CN102710393A - Interference alignment precoding method based on Stiefel manifold - Google Patents

Abstract

The invention discloses an interference alignment precoding method for transmitter unilateral optimization based on a Stiefel manifold, which is characterized in that the interference intensity in a minimum space signal is taken as an objective function, the objective function is reconfigured on the Stiefel manifold, an orthogonality constraint is removed, the inner product, the projection and steepest descent edge direction of the objective function on the Stiefel manifold are calculated, and interference aligned precoding is realized through a steepest descent edge algorithm based on the Stiefel manifold. As the dimensionality of the Stiefel manifold adopted in the interference alignment precoding method is smaller than that of a multidimensional complex space existing in a traditional optimization algorithm, the interference alignment precoding method has the advantages of quick convergence rate, low complexity, high system capacity and the like; and meanwhile, as only a transmitter needs to participate in the interference alignment precoding method, repeated iteration between a front communication link and a reverse communication link caused by the reason that transmitting and receiving ends must participate in an alignment as well as accompanied huge synchronism, feedback and other system overhead are avoided. The interference alignment precoding method is applicable to time-division and frequency duplex communication systems.

Description

A kind of interference alignment method for precoding based on Stifel stream shape

Technical field

The invention belongs to the multi-antenna transmitting transferring technology field of radio communication, be specifically related to method based on the interference alignment precoding of Stifel stream shape.

Background technology

Disturbing alignment is a kind of multi-antenna transmitting transferring technology that rises recently, can increase substantially the capacity of wireless communication system, thereby obtain paying attention to widely and using.According to " international electronics and The Institution of Electrical Engineers's information theory transactions " (IEEE Transactions on Information Theory, Vol.54, No.8; 2008; Page (s): 3425-3441) introduce, disturb the core of alignment techniques to be the interference signal that comes from different data streams is alignd in one direction, to reduce the dimension of interference space; And the degree of freedom (Degree ofFreedom, notion DoF) have been introduced.Utilize precoding to realize disturbing alignment to obtain extensive studies at present as a kind of practicable method.According to " the global communication annual meeting in 2008 of international electronics and The Institution of Electrical Engineers " (IEEE Global Communications Conference; Nov.30-Dec.4; 2008, Page (s): 1-6) introduce, disturb the method for precoding that aligns can utilize the reciprocity of channel; Through reducing the interference strength in the signal subspace, iterating in forward direction and reverse communication links obtains.But, make its scope of application only be confined to tdd communication systems and can not be applied to frequency-division duplex communication system because this method depends on channel reciprocity; Iterating of forward direction and reverse communication links needs transmitter and receiver to participate in simultaneously and strict synchronous and feedback simultaneously, can consume a large amount of telecommunication system resources like this; Particularly because this method adopts is traditional optimization methods, defective such as have inevitably that convergence rate is slow, complexity is high and power system capacity is low.

Summary of the invention

The objective of the invention is to propose the interference alignment method for precoding based on Stifel (Stiefel) stream shape of the monolateral optimization of a kind of transmitter; To overcome that conventional method relies on the reciprocity of channel and the defective of bringing the huge overhead that forward direction and reverse communication links iterate, the purpose that reach fast convergence rate, complexity is low and power system capacity is high, can both be suitable for for time division duplex and frequency-duplex communication system.

The present invention is based on the interference alignment method for precoding of Stifel stream shape, is that M, reception antenna number are that the reception signal indication of K user under interference channel of N is with number of transmit antennas in the multiple-input-multiple-output communication system earlier: $Y^{\left[k\right]}=\underset{j=1}{\overset{K}{\mathrm{\Σ}}}{H}^{\left[\mathrm{Kj}\right]}{V}^{\left[j\right]}{S}^{\left[j\right]}+W^{\left[k\right]},$ In the formula, the span of the first subscript k from 1 to K, the span of the second subscript j also from 1 to K; H ^[kj]Be the channel coefficient matrix of j transmitter to k receiver; S ^[j]Be the data of needs emission, V ^[j]Be corresponding pre-coding matrix, W ^[k]White Gaussian noise for the zero-mean unit variance; Then the interference covariance matrix of k receiving terminal is: $Q^{\left[k\right]}=\underset{\begin{array}{c}j=1\\ j\≠k\end{array}}{\overset{K}{\mathrm{\Σ}}}\frac{P^{\left[j\right]}}{d^{\left[j\right]}}{H}^{\left[\mathrm{Kj}\right]}{V}^{\left[j\right]}{V}^{\left[j\right]*}{H}^{\left[\mathrm{Kj}\right]*},$ In the formula, P ^[j]Be transmitted power, d ^[j]Be the data flow number of sending, subscript symbol * represents the conjugate transpose operation of matrix; To Q ^[k]Carry out characteristic value decomposition, obtain a series of characteristic value

Its subscript Q ^[k]Represent the corresponding interference covariance matrix of this characteristic value, the sequence number of characteristic value is arranged in its subscript i representative with ascending order; Interference strength in the minimize signal space is equivalent to and minimizes d ^[k]Individual characteristic value

Thereby set up mapping relations, obtain required target function from the multidimensional complex number space to the arithmetic number space

Constraints is V ^[j]* V ^[j]=I, I are unit vector;

It is characterized in that: f flows shape at Stifel with target function

Last reorganization is in the formula

Representing dimension is the complex number space that n multiply by p; Result after the reorganization of Stifel stream shape is: remove constraints V ^{[j] *}V ^[j]=I keeps

Minimize target; Then with complex matrix V ^[j]Be decomposed into real part

And imaginary part

Target function f respectively to real part and imaginary part differentiate, is obtained corresponding two Jacobi (Jacobian) determinant:

With

Then have $\mathrm{df}=[D_R^{\left[1\right]}....D_R^{\left[j\right]}]\left[\begin{array}{c}{\mathrm{dV}}_R^{\left[1\right]}\\ \·\\ \·\\ {\mathrm{dV}}_R^{\left[j\right]}\end{array}\right]+[D_I^{\left[1\right]}....D_I^{\left[j\right]}]\left[\begin{array}{c}{\mathrm{dV}}_I^{\left[1\right]}\\ \·\\ \·\\ {\mathrm{dV}}_I^{\left[j\right]}\end{array}\right];$ Obtain the first derivative of target function f to pre-coding matrix V: D thus _V=(D _R+ iD _I) ^T, in the formula () ^TRepresent the transpose of a matrix operation; And then obtain steepest descent on the Stifel stream shape along expression formula:

Then target function reduces along the direction of Z, is expressed as f (V+ α Z), and wherein the size of steps obtains through A Mihuo (Armijo) rule, specifically adopts following two criterions: if

Then the steps in the formula is replaced with 2 α and bring inequality into and judge, repeating above-mentioned steps is false up to inequality, wherein

Real part is got in representative, and tr () represents the matrix trace operation,

Inner product expression formula for Stifel stream shape; If

Then the steps in the formula is replaced into

Bring inequality into and judge, repeating above-mentioned steps is false up to inequality, finally obtains suitable steps; Then V+ α Z is carried out projection operation: π (V+ α Z)=UI _{N * p}R ^*, wherein U and R are respectively the left and right unitary matrice of the singular value decomposition generation of V+ α Z, i.e. V+ α Z=U ∑ R ^*, wherein ∑ is the singular value diagonal matrix; Successively to the pre-coding matrix V of K transmitter ^[j]Carry out iteration, wherein the span of subscript j when target function f iteration convergence, has just obtained the pre-coding matrix V of a series of interference alignment from 1 to K.

The principle that the present invention is based on the interference alignment method for precoding institute foundation of Stifel stream shape is:

Stifel stream shape:

can regard the set of the matrix that satisfies the quadrature constraint as, and the dimension of Stifel stream shape does $\dim \left(\mathrm{St}(n,p)\right)=\mathrm{np}-\frac{1}{2}p(p+1),$ And the dimension of the multidimensional complex number space at traditional optimization place does

Therefore with target function after reorganization on the Stifel stream shape, not only can remove condition of orthogonal constraints V ^{[j] *}V ^[j]The restriction of=I becomes constrained optimization problem into unconfined optimization problem, and because reduced the Spatial Dimension at optimization problem place, so simplified the complexity of algorithm greatly; Do not change simultaneously

Analytical expression.Because interference covariance matrix Q ^[k]Satisfy the definition of unitary matrice: Q ^[k]=Q ^{[k] *}So, its pairing a series of characteristic value

Be real number, that is to say that target function f is the multidimensional complex number space

To the one dimension real number space Mapping; Therefore must be the worry that sets the exam of real part and imaginary component: complex matrix V ^[j]Be decomposed into real part

And imaginary part

Target function f respectively to real part and imaginary part differentiate, is obtained two corresponding Jacobians:

With

Then have $\mathrm{df}=[D_R^{\left[1\right]}....D_R^{\left[j\right]}]\left[\begin{array}{c}{\mathrm{dV}}_R^{\left[1\right]}\\ \·\\ \·\\ {\mathrm{dV}}_R^{\left[j\right]}\end{array}\right]+[D_I^{\left[1\right]}....D_I^{\left[j\right]}]\left[\begin{array}{c}{\mathrm{dV}}_I^{\left[1\right]}\\ \·\\ \·\\ {\mathrm{dV}}_I^{\left[j\right]}\end{array}\right];$ Obtain the first derivative of target function f to pre-coding matrix V: D _V=(D _R+ iD _I) ^T, in the formula () ^TRepresent the transpose of a matrix operation; The steepest descent of Stifel stream shape along the principle of algorithm is: at first define the π of projection operation () on the Stifel stream shape, suppose that Y is the capable p row of n, and order is any matrix of p that then Y is defined as to the projection that Stifel flows shape

In the formula || || being Euclid norm, is Y=U ∑ R if further can obtain the singular value decomposition of Y ^*, π (Y)=UI then _{N * p}R ^*A point X ∈ St in the consideration stream shape (n, p), and its disturbance point π (X+ ε N), wherein N is any matrix of the capable p row of n, ε is any real number; If satisfy π (X+ ε N)=π (X)+o (ε ²), o (ε wherein ²) be high-order in a small amount, be the definition space at the set of such matrix N place Stifel stream shape at normal direction space N that X is ordered then _X(n p), can be known by the definition in normal direction space, if N ∈ N _X(n, p), then X+ ε N can not depart from X along with the increase of ε; Space, the tangential T that X is ordered _X(n p) is defined as normal direction space N _X(n, orthogonal complement space p), its rigorous mathematical definition are the space of satisfying the set of matrices place of following condition: if Z ∈ T _X(n, p), then

Wherein

Be satisfied [X X _⊥] ^*[X _⊥X]=matrix of I; Steepest descent needs the compute gradient direction along algorithm; But the expression formula of Stifel stream shape gradient direction; Must be just meaningful under the given situation of tangent space inner product expression formula; That is to say that inner product and gradient must be defined in the same topological structure, therefore will define Euclid's inner product of the tangent space of Stifel stream shape earlier:

Z in the formula ₁, Z ₂∈ T _X(n, p), X ∈ St (n, p); On the basis of this given inner product, the steepest descent that obtains target function f along the expression formula of Z is: D in the formula _VBe the first derivative of function f at V.

Because used the A Mihuo rule to obtain selecting suitable convergence steps in the inventive method, its derivation principle is following:

If Z is f (X) in the steepest descent of X along direction, then optional getting held back steps, makes it satisfy following two inequality:

$f\left(X\right)-f(X+\mathrm{\αZ})\≥\frac{1}{2}\mathrm{\α}\⟨Z,Z\⟩---\left(1\right)$

f(X)-f(X+2αZ)＜α<Z,Z>（2）

Inequality above utilizing can obtain suitable α, and concrete grammar is: constantly attempt α is enlarged twice, no longer set up up to inequality (2); Again the value of α is at this moment constantly attempted dwindling twice, up to satisfying inequality (1); The steepest descent that the A Mihuo rule is applied to Stifel stream shape replaces to pre-coding matrix V with independent variable in selecting along the algorithmic statement step-length, inner product is replaced to the inner product form of Stifel stream shape again; Finally can obtain two following step-length selection criterions: if

then the steps in the formula is replaced with 2 α bring inequality into and judge; Repeating above-mentioned steps is false up to inequality, and this step has guaranteed that the α that selects can reduce target function as much as possible significantly $\underset{V^{\left[1\right]},...,V^{\left[K\right]}}{\min }f\left(V\right)=\underset{k=1}{\overset{K}{\mathrm{\&Sigma;}}}\underset{i=1}{\overset{d^{\left[k\right]}}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_{Q^{\left[k\right]}}^i;$ if

then the steps in the formula is replaced into and brings inequality into and judge; Repeating above-mentioned steps is false up to inequality, and this step has guaranteed can not cause missing potential optimum point because the steps of selecting is too big; Finally obtain suitable steps through above two criterions.

Notice simultaneously; When the direction of V along steepest descent along Z; Be step-length when moving to V+ α Z with α, can regard α Z as disturbance the some V in the Stifel stream shape, therefore in fact V+ α Z not in Stifel stream shape; Must adopt projection operation V+ α Z to be limited in the stream shape again projection operation: π (V+ α Z)=UI _{N * p}R ^*, wherein U and R are respectively the left and right unitary matrice of singular value decomposition (SVD) generation of V+ α Z, i.e. V+ α Z=U ∑ R ^*, wherein ∑ is the singular value diagonal matrix.

Can know by top analysis, the interference alignment method for precoding based on Stifel stream shape of the monolateral optimization of transmitter of the present invention, with the interference strength in the minimize signal space as target function; Target function is recombinated on Stifel stream shape; Remove the quadrature constraint; Inner product, projection and the steepest descent of calculating target function on Stifel stream shape is along direction; With the A Mihuo rule is the step-length choice mechanism, finally forms this steepest descent based on Stifel stream shape realizes disturbing alignment along algorithm method for precoding.Because the dimension of the Stifel that adopts among the present invention stream shape is much littler than the multidimensional complex number space dimension at traditional optimal algorithm place, so adopt the inventive method to have fast convergence rate, complexity is low, the power system capacity advantages of higher.Simultaneously because the interference that the present invention proposes alignment method for precoding; Only need transmitter to participate in; Iterating between forward direction that has caused so avoided transmitting-receiving two-end all must participate in algorithm and the reverse communication links; And the huge overheads such as synchronous and feedback that thereupon bring, all be suitable for for time division duplex and frequency-duplex communication system.

Description of drawings

Fig. 1 is the system block diagram that the present invention is based on the interference alignment method for precoding of Stifel stream shape.

Fig. 2 is following 3 the user's multiple-input-multiple-output communication system model sketch mapes of interference channel.

Fig. 3 is that tradition disturbs alignment method for precoding and Stifel to flow the comparison diagram of interference subspace angle of 3 each receivers of user of shape method for precoding.

Fig. 4 is that tradition disturbs alignment method for precoding and Stifel stream shape to disturb alignment method for precoding target function constringency performance comparison diagram.

Fig. 5 is that tradition disturbs alignment method for precoding and Stifel stream shape to disturb the method for precoding that aligns, and time division multiplexing etc. is based on orthogonalized transmission mechanism power system capacity comparison diagram.

Embodiment

Below in conjunction with the description of drawings embodiments of the invention.

Embodiment 1:

Fig. 1 has provided the flow process schematic block diagram of the system block diagram of the interference alignment method for precoding that the present invention is based on Stifel stream shape.The inventive method with the interference strength in the minimize signal space as target function; Calculate inner product expression formula and the gradient direction of this target function on Stifel stream shape; Obtain the steepest gradient algorithm on the Stifel stream shape, the final scheme that realizes disturbing the alignment precoding.

Present embodiment is as shown in fig. 1 based on the concrete operations step of the interference alignment method for precoding of Stifel stream shape:

The 1st step. initialization pre-coding matrix: V arbitrarily at first ^[1]..., V ^[K]

The 2nd step. calculate covariance matrix Q ^[k]: with number of transmit antennas in the multiple-input-multiple-output communication system is that M, reception antenna number are that the reception signal indication of K user under interference channel of N is: $Y^{\left[k\right]}=\underset{j=1}{\overset{K}{\mathrm{\&Sigma;}}}{H}^{\left[\mathrm{Kj}\right]}{V}^{\left[j\right]}{S}^{\left[j\right]}+W^{\left[k\right]},$ In the formula, the span of the first subscript k from 1 to K, the span of the second subscript j also from 1 to K; H ^[kj]Be the channel coefficient matrix of j transmitter to k receiver; S ^[j]Be the data of needs emission, V ^[j]Be corresponding pre-coding matrix, W ^[k]White Gaussian noise for the zero-mean unit variance; Then the interference covariance matrix of k receiving terminal is: $Q^{\left[k\right]}=\underset{\begin{array}{c}j=1\\ j\&NotEqual;k\end{array}}{\overset{K}{\mathrm{\&Sigma;}}}\frac{P^{\left[j\right]}}{d^{\left[j\right]}}{H}^{\left[\mathrm{Kj}\right]}{V}^{\left[j\right]}{V}^{\left[j\right]*}{H}^{\left[\mathrm{Kj}\right]*},$ In the formula, P ^[j]Be transmitted power, d ^[j]Be the data flow number of sending, subscript symbol * represents the conjugate transpose operation of matrix;

The 3rd step. calculate Q ^[k]Characteristic value, set up target function: to Q ^[k]Carry out characteristic value decomposition, obtain a series of characteristic value

Its subscript Q ^[k]Represent the corresponding interference covariance matrix of this characteristic value, the sequence number of characteristic value is arranged in its subscript i representative with ascending order; Interference strength in the minimize signal space is equivalent to and minimizes d ^[k]Individual characteristic value

Thereby set up mapping relations, obtain required target function from the multidimensional complex number space to the arithmetic number space $\underset{V^{\left[1\right]},...,V^{\left[K\right]}}{\min }f=\underset{k=1}{\overset{K}{\mathrm{\&Sigma;}}}\underset{i=1}{\overset{d^{\left[k\right]}}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_{Q^{\left[k\right]}}^i,$ Constraints is V ^{[j] *}V ^[j]=I, I are unit vector;

The 4th step. f flows shape at Stifel with target function

Last reorganization is in the formula

Represent dimension n to multiply by the complex number space of p; Result after the reorganization of Stifel stream shape is: remove constraints V ^{[j] *}V ^[j]=I keeps Minimize target; Then with complex matrix V ^[j]Be decomposed into real part And imaginary part Target function f respectively to real part and imaginary part differentiate, is obtained two corresponding Jacobians With

Then have $\mathrm{Df}=[D_R^{\left[1\right]}....D_R^{\left[j\right]}]\left[\begin{array}{c}{\mathrm{DV}}_R^{\left[1\right]}\\ \&CenterDot;\\ \&CenterDot;\\ {\mathrm{DV}}_R^{\left[j\right]}\end{array}\right]+[D_I^{\left[1\right]}....D_I^{\left[j\right]}]\left[\begin{array}{c}{\mathrm{DV}}_I^{\left[1\right]}\\ \&CenterDot;\\ \&CenterDot;\\ {\mathrm{DV}}_I^{\left[j\right]}\end{array}\right],$ And then obtain the first derivative of target function f to V [j]: $D_V^{\left[j\right]}={(D_R^{\left[j\right]}+{\mathrm{ID}}_I^{\left[j\right]})}^T;$

The 5th step. calculate the steepest gradient direction on the Stifel stream shape: $Z^{\left[j\right]}=V^{\left[j\right]}{D}_V^{\left[j\right]}{V}^{\left[j\right]}-D_V^{\left[j\right]},$ Then target function reduces along the direction of Z, is expressed as f (V ^[j]+ α ^[j]Z ^[j]);

The 6th step. utilize the A Mihuo criterion to calculate f (V ^[j]+ α ^[j]Z ^[j]) in steps ^[j], adopt following two criterions, criterion one: if

Then with the steps in the formula ^[j]Replace with 2 α ^[j]Bring inequality into and judge, in the formula

Be the inner product expression-form of Stifel stream shape, repeating above-mentioned steps is false up to inequality; Criterion two: if Then with the steps in the formula ^[j]Be replaced into

Bring inequality into and judge, repeating above-mentioned steps is false up to inequality, final to suitable convergence steps ^[j]

The 7th step. projection operation retrains the result again and to get back to Stifel stream shape: π (V ^[j]+ α ^[j]Z ^[j])=UI _{N * p}R ^*, wherein U and R are respectively V ^[j]+ α ^[j]Z ^[j]The left and right unitary matrice that produces of singular value decomposition, i.e. V ^[j]+ α ^[j]Z ^[j]=U ∑ R ^*, ∑ is the singular value diagonal matrix in the formula;

The 8th step. iteration, restrain up to target function.

Fig. 2 has provided the inventive method has been applied in following 3 the user's multiple-input-multiple-output communication system model sketch mapes of interference channel; Solid arrow is represented communication link among the figure; Dotted arrow is represented interfering link; The antenna number of transmitter and receiver is 2, and 1 message of the each emission of each transmitter is given corresponding receiver, produces at other receivers simultaneously and disturbs.

Fig. 3 has provided under the represented scene of Fig. 2, the interference subspace angle comparison diagram of 3 each receivers of user.Can to expression tradition among the figure disturb the interference subspace angle of the receiver 1 under the alignment method for precoding along with the interference subspace angle of the convergence curve A1 of iterations, receiver 2 along with the interference subspace angle of the convergence curve B1 of iterations, receiver 3 convergence curve C1 along with iterations; Align with the interference of Stifel stream shape the receiver 1 under the method for precoding interference subspace angle along with the interference subspace angle of the convergence curve D1 of iterations, receiver 2 along with the interference subspace angle of the convergence curve E1 of iterations, the receiver 3 convergence curve F1 along with iterations compares intuitively; Thereby clearly find: it is faster that the interference alignment method for precoding that the present invention is based on Stifel stream shape disturbs the subspace angle to converge to zero speed than conventional method; And disturb the subspace angle is the zero interference signal complete matching that means; From disturbing the subspace angle, adopt the inventive method faster with this than conventional method convergence rate.

Fig. 4 has provided under the represented scene of Fig. 2, and tradition disturbs alignment method for precoding and Stifel stream shape to disturb alignment method for precoding target function constringency performance relatively.As can be seen from the figure; The target function that employing the present invention is based on the interference alignment method for precoding of Stifel stream shape disturbs the target function convergence curve B2 of the method for precoding that aligns to compare with the convergence curve A2 of iterations with tradition, and the inventive method is superior to conventional method on the target function constringency performance.

Fig. 5 has provided the power system capacity comparison diagram.The power system capacity that from figure, will the present invention is based on the interference alignment method for precoding of Stifel stream shape is disturbed and is alignd and the single user's of power system capacity curve C 3,2 antenna users of coding method power system capacity curve B 3, time division multiplexing (TDMA) method the power system capacity curve D 3 and the power system capacity curve E3 of precoding at random carry out relatively can finding out intuitively along with the curve A 3 of transmitter power and tradition, and the interference that the present invention is based on Stifel stream shape method for precoding that aligns is superior to other conventional methods on power system capacity.

Claims (1)

1. interference alignment method for precoding based on Stifel stream shape is that M, reception antenna number are that the reception signal indication of K user under interference channel of N is with number of transmit antennas in the multiple-input-multiple-output communication system earlier: $Y^{\left[k\right]}=\underset{j=1}{\overset{K}{\mathrm{\&Sigma;}}}{H}^{\left[\mathrm{Kj}\right]}{V}^{\left[j\right]}{S}^{\left[j\right]}+W^{\left[k\right]},$ In the formula, the span of the first subscript k from 1 to K, the span of the second subscript j also from 1 to K; H ^[kj]Be the channel coefficient matrix of j transmitter to k receiver; S ^[j]Be the data of needs emission, V ^[j]Be corresponding pre-coding matrix, W ^[k]White Gaussian noise for the zero-mean unit variance; Then the interference covariance matrix of k receiving terminal is: $Q^{\left[k\right]}=\underset{\begin{array}{c}j=1\\ j\&NotEqual;k\end{array}}{\overset{K}{\mathrm{\&Sigma;}}}\frac{P^{\left[j\right]}}{d^{\left[j\right]}}{H}^{\left[\mathrm{Kj}\right]}{V}^{\left[j\right]}{V}^{\left[j\right]*}{H}^{\left[\mathrm{Kj}\right]*},$ In the formula, P ^[j]Be transmitted power, d ^[j]Be the data flow number of sending, subscript symbol * represents the conjugate transpose operation of matrix; To Q ^[k]Carry out characteristic value decomposition, obtain a series of characteristic value

Its subscript Q ^[k]Represent the corresponding interference covariance matrix of this characteristic value, the sequence number of characteristic value is arranged in its subscript i representative with ascending order; Interference strength in the minimize signal space is equivalent to and minimizes d ^[k]Individual characteristic value

Thereby set up mapping relations, obtain required target function from the multidimensional complex number space to the arithmetic number space Constraints is V ^{[j] *}V ^[j]=I, I are unit vector;

It is characterized in that: f flows shape at Stifel with target function

Last reorganization is in the formula

Representing dimension is the complex number space that n multiply by p; Result after the reorganization of Stifel stream shape is: remove constraints V ^{[j] *}V ^[j]=I keeps

Minimize target; Then with complex matrix V ^[j]Be decomposed into real part

And imaginary part

Target function f respectively to real part and imaginary part differentiate, is obtained two corresponding Jacobians:

With

Then have

$\mathrm{df}=[D_R^{\left[1\right]}....D_R^{\left[j\right]}]\left[\begin{array}{c}{\mathrm{dV}}_R^{\left[1\right]}\\ \&CenterDot;\\ \&CenterDot;\\ {\mathrm{dV}}_R^{\left[j\right]}\end{array}\right]+[D_I^{\left[1\right]}....D_I^{\left[j\right]}]\left[\begin{array}{c}{\mathrm{dV}}_I^{\left[1\right]}\\ \&CenterDot;\\ \&CenterDot;\\ {\mathrm{dV}}_I^{\left[j\right]}\end{array}\right];$

Obtain the first derivative of target function f to pre-coding matrix V: D thus _V=(D _R+ iD _I) ^T, in the formula () ^TRepresent the transpose of a matrix operation; And then obtain steepest descent on the Stifel stream shape along expression formula:

Then target function reduces along the direction of Z, is expressed as f (V+ α Z), and wherein the size of steps obtains through the A Mihuo rule, specifically adopts following two criterions: if

Then the steps in the formula is replaced with 2 α and bring inequality into and judge, repeating above-mentioned steps is false up to inequality, wherein

Real part is got in representative, and tr () represents the matrix trace operation, Inner product expression formula for Stifel stream shape; If Then the steps in the formula is replaced into

Bring inequality into and judge, repeating above-mentioned steps is false up to inequality, finally obtains suitable steps; Then V+ α Z is carried out projection operation: π (V+ α Z)=UI _{N * p}R ^*, wherein U and R are respectively the left and right unitary matrice of the singular value decomposition generation of V+ α Z, i.e. V+ α Z=U ∑ R ^*, wherein ∑ is the singular value diagonal matrix; Successively to the pre-coding matrix V of K transmitter ^[j]Carry out iteration, wherein the span of subscript j when target function f iteration convergence, has just obtained the pre-coding matrix V of a series of interference alignment from 1 to K.

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