reflection coefficient is not take into account when the calculation of the normalized correlation function.
The behavior of channel coefficients in time is important when estimating channel coefficients and capacity of MIMO data transmission system.In general, changes are specified correlation function of each of coefficients, which reflects the rate of change over time and especially changes of module and argument of complex channel coefficients: Because phase shift introduced by each scatterer, are independent and identically distributed random variables with uniform distribution law in the interval [ ) , have the equality:

Development of Channel Coefficients Model And Filtering
Values of the MIMO communication system channel coefficients matrix correlation ( ) , obtained in accordance with (1).  .We also believe that relocation of mobile object is insignificantly during the filtration interval, which suggests that channel coefficients are stationary.
To ensure the independence of channel coefficients the size of discrete noise column vector k w is equal to the number of different channel coefficients RX TX N N .Submission process being evaluated in the form of ( 2) is widely used in optimal filtration theory [7][8][9].

Matrix of forming filter of size
can be calculated from measurements of correlation matrices with different delays into time [10,11]: Assuming a stationary random process matrix G is solution of the following equation by Cholesky decomposition: ( ) Thus completely determined stochastic differential equation ( 2), specifies time variation of channel coefficients.

Priory correlation matrix of stationary channel coefficients is equal [10]
, where priory variances matrix determined from the linear algebraic equation: Single out the terms containing the matrix posteriori variances: To solve this equation, it is advisable to introduce a matrices P , A in vector form in rows: Represent equation (3) in a vectorized form: , where vectorized matrix is the Kronecker product . Solution vectorized equation is: For optimal criterion of minimum mean squared error channel coefficients estimates used the method of Markov filtering of discrete random sequences [7][8][9], using developed dynamic model of channel coefficients in the form of a stochastic difference equation ( 2).The measured values of channel coefficients at each discrete point in time contain measurement error k Ε , which is assumed to be uncorrelated Gaussian random variable with correlation matrix Recursive filtering algorithm is given by [8]: ( ) where a posteriori variance of channel coefficients estimation determined by the difference equation: Expressions ( 4) and ( 5) compose optimal filtering algorithm, and matrix k V sets the channel coefficients posteriori dispersion, and its diagonal coefficients are estimation error variance.

Analysis of Channel Coefficients Estimating Errors
If channel coefficients are formed by difference equation ( 2), the channel coefficient estimation error variance determined by diagonal matrix elements k V .Variances matrix of channel coefficient estimation errors can also be obtained by means of statistical modeling filtration equation ( 4) and averaging over S N realizations: For more comprehensive taking into account the properties of channel coefficients can, it is necessary to use values H for recording the measured values at discrete time , and equation ( 4) when the simulation.Matrix of posteriori variances ( )( ) is obtained by the simulation and it is more accurate than x ps _

D
, since more fully take into account the form of the correlation function (1).When technical implementation of channel coefficients estimation algorithms (4) and ( 5) must perform a certain number of arithmetic operations in real time.This requires computing power of embedded processor: computing speed and amount of memory.Equations ( 5) can be solved in advance and results stored in memory, the volume of which is equal ( ) 2 / 1 − L L complex words.To compute channel coefficients vector estimates (4) is necessary to make ( )  1), which is about 50 μs, we conclude that technical implementation of proposed algorithm is possible.

Conclusion
Investigated Markov model of MIMO data transmission system channel coefficients from mobile object, taking into account different conditions of radio waves propagation and reflections.Application of channel coefficients filtering algorithm provides potential estimation accuracy.However, for implementation of optimal estimation algorithm need to know correlation properties of channel coefficients that as shown in this work depend on specific practical situations.Computational costs for technical implementation of proposed channel coefficients estimating algorithm are realized with modern signal processors.
Research performed in part of grant of the Russian Scientific Foundation, project № 14-19-01263.
from the transmitting antenna to one of scatterers, shift is determined by radial component of the moving object speed in the direction of the terrestrial antenna system, -the speed of wave propagation.Phase shift k θ , introduced by each scatterer, are independent and identically distributed random variables with uniform distribution law in the interval

[
objects for urban environment and indoor as well.Represent channel coefficients model in the form of a Markov sequence

Figure 2 .
Figure 2. Channel coefficient correlation function Given M correlation matrix values j K , M j ,..., 0 = , state vector of the dynamic system ,.., 2 , 1 , = k k x forming all channel coefficients.Dimension of channel coefficients vector is equal R T N N L =.We also believe GHz clock speed, it can be shown that necessary computing time can be 68 ns.Comparing this time with time interval necessary to update channel coefficients (Figure