CN106817155A - A kind of extensive MIMO low complexity channel estimation methods based on weighting Kapetyn series expansions - Google Patents

Abstract

The invention discloses a kind of extensive MIMO low complexity channel estimation methods based on weighting Kapetyn series expansions.Approximate expansion is carried out to the channel covariancc inverse matrix in Bayesian MMSE channel estimation expression formulas using Kapteyn Series Expansion Methods first, matrix inversion operation is converted into matrix multiplication and addition of matrices computing, then polynomial expansion is optimized using weighting scheme per term coefficient to multinomial, setting up model to weight vector α and β solve the mean square error minimum for causing to estimate, the solving result using α and β is estimated channel matrix.Test result indicate that with the increase of polynomial order, the MSE that the channel estimation methods based on weighting Kapetyn series expansions are obtained can converge on MMSE method, but computation complexity is less than MMSE method.Contrast with tradition Taylor MMSE and Kapetyn series expansion channel estimation methods, based on weight Kapetyn series expansion methods converge to MMSE method speed faster.

Description

A kind of extensive MIMO low complex degrees channel based on weighting Kapetyn series expansions Method of estimation

Technical field

The invention belongs to moving communicating field, the low complex degree channel estimation related generally in extensive mimo system is calculated Method.Extensive mimo system channel condition information is estimated using the Kapetyn series expansion algorithms based on weighting specifically Meter, sets up optimal solution of the model using the multinomial coefficient of solution by iterative method series expansion so that the MSE that algorithm is estimated is faster Converge to MMSE algorithms.

Background technology

Extensive mimo system is one of core technology of the 5th Generation Mobile Communication System, and base station is equipped with big in such a system The antenna (being more than 100) of amount is serviced for more mobile subscribers, to obtain spectrum efficiency higher, message transmission rate and handle up Amount and preferably communication quality.Efficient CSI acquisitions are the key issues that extensive mimo system realizes performance advantage, Each mobile subscriber is to Base Transmitter training signal in time division duplex (TDD, time-division duplexing) system, The CSI of up-link is estimated with MMSE algorithms after base station received signal, then using the conjugate transposition of estimate under The CSI of line link.

Traditional MMSE algorithm for estimating is estimated CSI, it is necessary to calculated the inverse of higher dimensional matrix, complexity is very high, is utilized Kapetyn Series Expansion Methods carry out estimation and can significantly reduce computation complexity instead of matrix inversion to channel, work as order of a polynomial When number N and Kapteyn exponent numbers K tends to infinite, Kapetyn series expansions estimate that the MSE (mean square error) for obtaining can be converged on MMSE method.But in practice, for fixed N and K, Kapetyn series expansion evaluated errors are bigger than normal.Can utilize The Kapetyn series expansion algorithms of weighting replace matrix inversion and channel are estimated, accelerate algorithm the convergence speed.

The content of the invention

The present invention is estimated higher dimensional matrix inversion process using the Kapetyn series expansion algorithms of weighting, main to think Think it is the Representation theorem coefficient weighting to Kapetyn series expansions, propose that a kind of iterative algorithm solves optimal system of polynomials Number causes that the MSE for estimating is minimum, accelerates algorithm the convergence speed, realizes good to extensive mimo system channel low complex degree Estimate.

Technical scheme：

A kind of extensive MIMO low complexity channel estimation methods based on weighting Kapetyn series expansions, including step：

1) dock collection of letters model vector and channel estimate matrix is obtained using traditional MMSE channel estimation methods：

Wherein,The estimation of channel matrix is represented,R represents channel covariance matrices, and P is represented and led Frequency signal matrix, defining pilot matrix form isS is noise covariance matrix,Y represents reception letter Number, n represents noise signal；

2) binomial expansion is carried out to formula (1) using Kapteyn Series Expansion Methods and coefficient is weighted, obtained based on weighting Kapetyn series expansions estimator：

Wherein N represents polynomial order, and K represents kapteyn exponent numbers, and C represents covariance matrix, α=[α₀,...,α_N]^TWith β=[β₀,...,β_K]^TIt is weight coefficient；

3) unconfined Non-linear Optimal Model is set up：

And formula (3) solve obtain optimal coefficient α and β；

4) by step 3) α that obtains and β substitutes into step 2) can try to achieve based on the Kapetyn for weighting in the formula (2) that obtains The channel estimation value of series expansion algorithms.

The step 3) solve unconfined Non-linear Optimal Model and be specially：

Formula (3) is resolved into two linear optimization subproblems：

31) the sub- linear optimization problem based on fixing Beta：

For each fixing Beta, formula (3) formula is changed into following optimization problem：

Formula (4) is sought into local derviation to each coefficient, is 0 with season derivative, try to achieve optimal coefficient α；

32) the sub- linear optimization problem based on fixed α：

α is set to fixed value α=α °, wushu (3) is decomposed into sub- linear optimization problem that another is based on fixed α；Should Sub- optimization problem is described as follows：

Similarly, partial derivative is sought formula (5), and it is 0 to make local derviation, you can obtain optimal coefficient β；

33) for giveBy step 31) obtain optimal coefficient α^*；And update α：

34) for giveBy step 32) obtain optimal coefficient β^*；And update β：

35) repeat step 33), 34) until object function MSE (α, β) is minimum, obtain optimal coefficient α and β.

The step 31) in solve optimal coefficient α specific as follows：

Object function asks the formula after partial derivative to be defined as follows each coefficient：

To each n=0 ..., N, make formula (6) equal to 0, obtain N+1 system of linear equations：

A and b is tried to achieve using formula (7), object function in formula (4) is converted into following matrix form：

MSE (α)=tr (R)-b^Hα-α^Hb+α^HAα (8)

Simultaneously in optimal coefficient α^*=A^-1Under b, formula (7) is converted into MSE (α^*)=tr (R)-b^HA^-1b；Existed using this equation The MSE of least estimated is obtained under fixed coefficient β.

A is tried to achieve using formula (7) and b is as follows：

Receive relationship description between signal and covariance matrix C as follows：

Y in formula (9)_tRepresent the reception signal in moment t；As T ＞ ＞ BN, covariance matrix C is equal to A and b in formula (7) can be reduced to following form：

Notice that A and b have following simplified form of calculation simultaneously：

The value of A and b is obtained using formula (11).

Beneficial effect：Test result indicate that, calculated compared to traditional Taylor-MMSE algorithms and Kapetyn series expansions Method, the Kapetyn series expansion algorithms based on weighting can effectively accelerate convergence rate, in the feelings that polynomial order N is limited Less MSE can also be obtained under condition, the accuracy of channel estimation is improve.

Brief description of the drawings

Fig. 1 is that the present invention is 0 in pilot pollution factor beta, and transmission antenna number is 10, and reception antenna number is 100, kapteyn Cut sets order exponent number be equal to 5dB under conditions of, to based on weighting Kapetyn series expansion algorithms obtain channel estimation with do not have The channel matrix that the Kapteyn-MMSE algorithms and Taylor-MMSE algorithms for having weighting are obtained is compared.

Specific embodiment

The present invention is further described below in conjunction with the accompanying drawings.

Specific steps of the present invention include：

1) set receipt signal model as：

Y=HP+N

Wherein Y represents reception signal, and H represents the mimo channel matrix under the conditions of quasistatic flat fading channel,Wherein N_rRepresent the reception antenna number of base station side, N_tTransmitting antenna number is represented,R generations Table channel covariance matrices,P represents the pilot signal matrix of transmitting terminal,N is additive noise signal, Obey Cyclic Symmetry multiple Gauss random distribution.

Docking collection of letters vectorization, defining pilot matrix form isThen above formula can be converted into as it is following to Amount form：

2) can derive that channel estimate matrix is as follows using traditional MMSE channel estimation methods：

WhereinThe estimation of channel matrix is represented,R represents channel covariance matrices, and P is represented and led Frequency signal matrix, defining pilot matrix form isS is noise covariance matrix,Y represents reception letter Number, n represents noise signal；

3) (1) formula is carried out into Kapteyn series expansions and coefficient is weighted, obtain following Kapetyn grades based on weighting The estimator that number launches：

Wherein N represents polynomial order, and K represents kapteyn exponent numbers, and C represents covariance matrix, α=[α in (2)₀,..., α_N]^TWith β=[β₀,...,β_K]^TIt is weight coefficient；

4) MSE estimated to find suitable α and β to cause is minimized, and sets up following unconfined Non-linear Optimal Model：

In order to avoid directly going to seek nonlinear optimal problem, below text in propose (3) are resolved into two linear optimization Problem, so as to solve the nonlinear optimal problem.

31) the sub- linear optimization problem based on fixing Beta

For each fixing Beta, MSE during α optimal values cause (3) formula can be obtained minimum.Therefore (3) formula can be changed into Following optimization problem：

In order to solve unconfined linear optimization problem (4), it is necessary to object function first is asked into local derviation to each coefficient, while It is 0 to make derivative, so as to try to achieve optimal coefficient α.Object function asks the formula after partial derivative to be defined as follows each coefficient：

To each n=0 ..., N, make (5) equal to 0, can obtain N+1 system of linear equations, the linear side is solved below Journey group is that can obtain optimal coefficient α.The system of linear equations form for obtaining is as follows：

Followed by the A and b that try to achieve, object function in (4) can be converted into following matrix form：

MSE (α)=tr (R)-b^Hα-α^Hb+α^HAα (7)

Simultaneously in optimal coefficient α^*=A^-1Under b, (7) can turn to MSE (α^*)=tr (R)-b^HA^-1b.Can be with using this equation The MSE of least estimated is obtained under fixed coefficient β.It is specifically described below and how solves A and b.

A and b is calculated in order to quick, can be believed to use to receive by receiving relation between signal and covariance matrix C Number estimate covariance matrix.Receive relationship description between signal and covariance matrix C as follows：

(8) y in_tRepresent the reception signal in moment t；As T ＞ ＞ BN, covariance matrix C is approximately equal toA and b in (6) can be reduced to following form below：

Notice that A and b have following simplified form of calculation simultaneously：

(10) formula of utilization can reduce the complexity for calculating A and b, obtain the value of A and b, and then can be in the hope of optimal coefficient α^*。

32) the sub- linear optimization problem based on fixed α

α is set to fixed value α=α °, (3) another sub- linear optimization problem for being based on fixed α can be decomposed into. The sub- optimization problem is described as follows：

Similarly, it is a linear optimization problem on β (11) to be can be seen that from object function, only need to ask inclined to it Derivative, it is 0 then to make local derviation, you can obtain optimal coefficient β.The formula that (11) seek factor beta partial derivative is given below：

Then, to each k=0 ..., K, it is 0 to make (11), can obtain K+1 on β_kSystem of linear equations.Then lead to Cross and solve the system of linear equations, you can obtain the optimal coefficient β when factor alpha is fixed.The system of linear equations form is given below such as Under：

(13) system of linear equations described by formula is similar with the problem that formula (6) is described, therefore solution (13) can be by upper The solution of section, and then optimal coefficient β can be obtained.Under fixed coefficient α, object function in (11) can be converted into MSE (β^*)=tr (R)-b^HA^-1b。

By combining (1) and (2) two trifles to two solutions of linear optimization subproblem, a kind of iteration is set forth below Algorithm solves nonlinear optimal problem in (3).From the algorithm of table 1, it can be seen that non-linear objective function in each iteration MSE (α, β) is nonincremental.It is described in detail below：

Table 1 solves the iterative algorithm of α and β

Optimal coefficient α that iterative algorithm is obtained and_βSubstitute into formula (2), you can try to achieve the Kapetyn series based on weighting The channel estimation value of deployment algorithm.

The above is only the preferred embodiment of the present invention, it should be pointed out that：For the ordinary skill people of the art For member, under the premise without departing from the principles of the invention, some improvements and modifications can also be made, these improvements and modifications also should It is considered as protection scope of the present invention.

Claims (4)

1. a kind of based on the extensive MIMO low complexity channel estimation methods for weighting Kapetyn series expansions, it is characterised in that： Including step：

1) dock collection of letters model vector and channel estimate matrix is obtained using traditional MMSE channel estimation methods：

${\hat{h}}_{mmse}=\hat{h}+R{\tilde{P}}^H{(\tilde{P}R{\tilde{P}}^H+S)}^{-1}(y-\tilde{P}\stackrel{\‾}{h}-\stackrel{\‾}{n})---\left(1\right)$

Wherein,The estimation of channel matrix is represented,R represents channel covariance matrices, and P represents that pilot tone is believed Number matrix, defining pilot matrix form isS is noise covariance matrix,Y represents reception signal, n Represent noise signal；

2) binomial expansion is carried out to formula (1) using Kapteyn Series Expansion Methods and coefficient is weighted, obtained based on weighting The estimator of Kapetyn series expansions：

$\left\{\begin{array}{c}{\hat{h}}_{w-kapteyn}=\stackrel{\‾}{h}+R{\tilde{P}}^H\underset{n=0}{\overset{N}{\Σ}}{d}_w^{n+1}{q}_n{\mathrm{\α}}_n{C}^n{Y}_{n,K}(y-\tilde{P}\stackrel{\‾}{h}-\stackrel{\‾}{n})\\ Y_{n,K}=\underset{k=0}{\overset{K}{\Σ}}{d}_w^{2k}{J}_n\left(k\right){\mathrm{\β}}_k{C}^{2k}\end{array}\right.---\left(2\right)$

Wherein N represents polynomial order, and K represents kapteyn exponent numbers, and C represents covariance matrix, α=[α₀,...,α_N]^TWith β= [β₀,...,β_K]^TIt is weight coefficient；

3) unconfined Non-linear Optimal Model is set up：

$\underset{\mathrm{\α},\mathrm{\β}}{min}:MSE(\mathrm{\α},\mathrm{\β})=E\left\{\right||h-{\hat{h}}_{w-kapteyn}|{|}_F^2\}---\left(3\right)$

And formula (3) solve obtain optimal coefficient α and β；

4) by step 3) α that obtains and β substitutes into step 2) can try to achieve based on the Kapetyn series for weighting in the formula (2) that obtains The channel estimation value of deployment algorithm.

2. extensive MIMO low complexity channel estimation methods according to claim 1, it is characterised in that：The step 3) Unconfined Non-linear Optimal Model is solved to be specially：

Formula (3) is resolved into two linear optimization subproblems：

31) the sub- linear optimization problem based on fixing Beta：

For each fixing Beta, formula (3) formula is changed into following optimization problem：

Formula (4) is sought into local derviation to each coefficient, is 0 with season derivative, try to achieve optimal coefficient α；

32) the sub- linear optimization problem based on fixed α：

α is set to fixed value α=α °, wushu (3) is decomposed into sub- linear optimization problem that another is based on fixed α；The son is excellent Change problem is described as follows：

Similarly, partial derivative is sought formula (5), and it is 0 to make local derviation, you can obtain optimal coefficient β；

33) for giveBy step 31) obtain optimal coefficient α^*；And update α：

34) for giveBy step 32) obtain optimal coefficient β^*；And update β：

35) repeat step 33), 34) until object function MSE (α, β) is minimum, obtain optimal coefficient α and β.

3. extensive MIMO low complexity channel estimation methods according to claim 2, it is characterised in that：The step 31) optimal coefficient α is solved in specific as follows：

Object function asks the formula after partial derivative to be defined as follows each coefficient：

$\frac{\∂}{\∂{\mathrm{\α}}_n}MSE\left(\mathrm{\α}\right)=-d_w^{n+1}{q}_{n}tr\left\{R{\tilde{P}}^H{C}^n{Y}_{n,K}\tilde{P}R\right\}+\underset{n_2=0}{\overset{N}{\Σ}}{\mathrm{\α}}_{n_2}^{*}tr\left\{R{\tilde{P}}^H{d}_w^{n_1+n_2+2}{q}_{n_1}{q}_{n_2}{Z}_2\tilde{P}R\right\}---\left(6\right)$

To each n=0 ..., N, make formula (6) equal to 0, obtain N+1 system of linear equations：

$\left\{\begin{array}{c}A\mathrm{\α}=b\\ \[A_{i,j}\]=d_w^{i+j}{q}_{i-1}{q}_{j-1}tr\left\{R{\tilde{P}}^H{C}^{i-1}{Y}_{i-1,K}{\mathrm{CY}}_{j-1,K}{C}^{j-1}\tilde{P}R\right\}\\ \[b_i\]=d_w^i{q}_{i-1}tr\left\{R{\tilde{P}}^H{C}^{i-1}{Y}_{i-1,K}\tilde{P}R\right\}\end{array}\right.---\left(7\right)$

A and b is tried to achieve using formula (7), object function in formula (4) is converted into following matrix form：

MSE (α)=tr (R)-b^Hα-α^Hb+α^HAα (8)

Simultaneously in optimal coefficient α^*=A^-1Under b, formula (7) is converted into MSE (α^*)=tr (R)-b^HA^-1b；It is fixed using this equation The MSE of least estimated is obtained under number β.

4. extensive MIMO low complexity channel estimation methods according to claim 3, it is characterised in that：Using formula (7) Try to achieve A and b is as follows：

Receive relationship description between signal and covariance matrix C as follows：

$C=(\tilde{P}R{\tilde{P}}^H+S)=E\left\{{\mathrm{yy}}^H\right\}=\underset{T\→\mathrm{\∞}}{\lim }\frac{1}{T}\underset{t=1}{\overset{T}{\Σ}}{y}_t{y}_t^H---\left(9\right)$

Y in formula (9)_tRepresent the reception signal in moment t；As T ＞ ＞ BN, covariance matrix C is equal toBy formula (7) A and b can be reduced to following form in：

Notice that A and b have following simplified form of calculation simultaneously：

$\begin{array}{c}tr\left\{R{\tilde{P}}^H{C}^n\tilde{P}R\right\}=tr\left\{R{\tilde{P}}^H{(\tilde{P}R{\tilde{P}}^H+S)}^n\tilde{P}R\right\}\\ =tr\left\{R{\tilde{P}}^H{(\tilde{P}R{\tilde{P}}^H+S)}^{n-1}\left(\frac{1}{T}\underset{t=1}{\overset{T}{\Σ}}{y}_t{y}_t^H\right)\tilde{P}R\right\}\\ =\frac{1}{T}\underset{t=1}{\overset{T}{\Σ}}{y}_t^H\left(\tilde{P}{R}^2{\tilde{P}}^H{\left(\tilde{P}R{\tilde{P}}^H+S\right)}^{n-1}\right)y_t\end{array}---\left(11\right)$

The value of A and b is obtained using formula (11).