CN104618006A - Distributed multi-user downstream MIMO robust beam forming method - Google Patents

Abstract

The invention provides a distributed multi-user downstream MIMO (multiple input multiple output) robust beam forming method. Specifically, aiming at estimation, quantization and delay error which coexist in CSI (channel state information) fed back to a BS (base station) by mobile users in practical mobile communication application of a multi-user MIMO FDD (frequency division duplexing) system, and based on the channel conditions, the robust beam forming method based on signal leakage MMSE (minimum mean-square error) and having distributed characteristic is provided. The forming method can be applied to a more practical distributed mobile communication system, can independently design beam forming matrix for a single mobile user, and can better meet the requirements of the mobile users on quality of service (QoS) of communication.

Description

A kind of distributed multi-user descending MIMO robust beam-forming method

Technical field

The present invention relates to multiple-input, multiple-output (MIMO, multiple input multiple output) communication system, particularly multiuser downstream MIMO robust beam forming optimal design.

Background technology

MIMO technology, because it can effectively improve the advantage such as spectrum efficiency and transmission rate, has been widely used in Modern Communication System, as multiple cell multi-user cell mobile communication systems etc.But in MIMO downlink system, multiple user and BS, at the enterprising Serial Communication of same running time-frequency resource, inevitably exist interference mutually between user, therefore in order to solve problems and eliminate interference, general beam forming technique of selecting is applied to BS end.

Early stage beam-forming method all supposes that BS can obtain desirable CSI.But in practical communication application, BS cannot obtain desirable CSI.As in time division duplex (TDD, time division duplexing) system, BS obtains CSI to channel estimating can produce channel estimation errors; And at Frequency Division Duplexing (FDD) (FDD, frequencydivision duplexing) in system, it is channel estimating by user side and Limited Feedback that BS obtains the mode of CSI, and wherein Limited Feedback can bring quantization error, and channel feedback procedure can cause delay time error.Therefore mobile subscriber feeds back in the CSI of BS and often there is estimation simultaneously, quantizes and delay time error, and under channel condition based on this kind of imperfect CSI, multiuser downstream robust beamforming design becomes current study hotspot.

Find from existing China and foreign countries detailed investigation of related literatures, although there are some robust beamforming design to MIMO downlink user, all based on rationality CSI channel condition.In addition, in existing document, also existing correlative study enters singly to go out (MISO based on the multiuser downstream under Limited Feedback channel more, multiple input singleoutput) under there is the CSI of delay time error in robust beam forming problem or consideration, based on the multiuser downstream MISO robust beam forming problem of Limited Feedback channel, but above-mentioned carried research is also all hypothesis channel estimating is desirable, does not namely have channel estimation errors.And in actual FDD communication system, channel estimation errors is inevitable, and on performance of mobile communication system, there is very important impact.

Based on this, the present invention consider exist at the same time estimations, quantification and delay time error imperfect CSI condition under, using BS transmitting power as constraints, MMSE is as majorized function, according to convex optimum theory, set up the target function based on the MMSE of leakage signal, derive a kind of method of distributed downlink MIMO robust beam forming, thus effective elevator system performance.

Summary of the invention

Goal of the invention: in multi-user MIMO communication system, based on proposing the descending MIMO beamforming design method of a kind of distributed multi-user under imperfect CSI condition.The method takes into full account that channel quantitative, time delay and estimation three factors are on the impact of beamforming matrix, based on the MMSE criterion design beamforming matrix of leakage signal, finally derives distributed system optimal beam forming matrix and closed solutions thereof.

Technical scheme of the present invention:

Imperfect CSI: estimate, quantize and delay time error.

System model is multiuser downstream mimo system, and wherein BS is K user's service on same running time-frequency resource, and BS configures M root transmitting antenna, and each user configures N root reception antenna, as shown in Figure 1.

The signal that a kth user receives can be expressed as:

$y_k=G_k^H\mathrm{Wx}+n_k---\left(1\right)$

Wherein, represent from BS to the channel matrix of user k, its element is separate and be the multiple Gaussian random variable of zero-mean, unit variance, represent the Gauss's additive white noise vector received by user k, its element is separate and be the multiple Gaussian random variable of zero-mean, unit variance, with be respectively pre-coding matrix and transmission signal vector, () ^hthe conjugate transpose of representing matrix.

The signal that all K user receives is:

y＝GWx+n (2)

Wherein, g=(G ₁..., G _k) ^h, () ^tthe transposition of representing matrix.Notice E [nn ^h]=1, E [xx ^h, if BS transmitting power is restricted to P, then there is tr (WW]=1 ^hthe mark of)≤P, tr () representing matrix.

In the simultaneous situation of channel estimating, quantification and delay time error, the channel matrix G of the n-th moment BS to user k _k[n] is modeled as:

$\begin{array}{c}{G}_k\left[n\right]={\mathrm{\ρ}}_k\left(H_k\right[n-1\left]U_k\right[n-1]V{[n-1]}_k+S_k[n-1\left]Z_k\right[n-1\left]\right)\sqrt{{\mathrm{\Λ}}_k[n-1]}\\ +E_k\left[n\right]+{\mathrm{\Δ}}_k[n-1]\end{array}---\left(3\right)$

Write (3) formula as matrix form, supposed G=(G ₁... .., G _k) ^h, obtain:

G＝AH+BS+E+Δ (4)

Wherein, H=[H ₁[n-1] ..., H _k[n-1]] ^h, E=[E ₁[n] ..., E _k[n]] ^h, D=diag (ρ ₁..., ρ _k),

$C=\mathrm{diag}\left(V_1^H\right[n-1\left]U_1^H\right[n-1],...,V_K^H[n-1\left]U_K^H\right[n-1\left]\right),S={\left[S_1\right[n-1],...,S_K[n-1\left]\right]}^H$

$\mathrm{\Λ}=\mathrm{diag}(\sqrt{{\mathrm{\Λ}}_1^H[n-1]},...,\sqrt{{\mathrm{\Λ}}_K^H[n-1]}),Z=\mathrm{diag}\left(Z_1^H\right[n-1],...,Z_K^H[n-1\left]\right),A=\mathrm{D\ΛC},$

B=D Λ Z, diag () represent diagonal matrix.

Distributed beams form finding design:

Namely, in practical communication system, BS is that each user designs pre-coding matrix separately.The present invention is based on the MMSE expression formula of leakage signal, consider the imperfect CSI situation that BS obtains, propose the distributed beams manufacturing process with robustness.

Write (1) formula as following form:

Wherein, y' _kfor useful signal and the noise item of user, Section 2 is the multi-user interference that user receives.In practical communication system, we wish that the interference that user receives is little, i.e. primary signal x _kwith y' _krelatively.Based on this thought, and the distributed nature that binding signal is revealed, the MMSE that we define based on leakage signal under multiuser downstream MIMO is:

$\begin{array}{c}\mathrm{\ϵ}(W_k,\mathrm{\β})=E_{x,n}{\left|\right|x_k-{\mathrm{\β}}^{-1}{y}_k^{\′}\left|\right|}^2+E_{x,n}\left[\underset{m=1,m\≠k}{\overset{K}{\mathrm{\Σ}}}{\left|\right|{\mathrm{\β}}^{-1}{G}_m^H{w}_k{x}_k\left|\right|}^2\right]\\ ={\mathrm{\β}}^{-2}\underset{m=1}{\overset{K}{\mathrm{\Σ}}}{\left|\right|G_m^H{W}_k\left|\right|}_F^2-2{\mathrm{\β}}^{-1}\mathrm{Tr}\left[\mathrm{Re}\left(G_k^H{W}_k\right)\right]+({\mathrm{\β}}^{-2}+1)N\end{array}---\left(6\right)$

Thus, we obtain the MMSE robust beamforming design problem based on leakage signal:

min E _<G|H>ε(W _k,β)

$\begin{array}{cc}s.t.& \underset{k=1}{\overset{K}{\mathrm{\&Sigma;}}}\mathrm{tr}\left(W_k{W}_k^H\right)\&le;P\end{array}$ (7)

Wherein E _<G|H>ε (W _k, β) reached by following formula:

$\begin{array}{c}{E}_{<G|H>}\mathrm{\&epsiv;}(W_k,\mathrm{\&beta;})\\ ={\mathrm{\&beta;}}^{-2}{E}_{<G|H>}\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{{\left|\right|G_m^H{W}_k\left|\right|}_F}^2-2{\mathrm{\&beta;}}^{-1}{E}_{<G|H>}\left\{\mathrm{tr}\right[\mathrm{Re}\left(G_k^H{W}_k\right)\left]\right\}+({\mathrm{\&beta;}}^{-2}+1)N\\ ={\mathrm{\&beta;}}^{-2}[M(1-\frac{\mathrm{\&delta;N}}{N})-\frac{\mathrm{M\&delta;}}{M-N}]\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{\left|\right|{\mathrm{DH}}_m{W}_k\left|\right|}_F^2+{\mathrm{\&beta;}}^{-2}[K+(\frac{\mathrm{M\&delta;}}{M-N}-1){\left|\right|D\left|\right|}_F^2+\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{\mathrm{\&sigma;}}_m^2]\mathrm{tr}\left(W_k{W}_k^H\right)\\ -2{\mathrm{\&beta;}}^{-1}\mathrm{\&alpha;}{\mathrm{\&rho;}}_k\mathrm{Re}\left[\mathrm{tr}\left(H_k{E}_k\right)\right]+({\mathrm{\&beta;}}^{-2}+1)N\end{array}---\left(8\right)$

Problem (7) is convex optimization problem, and KKT condition can be utilized to solve, and builds Lagrange's equation as follows:

$l(W_k,\mathrm{\&beta;},{\mathrm{\&lambda;}}_k)=E_{<G|H>}\mathrm{\&epsiv;}(W_k,\mathrm{\&beta;})+{\mathrm{\&lambda;}}_k[\underset{k=1}{\overset{K}{\mathrm{\&Sigma;}}}\mathrm{tr}\left(W_k{W}_k^H\right)-P]---\left(9\right)$

(9) l (W in formula _k, β, λ _k) to W _k, β asks partial derivative respectively:

$\frac{\&PartialD;l(W,\mathrm{\&beta;},\mathrm{\&lambda;})}{\&PartialD;W}={\mathrm{\&beta;}}^{-2}[M(1-\frac{\mathrm{\&delta;}}{N})-\frac{\mathrm{M\&delta;}}{M-N}]\underset{m-1}{\overset{K}{\mathrm{\&Sigma;}}}{D}^2{H}_m^H{H}_m{W}_k+{\mathrm{\&beta;}}^{-2}[K+(\frac{\mathrm{M\&delta;}}{M-N}-1){\left|\right|D\left|\right|}_F^2+\underset{m-1}{\overset{K}{\mathrm{\&Sigma;}}}{\mathrm{\&sigma;}}_m^2]W_k-\mathrm{\&alpha;}{\mathrm{\&beta;}}^{-1}{\mathrm{\&rho;}}_k{H}_k^H+{\mathrm{\&lambda;}}_k{W}_k=0---\left(10\right)$

$\frac{\&PartialD;l(W,\mathrm{\&beta;},\mathrm{\&lambda;})}{\&PartialD;\mathrm{\&beta;}}=[M(1-\frac{\mathrm{\&delta;}}{M-N})\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{\left|\right|{\mathrm{DH}}_m{W}_k\left|\right|}_F^2+[K+(\frac{\mathrm{M\&delta;}}{M-N}-1){\left|\right|D\left|\right|}_F^2+\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{\mathrm{\&sigma;}}_m^2]\mathrm{tr}\left(W_k{W}_k^H\right)-\mathrm{\&alpha;\&beta;}{\mathrm{\&rho;}}_k\mathrm{Re}[\mathrm{tr}\left]\left(H_k{W}_k\right)\right]+N=0---\left(11\right)$

According to (10) formula, the beamforming matrix W of user k can be obtained _k:

$W_k={\mathrm{\&alpha;\&beta;\&rho;}}_k{\left\{\right[M(1-\frac{\mathrm{\&delta;}}{N})-\frac{\mathrm{M\&delta;}}{M-N}]\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{D}^2{H}_m^H{H}_m+[K+(\frac{\mathrm{M\&delta;}}{M-N}-1){\left|\right|D\left|\right|}_F^2+\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{\mathrm{\&sigma;}}_m^2+{\mathrm{\&lambda;}}_k{\mathrm{\&beta;}}^2\left]I_N\right\}}^{-1}{H}_k^H---\left(12\right)$

From (12) formula, as long as know λ _kβ ², can W be obtained _kclosed solutions.In order to obtain λ _kβ ², to (11) formula premultiplication and track taking obtains:

$[M(1-\frac{\mathrm{\&delta;}}{N})-\frac{\mathrm{M\&delta;}}{M-N}]\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{\left|\right|{\mathrm{DH}}_m{W}_k\left|\right|}_F^2+[K+(\frac{\mathrm{M\&delta;}}{M-N}-1){\left|\right|D\left|\right|}_F^2+\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{\mathrm{\&sigma;}}_m^2]\mathrm{tr}\left(W_k{W}_k^H\right)+{\mathrm{\&lambda;}}_{k2}{\mathrm{\&beta;}}^2\mathrm{tr}\left(W_k^H{W}_k\right)-\mathrm{\&alpha;\&beta;}{\mathrm{\&rho;}}_k\mathrm{tr}\left(H_k^H{W}_k\right)=0---\left(13\right)$

Simultaneous (10) formula and (13) formula can obtain: again by BS transmitting power restrictive condition $\underset{k=1}{\overset{K}{\mathrm{\&Sigma;}}}\mathrm{tr}\left(W_k{W}_k^H\right)=P,$ Can obtain:

${\mathrm{\&lambda;}}_k{\mathrm{\&beta;}}^2=\frac{\mathrm{NK}}{\underset{k=1}{\overset{K}{\mathrm{\&Sigma;}}}\mathrm{tr}\left(W_k{W}_k^H\right)}=\frac{\mathrm{NK}}{P}---\left(14\right)$

(14) formula is substituted into (12) formula obtain:

$W_k={\mathrm{\&alpha;\&beta;\&rho;}}_k{\left\{\right[M(1-\frac{\mathrm{\&delta;}}{N})-\frac{\mathrm{M\&delta;}}{M-N}]\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{D}^2{H}_m^H{H}_m+[K+(\frac{\mathrm{M\&delta;}}{M-N}-1){\left|\right|D\left|\right|}_F^2+\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{\mathrm{\&sigma;}}_m^2+\frac{\mathrm{NK}}{P}\left]I_N\right\}}^{-1}{H}_k^H---\left(15\right)$

In order to try to achieve β, order

W _k=β W _k' (16) wherein, W _k' provided by (19) formula, (16) formula is substituted into power limitation condition, can obtain

$\mathrm{\&beta;}=\sqrt{\frac{P}{\underset{k=1}{\overset{K}{\mathrm{\&Sigma;}}}\mathrm{tr}\left(W_k^{\&prime;}{W}_k^{\&prime;H}\right)}}---\left(17\right)$

Therefore, distributed system optimal beam forming matrix is:

$W_k^{*}=\sqrt{\frac{P}{\underset{k=1}{\overset{K}{\mathrm{\&Sigma;}}}\mathrm{tr}{\left(W_k^{\&prime;}{W}_k^{\&prime;}\right)}^H}}{W}_k^{\&prime;}---\left(18\right)$

Wherein:

$W_k^{\&prime;}=\mathrm{\&alpha;}{\mathrm{\&rho;}}_k{\left\{\right[M(1-\frac{\mathrm{\&delta;}}{N})-\frac{\mathrm{M\&delta;}}{M-N}]\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{D}^2{H}_m^H{H}_m+[K+(\frac{\mathrm{M\&delta;}}{M-N}-1){\left|\right|D|}_F^2+\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{\mathrm{\&sigma;}}_m^2+\frac{\mathrm{NK}}{P}\left]I_N\right\}}^{-1}{H}_k^H---\left(19\right)$

Accompanying drawing explanation

Fig. 1 multiuser downstream mimo system block diagram;

Fig. 2 distributed multi-user descending MIMO robust beam-forming method flow chart.

Embodiment

Below embodiments of the present invention are described.Distributed multi-user descending MIMO robust beam-forming method flow chart is as Fig. 2, and objective for implementation is a kth user in community,

Step 10, starts;

Step 20, in the simultaneous situation of channel estimating, quantification and delay time error, sets up the channel model matrix of a kth user, such as formula (3);

Step 30, based on the MMSE expression formula of leakage signal, releases robust beamforming design problem, such as formula (7);

Step 40, utilizes KKT condition to solve, and builds Lagrange's equation, asks partial derivative, obtain the beamforming matrix W of user k _k, such as formula (12);

Step 50, solves λ _kβ ², such as formula (14);

Step 60, solves β, such as formula (17);

Step 70, according to above-mentioned steps, derives distributed system optimal beam forming matrix such as formula (18);

Step 80, terminates.

As mentioned above, the present invention simulates in actual mobile communication the combined channel error model that simultaneously there is channel estimating, quantification and delay time error, and on the basis of this system model, propose the Robust distributed beamforming design method of the MMSE based on leakage signal, and try to achieve the closed solutions of beamforming matrix.The method can be applied in more practical distributed mobile communication system, and namely BS can be single mobile subscriber's independent design beamforming matrix, more can meet mobile subscriber to telecommunication service quality QoS demand.

In this description, the present invention describes with reference to specific embodiment.But, still can make various amendment and conversion obviously and not deviate from the spirit and scope of the present invention.Therefore, specification and accompanying drawing are regarded in an illustrative, rather than a restrictive.

Claims (3)

1. the present invention proposes a kind of distributed multi-user descending MIMO robust beam-forming method, be specially: for multiple-input, multiple-output (MIMO, multiple input multiple output) Frequency Division Duplexing (FDD) (FDD, frequencydivision duplexing) system mobile communication practical application in, mobile subscriber feeds back to base station (BS, basestation) channel condition information (CSI, channel state information) there is estimation simultaneously, quantize and delay time error, and based on this channel condition, least mean-square error (MMSE based on leakage signal is proposed, minimum mean-square error) the robust beam-forming method with distributed nature.

2. a kind of distributed multi-user according to claim 1 descending MIMO robust beam-forming method, is characterized in that:

For in multi-user MIMO communication system, when BS cannot obtain desirable CSI, the descending MIMO beamforming design method of distributed multi-user is proposed;

System model is multiuser downstream mimo system, and wherein BS is K user's service on same running time-frequency resource, and BS configures M root transmitting antenna, and each user configures N root reception antenna;

The signal that a kth user receives can be expressed as:

$y_k=G_k^H\mathrm{Wx}+n_k---\left(1\right)$

Wherein, represent from BS to the channel matrix of user k, its element is separate and be the multiple Gaussian random variable of zero-mean, unit variance, represent the Gauss's additive white noise vector received by user k, its element is separate and be the multiple Gaussian random variable of zero-mean, unit variance, with be respectively pre-coding matrix and transmission signal vector, () ^hthe conjugate transpose of representing matrix;

The signal that all K user receives is:

y＝GWx+n (2)

Wherein, g=(G ₁..., G _k) ^h, () ^tthe transposition of representing matrix.Notice E [nn ^h]=1, E [xx ^h, if BS transmitting power is restricted to P, then there is tr (WW]=1 ^hthe mark of)≤P, tr () representing matrix;

In the simultaneous situation of channel estimating, quantification and delay time error, the channel matrix G of the n-th moment BS to user k _k[n] is modeled as:

$\begin{array}{c}{G}_k\left[n\right]={\mathrm{\&rho;}}_k\left(H_k\right[n-1]U_k{[n-1]}_k+S_k[n-1\left]Z_k\right[n-1\left]\right)\sqrt{{\mathrm{\&Lambda;}}_k[n-1]}\\ +E_k\left[n\right]+{\mathrm{\&Delta;}}_k[n-1\end{array}---\left(3\right)$

Write (3) formula as matrix form, supposed G=(G ₁... .., G _k) ^h, obtain:

G＝AH+BS+E+Δ (4)

Wherein, H=[H ₁[n-1] ..., H _k[n-1]] ^h, E=[E ₁[n] ..., E _k[n]] ^h, D=diag (ρ ₁..., ρ _k), $C=\mathrm{ding}\left(V_1^H\right[n-1\left]U_1^H\right[n-1],..,V_K^H[n-1\left]U_K^H\right[n-1\left]\right),$ S＝[S ₁[n-1],...,S _K[n-1]] ^H， $\mathrm{\&Lambda;}=\mathrm{diag}(\sqrt{{\mathrm{\&Lambda;}}_1^H[n-1]},..,\sqrt{{\mathrm{\&Lambda;}}_K^H[n-1]}),$ $Z=\mathrm{diag}\left(Z_1^H\right[n-1],...,Z_K^H[n-1\left]\right),$ A=D Λ C, B=D Λ Z, diag () represents diagonal matrix;

Distributed beams form finding design:

Namely, in practical communication system, BS is that each user designs pre-coding matrix separately; The present invention is based on the MMSE expression formula of leakage signal, consider the imperfect CSI situation that BS obtains, propose the distributed beams manufacturing process with robustness;

Write (1) formula as following form:

Wherein, y' _kfor useful signal and the noise item of user, Section 2 is the multi-user interference that user receives.In practical communication system, we wish that the interference that user receives is little, i.e. primary signal x _kwith y' _krelatively.Based on this thought, and the distributed nature that binding signal is revealed, the MMSE that we define based on leakage signal under multiuser downstream MIMO is:

$\begin{array}{c}\mathrm{\&epsiv;}(W_k,\mathrm{\&beta;})=E_{x,n}{\left|\right|x_k-{\mathrm{\&beta;}}^{-1}{y}_k^{\&prime;}\left|\right|}^2+E_{x,n}\left[\underset{m=1,m\&NotEqual;k}{\overset{K}{\mathrm{\&Sigma;}}}{\left|\right|{\mathrm{\&beta;}}^{-1}{G}_m^H{W}_k{x}_k\left|\right|}^2\right]\\ ={\mathrm{\&beta;}}^{-2}\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{\left|\right|G_m^H{W}_k\left|\right|}_F^2-{2\mathrm{\&beta;}}^{-1}\mathrm{Tr}\left[\mathrm{Re}\left(G_k^H{W}_k\right)\right]+({\mathrm{\&beta;}}^{-2}+1)N\end{array}---\left(6\right)$

Thus, we obtain the MMSE robust beamforming design problem based on leakage signal:

$\begin{array}{c}\begin{array}{cc}\min & E_{\&lang;G|H\&rang;}\end{array},\mathrm{\&epsiv;}(W_k,\mathrm{\&beta;})\\ \begin{array}{cc}s.t.& \underset{k=1}{\overset{K}{\mathrm{\&Sigma;}}}\mathrm{tr}\left(W_k{W}_k^H\right)\&le;P\end{array}\end{array}---\left(7\right)$

Wherein E _<G|H>ε (W _k, β) reached by following formula:

$\begin{array}{c}{E}_{\&lang;G|H\&rang;}\mathrm{\&epsiv;}(W_k,\mathrm{\&beta;})\\ ={\mathrm{\&beta;}}^{-2}{E}_{\&lang;G|H\&rang;}\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{\left|\right|G_m^H{W}_k\left|\right|}_F^2{-2\mathrm{\&beta;}}^{-1}{E}_{\&lang;G|H\&rang;}\left\{\mathrm{tr}\right[\mathrm{Re}\left(G_k^H{W}_k\right)\left]\right\}+({\mathrm{\&beta;}}^{-2}+1)N\\ ={\mathrm{\&beta;}}^{-2}[M(1-\frac{\mathrm{\&delta;N}}{N})-\frac{\mathrm{M\&delta;}}{M-N}]\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{\left|\right|{\mathrm{DH}}_m{W}_k\left|\right|}_F^2+{\mathrm{\&beta;}}^{-1}[K+(\frac{\mathrm{M\&delta;}}{M-N}-1){\left|\right|D\left|\right|}_F^2+\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{\mathrm{\&sigma;}}_m^2]\mathrm{tr}\left(W_k{W}_k^H\right)\\ -2{\mathrm{\&beta;}}^{-1}{\mathrm{\&alpha;\&rho;}}_k\mathrm{Re}\left[\mathrm{tr}\left(H_k{W}_k\right)\right]+({\mathrm{\&beta;}}^{-2}+1)N\end{array}---\left(8\right)$

Problem (7) is convex optimization problem, and KKT condition can be utilized to solve, and builds Lagrange's equation as follows:

$l(W_k,\mathrm{\&beta;},{\mathrm{\&lambda;}}_k)=E_{\&lang;G|H\&rang;}\mathrm{\&epsiv;}(W_k,\mathrm{\&beta;})+{\mathrm{\&lambda;}}_k[\underset{k=1}{\overset{K}{\mathrm{\&Sigma;}}}\mathrm{tr}\left(W_k{W}_k^H\right)-P]---\left(9\right)$

(9) l (W in formula _k, β, λ _k) to W _k, β asks partial derivative respectively:

$\frac{\&PartialD;l(W,\mathrm{\&beta;},\mathrm{\&lambda;})}{\&PartialD;W}={\mathrm{\&beta;}}^{-2}[M(1-\frac{\mathrm{\&delta;}}{N})-\frac{\mathrm{M\&delta;}}{M-N}]\underset{m-1}{\overset{K}{\mathrm{\&Sigma;}}}{D}^2{H}_m^H{H}_m{W}_k+{\mathrm{\&beta;}}^{-1}[K+(\frac{\mathrm{M\&delta;}}{M-N}-1){\left|\right|D\left|\right|}_F^2+\underset{m-1}{\overset{K}{\mathrm{\&Sigma;}}}{\mathrm{\&sigma;}}_m^2]W_k-{\mathrm{\&alpha;\&beta;}}^{-1}{\mathrm{\&rho;}}_k{H}_k^H+{\mathrm{\&lambda;}}_k{W}_k=0---\left(10\right)$

$\frac{\&PartialD;l(W,\mathrm{\&beta;},\mathrm{\&lambda;})}{\&PartialD;\mathrm{\&beta;}}=[M(1-\frac{\mathrm{\&delta;}}{N})-\frac{\mathrm{M\&delta;}}{M-N}]\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{\left|\right|{\mathrm{DH}}_m{W}_k\left|\right|}_F^2+[K+(\frac{\mathrm{M\&delta;}}{M-N}-1){\left|\right|D\left|\right|}_F^2+\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{\mathrm{\&sigma;}}_m^2]\mathrm{tr}\left(W_k{W}_k^H\right)-{\mathrm{\&alpha;\&beta;\&rho;}}_k\mathrm{Re}\left[\mathrm{tr}\left(H_k{W}_k\right)\right]+N=0---\left(11\right)$

According to (10) formula, the beamforming matrix W of user k can be obtained _k:

$W_k={\mathrm{\&alpha;\&beta;\&rho;}}_k{\left\{\right[M(1-\frac{\mathrm{\&delta;}}{N})-\frac{\mathrm{M\&delta;}}{M-N}]\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{D}^2{H}_m^H{H}_m+[K+(\frac{\mathrm{M\&delta;}}{M-N}-1){\left|\right|D\left|\right|}_F^2+\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{\mathrm{\&sigma;}}_m^2+{\mathrm{\&lambda;}}_k{\mathrm{\&beta;}}^2\left]I_N\right\}}^{-1}{H}_k^H---\left(12\right)$

From (12) formula, as long as know λ _kβ ², can W be obtained _kclosed solutions.In order to obtain λ _kβ ², to (11) formula premultiplication and track taking obtains:

$[M(1-\frac{\mathrm{\&delta;}}{N})-\frac{\mathrm{M\&delta;}}{M-N}]\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{\left|\right|{\mathrm{DH}}_m{W}_k\left|\right|}_F^2+[K+(\frac{\mathrm{M\&delta;}}{M-N}-1){\left|\right|D\left|\right|}_F^2+\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{\mathrm{\&sigma;}}_m^2]\mathrm{tr}\left(W_k{W}_k^H\right)+{\mathrm{\&lambda;}}_k{\mathrm{\&beta;}}^2\mathrm{tr}\left(W_k^H{W}_k\right)-{\mathrm{\&alpha;\&beta;\&rho;}}_k\mathrm{tr}\left(H_k^H{W}_k\right)=0---\left(13\right)$

Simultaneous (10) formula and (13) formula can obtain: bar is limited again by BS transmitting power

Part $\underset{k=1}{\overset{K}{\mathrm{\&Sigma;}}}\mathrm{tr}\left(W_k{W}_k^H\right)=P,$ Can obtain:

${\mathrm{\&lambda;}}_k{\mathrm{\&beta;}}^2=\frac{\mathrm{NK}}{\underset{k=1}{\overset{K}{\mathrm{\&Sigma;}}}\mathrm{tr}\left(W_k{W}_k^H\right)}=\frac{\mathrm{NK}}{P}---\left(14\right)$

(14) formula is substituted into (12) formula obtain:

$W_k={\mathrm{\&alpha;\&beta;\&rho;}}_k{\left\{\right[M(1-\frac{\mathrm{\&delta;}}{N})-\frac{\mathrm{M\&delta;}}{M-N}]\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{D}^2{H}_m^H{H}_m+[K+(\frac{\mathrm{M\&delta;}}{M-N}-1){\left|\right|D\left|\right|}_F^2+\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{\mathrm{\&sigma;}}_m^2+\frac{\mathrm{NK}}{P}\left]I_n\right\}}^{-1}{H}_k^H---\left(15\right)$

In order to try to achieve β, order

W _k＝βW _k' (16)

Wherein W _k' provided by (18) formula, (16) formula is substituted into power limitation condition, can obtain

$\mathrm{\&beta;}\sqrt{\frac{P}{\underset{k=1}{\overset{K}{\mathrm{\&Sigma;}}}\mathrm{tr}\left({W_k}^{\&prime;}{W_k}^{\&prime;H}\right)}}---\left(17\right)$

Therefore distributed system optimal beam forming matrix is:

$W_k^{*}=\sqrt{\frac{P}{\underset{k=1}{\overset{K}{\mathrm{\&Sigma;}}}\mathrm{tr}\left({W_k}^{\&prime;}{W_k}^{\&prime;H}\right)}}{W_k}^{\&prime;}---\left(18\right)$

Wherein:

${W_k}^{\&prime;}={\mathrm{\&alpha;\&rho;}}_k{\left\{\right[M(1-\frac{\mathrm{\&delta;}}{N})-\frac{\mathrm{M\&delta;}}{M-N}]\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{D}^2{H}_m^H{H}_m+[K+(\frac{\mathrm{M\&delta;}}{M-N}-1){\left|\right|D\left|\right|}_F^2+\underset{m=1}{\overset{K}{\mathrm{\&Sigma;}}}{\mathrm{\&sigma;}}_m^2+\frac{\mathrm{NK}}{P}\left]I_N\right\}}^{-1}{H}_k^H---\left(19\right)$

3. a kind of distributed multi-user according to claim 2 descending MIMO robust beam-forming method, is characterized in that:

Distributed multi-user descending MIMO robust beam-forming method idiographic flow is as follows, and objective for implementation is a kth user in community,

(1) start;

(2) in the simultaneous situation of channel estimating, quantification and delay time error, the channel model matrix of a kth user is set up, such as formula (3);

(3) based on the MMSE expression formula of leakage signal, robust beamforming design problem is obtained, such as formula (7);

(4) utilize KKT condition to solve, build Lagrange's equation, ask partial derivative, obtain the beamforming matrix W of user k _k, such as formula (12);

(5) λ is solved _kβ ², such as formula (14);

(6) β is solved, such as formula (17);

(7) according to above-mentioned steps, distributed system optimal beam forming matrix is derived such as formula (18);

(8) terminate.