CN105207701A - ICA-based decoding method for multi-cell multi-user multi-antenna system - Google Patents

Abstract

The invention relates to an ICA-based decoding method for a multi-cell multi-user multi-antenna system. Under the condition that a receiver does not know channel state information, the ICA-based decoding method can realize decoding in the multi-cell multi-user multi-antenna system by utilizing a small amount of non-orthogonal pilot frequency sequences. The ICA-based decoding method can be used to improve the data transmission speed and reduce distortion caused by pilot frequency pollution.

Description

For the coding/decoding method based on ICA in multiple cell multi-user multi-aerial system

Technical field

The invention belongs to wireless communication technology field, particularly radio communication multi-antenna technology field, specifically one is used in the coding/decoding method based on ICA (independent component analysis method) in multiple cell multi-user multi-aerial system.

Background technology

Multiple cell multi-user multi-aerial system is the key technology of future wireless system development, in multi-antenna systems the antenna of base station configuration One's name is legion, and these antenna serves relatively less single antenna mobile subscriber simultaneously.In communication process, if recipient knows channel condition information, multiaerial system can realize high data rate and energy efficiency very simply.But channel condition information is unknown, so be extremely important to the estimation of channel condition information in multi-antenna systems.

Before transmit leg transmission information, first send the pilot signal that one group of receiving-transmitting sides is appointed in advance, recipient estimates channel condition information by pilot signal.Pilot requirements is mutually orthogonal, is even re-used if pilot signal is nonopiate, can cause pilot pollution.The user of such as different districts uses same pilot tone in same frequency range, same time, and the performance that channel status is estimated can non-constant, and the performance of system decodes also can non-constant.

Summary of the invention

In order to reduce channel status estimate in the distortion that produced by pilot pollution, present invention proposes the coding/decoding method of a kind of signal in multiple cell multi-user multi-aerial system, namely a kind ofly make full use of the coding/decoding method of a small amount of non-orthogonal pilot signal based on ICA.

Assuming that system model is:

$Y=\sqrt{\mathrm{\ρ}}HBX+N---\left(1\right)$

Wherein, this system has L user, and base station side antenna number is M.Suppose that each user sends the long information sequence for T again, wherein each element takes from a certain QAM (quadrature amplitude modulation) constellation equably, and sequence front end all comprises the long pilot frequency sequence for τ.The transmission signal of L user is designated as the matrix X=[Φ S] of L × T, wherein the front τ row of X are pilot frequency sequence matrix Φ, the pilot frequency sequence that its i-th line display i-th user sends, (T-τ) is classified as the information matrix S of user afterwards, the information that its i-th line display i-th user sends.The Received signal strength of base station can be expressed as the Received signal strength matrix Y of M × T.H is a dimension is the channel matrix of M × L.ρ is signal to noise ratio, for diagonal matrix, β _l(1≤l≤L) represents that l user is to the path loss of base station and shadow fading gain coefficient, and N represents random noise matrix, makes G=HB.

Based on above-mentioned system model, the inventive method specifically comprises design pilot frequency sequence and decoding two parts.

1. about the method for designing of pilot frequency sequence, specific as follows:

According to the number of users L of this system, obtain an integer τ, require that τ satisfies condition 4 ^τ-2< L≤4 ^τ-1.The pilot frequency sequence matrix Φ of a structure L × τ, require that each element of the first row of Φ is 1, other element is taken from 4-QAM, and ensures that any two row vectors of Φ are not identical.

2. send signal matrix X about by solving in Received signal strength matrix Y, specific as follows:

It is Y=[y that 2-1 base station receives signal matrix ₁y ₂y _t], y _iit is i-th (1≤i≤T) individual column vector.The sample average calculating Y is covariance matrix E (yy can be obtained ^h| G) sample estimate:

$\mathrm{\&Sigma;}=\frac{1}{T}({\mathrm{YY}}^H-T{\stackrel{\&OverBar;}{yy}}^H)$

Wherein, the conjugate transpose of subscript H representing matrix.

2-2. carries out Eigenvalues Decomposition to Σ and makes wherein be the diagonal matrix of a M × M, and the element on diagonal is by descending, has it is the unitary matrice of a M × M.Order ${\hat{D}}_S=diag\left({\hat{d}}_1,{\hat{d}}_2,\mathrm{...},{\hat{d}}_L\right),{\hat{D}}_N=diag\left({\hat{d}}_{L+1},{\hat{d}}_{L+2},\mathrm{...},{\hat{d}}_M\right),$ ? resolve into $\hat{U}=\left[\begin{array}{cc}{\hat{U}}_S& {\hat{U}}_N\end{array}\right],$ Wherein be front L row, remaining M-L row.Decomposed by these, have $\mathrm{\&Sigma;}=\left[\begin{array}{cc}{\hat{U}}_S& {\hat{U}}_N\end{array}\right]\left[\begin{array}{cc}{\hat{D}}_S& 0\\ 0& {\hat{D}}_N\end{array}\right]\left[\begin{array}{c}{\hat{U}}_S^H\\ {\hat{U}}_N^H\end{array}\right],$ 0 represents null matrix.PCA (PCA) is utilized to obtain:

$V^{H}X=\frac{1}{2}{\left(\sqrt{{\hat{D}}_S-I_L}\right)}^{-1}{\hat{U}}_S^{H}Y---\left(2\right)$

Wherein, I _lfor the unit matrix on L rank, V ^hfor the unitary matrice of L × L.

2-3. is V in (2) formula ^hunknown with X, (2) formula is used to the ICA algorithm of plural number, obtain a weight matrix through iteration, be designated as the unitary matrice W of L × L ^h.Order obtain thus

$\hat{X}=\sqrt{\mathrm{\&rho;}}{\left(\frac{1}{\mathrm{\&rho;}}{I}_L+{\hat{G}}^H\hat{G}\right)}^{-1}{\hat{G}}^{H}Y$

2-4. order $\hat{X}=\left[\begin{array}{cc}\widehat{\mathrm{\&Phi;}}& \hat{S}\end{array}\right],$ Wherein with the matrix of L × τ and L × (T-τ) respectively.Will $\left[\begin{array}{cc}\widehat{\mathrm{\&Phi;}}& \hat{S}\end{array}\right]$ The amplitude quantizing of the 1st element of every a line is 1, then remembers that l (1≤l≤L) row the 1st element is again will $\left[\begin{array}{cc}\widehat{\mathrm{\&Phi;}}& \hat{S}\end{array}\right]$ Each element that l is capable is multiplied by each provisional capital performs this operation, and the matrix obtained after having operated is designated as then right other element except the 1st row, according to the principle the shortest with the constellation QAM distance adopted with 4-QAM, carries out adjudicating the matrix obtained respectively and is designated as $\left[\begin{array}{cc}\stackrel{\&OverBar;}{\mathrm{\&Phi;}}& \stackrel{\&OverBar;}{S}\end{array}\right].$ Get the 1st row of Φ, with every a line do inner product operation, the row at inner product that a line place maximum is as the solution of X the 1st row.Get the 2nd row of Φ again, with every a line do inner product operation, the row at inner product that a line place maximum is as the solution of X the 2nd row, and by parity of reasoning, finally obtains the solution of the every a line of X.Use the coding/decoding method based on ICA in multiple cell multi-user multi-aerial system in the present invention, solve from the Received signal strength matrix Y that base station receives and send signal matrix X.

Beneficial effect of the present invention: when recipient does not know channel condition information, this coding/decoding method, based on ICA, adopts a small amount of non-orthogonal pilot sequence, realizes the decoding in multiple cell multi-user multi-aerial system.This invention can improve message transmission rate and reduce the distortion brought by pilot pollution.

Accompanying drawing explanation

Fig. 1 be the present invention in the case of the embodiment, about decoding bit error rate analogous diagram.

Embodiment

Introduce the theoretical foundation of this method for designing below:

Base station is at T continuous slot, and the signal model received can be designated as:

$Y=\sqrt{\mathrm{\&rho;}}HBX+N$

Wherein, in system, number of users is L, at T continuous slot, sends the matrix that signal matrix X is L × T, can be expressed as X=[x ₁x ₂x _t], x _i(1≤i≤T) is i-th column vector of X, and each element takes from a certain constellation QAM equably.Base station side is furnished with M root antenna, and Received signal strength matrix is the matrix Y of M × T, can be expressed as Y=[y ₁y ₂y _t], y _ii-th column vector of Y.H is a dimension is the channel matrix of M × L, and its each element is all average is zero, and variance is the multiple Gaussian random variable of 1, and separate between two.N is the noise matrix of a M × T, and its each element is all average is zero, and variance is the multiple Gaussian random variable of 1, and separate between two, can be expressed as N=[n ₁n ₂n _t], n _ii-th column vector of N. diagonal matrix, β _irepresent that l (1≤l≤L) individual user is to the path loss of base station and shadow fading gain coefficient, make G=HB, ρ is signal to noise ratio.In said system model, suppose at a time t, transmission signal is x, and Received signal strength is y, and noise is n.Have according to signal model:

$y=\sqrt{\mathrm{\&rho;}}Gx+n---\left(3\right)$

The multiple Gaussian random variable that x and w is zero-mean, variance is 1, obviously has: E (xx ^h)=I _lwith E (nn ^h)=I _m.Suppose that G is constant in section sometime.In order to obtain theoretic signal subspace, the covariance matrix of y in peer-to-peer (3) is first wanted to carry out theoretical decomposition:

$\begin{array}{c}E\left({\mathrm{yy}}^H\right|G)=\mathrm{\&rho;}GE\left({\mathrm{xx}}^H\right)G^H+E\left({\mathrm{nn}}^H\right)\\ ={\mathrm{\&rho;GG}}^H+I_M\end{array}---\left(4\right)$

Singular value decomposition G=UDV is carried out to the G in above formula ^h, D is the diagonal matrix of a M × L, and the element on diagonal is the singular value of G and by descending, assuming that base station side antenna number is greater than community user number (M>=L), is designated as matrix D capable for L before D _s=diag (d ₁, d ₂..., d _l), have $D=\left[\begin{array}{c}{D}_S\\ 0\end{array}\right],$ 0 represents null matrix.U and V is the unitary matrice on M and L rank respectively, then U is resolved into $U=\left[\begin{array}{cc}{U}_S& U_N\end{array}\right],$ U _sfor L row before U, U _nremaining M-L row.Decomposed by these, have:

$\begin{array}{c}E\left({\mathrm{yy}}^H\right|G)={\mathrm{\&rho;UD}}^2{U}^H+I_M\\ =\left[\begin{array}{cc}{U}_S& U_N\end{array}\right]\left[\begin{array}{cc}{\mathrm{\&rho;D}}_S^2+I_L& 0\\ 0& I_{M-L}\end{array}\right]\left[\begin{array}{c}{U}_S^H\\ U_N^H\end{array}\right]\end{array}---\left(5\right)$

From PCA, the dominant eigenvalue that signal subspace is corresponding is element on diagonal, U _sin the corresponding principal eigenvector of column vector, remaining corresponds to noise subspace.

Next, actual Received signal strength be utilized E (yy ^h| G) estimate.Assuming that at T continuous slot, the observation sample y of base station ₁, y ₂, y _t, y _iit is the Received signal strength on i-th (1≤i≤T) individual time slot. sample average: y sample covariance is designated as Σ, can be calculated by following formula:

$\mathrm{\&Sigma;}=\frac{1}{T}({\mathrm{YY}}^H-T{\stackrel{\&OverBar;}{yy}}^H)---\left(6\right)$

When sample number is enough large, when namely T is enough large, from statistical theory:

Σ≈E(yy ^H|G)(7)

Next, Eigenvalues Decomposition is carried out to Σ wherein, be the diagonal matrix on M rank, the element on diagonal, by descending, has $\hat{D}=\left[\begin{array}{cc}{\hat{D}}_S& 0\\ 0& {\hat{D}}_N\end{array}\right],$ Wherein it is the unitary matrice of a M × M.? resolve into $\hat{U}=\left[\begin{array}{cc}{\hat{U}}_S& {\hat{U}}_N\end{array}\right],$ Wherein be front L row, remaining M-L row.Decomposed by these, Σ can be write as:

$\mathrm{\&Sigma;}=\left[\begin{array}{cc}{\hat{U}}_S& {\hat{U}}_N\end{array}\right]\left[\begin{array}{cc}{\hat{D}}_S& 0\\ 0& {\hat{D}}_N\end{array}\right]\left[\begin{array}{c}{\hat{U}}_S^H\\ {\hat{U}}_N^H\end{array}\right]---\left(8\right)$

By equation (5), (7) and (8), can obtain: $U_S\&ap;{\hat{U}}_S,D_S\&ap;\sqrt{\frac{1}{\mathrm{\&rho;}}({\hat{D}}_S-I_L)}.$ Therefore have:

$G\&ap;{\hat{U}}_S\sqrt{\frac{1}{\mathrm{\&rho;}}({\hat{D}}_S-I_L)}{V}^H---\left(9\right)$

Transmit-receive equation can be written as:

$Y\&ap;{\hat{U}}_S\sqrt{{\hat{D}}_S-I_L}{V}^{H}X+N---\left(10\right)$

For antenna for base station side, V in above formula ^hx and N is unknown, implements MMSE (least mean-square error) estimation criterion to formula (10), estimative to liking V ^hx, as follows:

$V^{H}X=\frac{1}{2}{\left(\sqrt{{\hat{D}}_S-I_L}\right)}^{-1}{\hat{U}}_S^{H}Y---\left(11\right)$

Send the element independence in signal X, take from QAM equably, therefore, it is feasible for applying plural ICA at equation (11).Be matrix Z on the right side of note formula (11) equal sign, the Fast ICA algorithm of application plural number solves, and can obtain the unitary matrice W of a L × L, make

$\tilde{X}=WZ---\left(12\right)$

the solution of equation (12), namely to the estimation of X.Here, ICA method is with basic uncertainty, and Here it is phase place and the order of middle independent element cannot be determined.Describe with mathematical linguistics and be exactly, the permutation matrix P of a L × L may be there is, wherein in P, any a line or any row have and only have one and are nonzero element, make wherein, in element be estimation to element independent in X, but obtain compared with the independent element in X, being first that order is different, is then phase θ _l.Therefore, determine the order that middle element is correct and phase place.In order to determine correct order and phase place, proposition is sent a small amount of pilot frequency sequence by user, base station these signals received decide order and phase place.

Order and phase place are decided by the pilot matrix Φ of L × τ in base station.τ is one and meets 4 ^τ-2< L≤4 ^τ-1integer, under given L, τ unique exists.Number L is larger in community, and τ increases slower.Further, when L >=3, τ < L.Then, the 1st each element arranged making Φ is 1, and the reason done like this is, if first element of every a line is all 1 when sending, then the initial phase of these elements is 0, and the change so receiving a certain line phase can obtain from the phase place change of this row the 1st element.The element of position (i, j) takes from 4-QAM all equably, wherein 1≤i≤L, 2≤j≤τ.And will ensure that in Φ, any two row vectors are all unequal, requiring each row vector unequal is row in order to the maximum method of inner product can be used after the decoding to find coupling.

Sum up above-mentioned.User's sending metrix is X=[Φ S], through operation PCA and plural ICA, can obtain the unitary matrice W of a L × L, compare equation (11) and (12), can W ^hbe used as V ^hsolution.Obtain the solution of channel matrix G thus, be designated as $\hat{G}={\hat{U}}_S\sqrt{\frac{1}{\mathrm{\&rho;}}({\hat{D}}_S-I_L)}{W}^H,$ Have:

$Y\&ap;\sqrt{\mathrm{\&rho;}}\hat{G}\hat{X}+N---\left(13\right)$

To above formula, estimate with MMSE estimation criterion have:

$\hat{X}\&ap;\sqrt{\mathrm{\&rho;}}{(\frac{1}{\mathrm{\&rho;}}{I}_L+{\hat{G}}^H\hat{G})}^{-1}{\hat{G}}^{H}Y---\left(14\right)$

Wherein $\hat{X}=\left[\begin{array}{cc}\widehat{\mathrm{\&Phi;}}& \hat{S}\end{array}\right],$ with the matrix of L × τ and L × (T-τ) respectively.

In order to correct $\hat{X}=\left[\begin{array}{cc}\widehat{\mathrm{\&Phi;}}& \hat{S}\end{array}\right]$ Phase place, be 1 by the amplitude quantizing of the 1st of its every a line the element, then remember that l (1≤l≤L) row the 1st element is then each capable for l element is multiplied by at this, a new matrix can be obtained except the 1st row, all the other elements are adjudicated according to shortest distance principle.Such as, the element in S takes from 4-QAM, if then in the real part of certain element or imaginary part be greater than zero, then this element real part or imaginary part judgement are otherwise judgement is obtain a new matrix thus $\left[\begin{array}{cc}\stackrel{\&OverBar;}{\mathrm{\&Phi;}}& \stackrel{\&OverBar;}{S}\end{array}\right].$

In order to correct $\left[\begin{array}{cc}\stackrel{\&OverBar;}{\mathrm{\&Phi;}}& \stackrel{\&OverBar;}{S}\end{array}\right]$ Order, get the 1st row of Φ, with every a line do inner product.Suppose l ₁row calculates inner product and does large value, then $\left[\begin{array}{cc}\stackrel{\&OverBar;}{\mathrm{\&Phi;}}& \stackrel{\&OverBar;}{S}\end{array}\right]$ L ₁row is as the solution of X the 1st row.Then, get the 2nd row of Φ, with every a line do inner product.Suppose l ₂row calculates inner product and does large value, then $\left[\begin{array}{cc}\stackrel{\&OverBar;}{\mathrm{\&Phi;}}& \stackrel{\&OverBar;}{S}\end{array}\right]$ L ₂row is as the solution of X the 2nd row.By parity of reasoning, finally obtains the solution of the every a line of X.

Embodiment

Suppose in antenna system, antenna for base station number is 128, has 8 users in system, and these 8 users, from identical or different communities, send the information sequence of T=200.B gets fixing battle array Β respectively ₁=diag ([10.90.80.70.40.30.20.1]) and B ₂=I ₈.

Suppose that each element of information sent is taken from 4-QAM equably, especially, the element (except the 1st row) of pilot frequency sequence takes from 4-QAM, according to method of estimation of the present invention, specifically implements according to the following steps:

Step 1. user sends signal matrix $X=\left[\begin{array}{cc}\mathrm{\&Phi;}& S\end{array}\right],$ Front 3 are classified as pilot matrix Φ:

${\left[\begin{array}{cccccccc}1& 1& 1& 1& 1& 1& 1& 1\\ \frac{1+j}{\sqrt{2}}& \frac{1-j}{\sqrt{2}}& \frac{1+j}{\sqrt{2}}& \frac{-1+j}{\sqrt{2}}& \frac{1+j}{\sqrt{2}}& \frac{1-j}{\sqrt{2}}& \frac{1-j}{\sqrt{2}}& \frac{-1+j}{\sqrt{2}}\\ \frac{1+j}{\sqrt{2}}& \frac{1+j}{\sqrt{2}}& \frac{1-j}{\sqrt{2}}& \frac{1+j}{\sqrt{2}}& \frac{-1+j}{\sqrt{2}}& \frac{1-j}{\sqrt{2}}& \frac{-1+j}{\sqrt{2}}& \frac{1-j}{\sqrt{2}}\end{array}\right]}^T$

The wherein transposition of subscript T representing matrix.The useful information battle array of user is S, and be 4 to 200 row of X, dimension is 8 × 197, and wherein each element takes from 4-QAM equably.

The signal matrix that step 2. base station receives is Y=[y ₁y ₂y ₂₀₀], y _iit is i-th (1≤i≤200) individual column vector.Calculating sample average is can in this channel G situation, covariance matrix E (yy ^h| G) estimation:

$\mathrm{\&Sigma;}=\frac{1}{200}({\mathrm{YY}}^H-200{\stackrel{\&OverBar;}{yy}}^H)$

Σ is carried out Eigenvalues Decomposition by step 3., 8 eigenvalue clusters diagonally battle array that Σ is maximum, and meets the unitary matrice of 128 × 8, in i-th (1≤i≤8) individual column vector be with the characteristic vector that on diagonal, i-th characteristic value is corresponding, can obtain:

$V^{H}X=\frac{1}{2}{\left(\sqrt{{\hat{D}}_S-I_8}\right)}^{-1}{\hat{U}}_S^{H}Y$

Step 4. is V in above formula ^hunknown with X, use plural fast ICA Algorithm algorithm to above formula, when being similar to negentropy in algorithm, use iteration function is a gets 0.1.A weight matrix W is obtained through algorithm ^h, it is the unitary matrice of 8 × 8.Obtain the solution of channel matrix G thus, be designated as

$\hat{G}={\hat{U}}_S\sqrt{\frac{1}{\mathrm{\&rho;}}({\hat{D}}_S-I_8)}{W}^H$

To system model n is noise matrix, estimates with MMSE estimation criterion

$\hat{X}\&ap;\sqrt{\mathrm{\&rho;}}{(\frac{1}{\mathrm{\&rho;}}{I}_8+{\hat{G}}^H\hat{G})}^{-1}{\hat{G}}^{H}Y$

Step 5. makes $\hat{X}=\left[\begin{array}{cc}\widehat{\mathrm{\&Phi;}}& \hat{S}\end{array}\right],$ Wherein with the matrix of 8 × 3 and 8 × 197 respectively.Will $\left[\begin{array}{cc}\widehat{\mathrm{\&Phi;}}& \hat{S}\end{array}\right]$ The amplitude quantizing of the 1st element of every a line is 1, then remembers that l (1≤l≤8) row the 1st element is then will $\left[\begin{array}{cc}\widehat{\mathrm{\&Phi;}}& \hat{S}\end{array}\right]$ L each element capable be multiplied by right $\left[\begin{array}{cc}\widehat{\mathrm{\&Phi;}}& \hat{S}\end{array}\right]$ Every a line perform this operation, obtain a new matrix then right except the 1st row, other element carries out judgement and obtains $\left[\begin{array}{cc}\stackrel{\&OverBar;}{\mathrm{\&Phi;}}& \stackrel{\&OverBar;}{S}\end{array}\right].$ In this example, the element in S all takes from 4-QAM, if then in the real part of certain element or imaginary part be greater than zero, then this element real part or imaginary part judgement are otherwise judgement is obtain a new matrix thus $\left[\begin{array}{cc}\stackrel{\&OverBar;}{\mathrm{\&Phi;}}& \stackrel{\&OverBar;}{S}\end{array}\right].$ Then, get the 1st row of Φ, with every a line do inner product operation, the row at inner product that a line place maximum is as the solution of X the 1st row.Get the 2nd row of Φ again, with every a line do inner product operation, the row at inner product that a line place maximum is as the solution of X the 2nd row, and by parity of reasoning, finally obtains the solution of the every a line of X.

About example decoded bit error rate as shown in Figure 1, B ₂compare B ₁the error rate lower, the interference namely between user is stronger, and decoding performance is better, illustrates that the present invention can not only reduce the interference of user between different districts, and can reduce the interference between same intra-cell users.When number of users L is given, the present invention uses a small amount of non-orthogonal pilot sequence, improves message transmission rate and reduces the distortion brought by pilot pollution.

Those of ordinary skill in the art will be appreciated that, above example is only used to the present invention is described, and not as limitation of the invention, as long as within the scope of the invention, to the change of above embodiment, distortion all will drop on protection scope of the present invention.

Claims (1)

1., for the coding/decoding method based on ICA in multiple cell multi-user multi-aerial system, initialization system model is:

$Y=\sqrt{\mathrm{\&rho;}}HBX+N---\left(1\right)$

Wherein, this system has L user, and base station side antenna number is M; Reset each user and send the long information sequence for T, wherein each element takes from a certain qam constellation equably, and sequence front end all comprises the long pilot frequency sequence for τ; The transmission signal of L user is designated as the matrix X=[Φ S] of L × T, wherein the front τ row of X are pilot frequency sequence matrix Φ, the pilot frequency sequence that its i-th line display i-th user sends, (T-τ) is classified as the information matrix S of user afterwards, the information that its i-th line display i-th user sends; The Received signal strength of base station can be expressed as the Received signal strength matrix Y of M × T; H is a dimension is the channel matrix of M × L; ρ is signal to noise ratio, for diagonal matrix, β _lrepresent that l user is to the path loss of base station and shadow fading gain coefficient, N represents random noise matrix, and 1≤l≤L, makes G=HB;

It is characterized in that the method comprises the following steps:

Step 1. designs pilot frequency sequence, specifically:

According to the number of users L of this system, obtain an integer τ, require that τ satisfies condition 4 ^τ-2<L≤4 ^τ-1; The pilot frequency sequence matrix Φ of a structure L × τ, require that each element of the first row of Φ is 1, other element is taken from 4-QAM, and ensures that any two row vectors of Φ are not identical;

Step 2. estimates X by Y, specific as follows:

It is Y=[y that 2-1. base station receives signal matrix ₁y ₂y _t], y _ii-th column vector, 1≤i≤T; The sample average calculating Y is covariance matrix E (yy can be obtained ^h| G) sample estimate:

$\mathrm{\&Sigma;}=\frac{1}{T}({\mathrm{YY}}^H-T{\stackrel{\&OverBar;}{yy}}^H)$

Wherein, the conjugate transpose of subscript H representing matrix;

2-2. carries out Eigenvalues Decomposition to Σ and makes wherein be the diagonal matrix of a M × M, and the element on diagonal is by descending, has it is the unitary matrice of a M × M; Order ${\hat{D}}_S=diag\left({\hat{d}}_1,{\hat{d}}_2,\mathrm{...},{\hat{d}}_L\right),$ ${\hat{D}}_N=diag\left({\hat{d}}_{L+1},{\hat{d}}_{L+2},\mathrm{...},{\hat{d}}_M\right),$ ? resolve into $\hat{U}=\left[\begin{array}{cc}{\hat{U}}_S& {\hat{U}}_N\end{array}\right],$ Wherein be front L row, remaining M-L row; Decomposed by these, have $\mathrm{\&Sigma;}=\left[\begin{array}{cc}{\hat{U}}_S& {\hat{U}}_N\end{array}\right]\left[\begin{array}{cc}{\hat{D}}_S& 0\\ 0& {\hat{D}}_N\end{array}\right]\left[\begin{array}{c}{\hat{U}}_S^H\\ {\hat{U}}_N^H\end{array}\right],$ 0 represents null matrix; PCA is utilized to obtain:

$V^{H}X=\frac{1}{2}{\left(\sqrt{{\hat{D}}_S-I_L}\right)}^{-1}{\hat{U}}_S^{H}Y---\left(2\right)$

Wherein, I _lfor the unit matrix on L rank, V ^hfor the unitary matrice of L × L;

2-3. is V in (2) formula ^hunknown with X, (2) formula is used to the ICA algorithm of plural number, obtain a weight matrix through iteration, be designated as the unitary matrice W of L × L ^h; Order obtain thus

$\hat{X}=\sqrt{\mathrm{\&rho;}}{(\frac{1}{\mathrm{\&rho;}}{I}_L+{\hat{G}}^H\hat{G})}^{-1}{\hat{G}}^{H}Y$

Order $\hat{X}=\left[\begin{array}{cc}\widehat{\mathrm{\&Phi;}}& \hat{S}\end{array}\right],$ Wherein with the matrix of L × τ and L × (T-τ) respectively; Will $\left[\begin{array}{cc}\widehat{\mathrm{\&Phi;}}& \hat{S}\end{array}\right]$ The amplitude quantizing of the 1st element of every a line is 1, then remembers that capable 1st element of l is again will $\left[\begin{array}{cc}\widehat{\mathrm{\&Phi;}}& \hat{S}\end{array}\right]$ Each element that l is capable is multiplied by each provisional capital performs this operation, and the matrix obtained after having operated is designated as then right other element except the 1st row, according to the principle the shortest with the constellation QAM distance adopted with 4-QAM, carries out adjudicating the matrix obtained respectively and is designated as $\left[\begin{array}{cc}\stackrel{\&OverBar;}{\mathrm{\&Phi;}}& \stackrel{\&OverBar;}{S}\end{array}\right];$ Get the 1st row of Φ, with every a line do inner product operation, the row at inner product that a line place maximum is as the solution of X the 1st row; Get the 2nd row of Φ again, with every a line do inner product operation, the row at inner product that a line place maximum is as the solution of X the 2nd row, and by parity of reasoning, finally obtains the solution of the every a line of X.