KR20090115822A - Eigenvalue decomposition and singular value decomposition of matrices using jacobi rotation - Google Patents

Abstract

Techniques for decomposing matrices using Jacobi rotation are described. Multiple iterations of Jacobi rotation are performed on a first matrix of complex values with multiple Jacobi rotation matrices of complex values to zero out the off-diagonal elements in the first matrix. For each iteration, a submatrix may be formed based on the first matrix and decomposed to obtain eigenvectors for the submatrix, and a Jacobi rotation matrix may be formed with the eigenvectors and used to update the first matrix. A second matrix of complex values, which contains orthogonal vectors, is derived based on the Jacobi rotation matrices. For eigenvalue decomposition, a third matrix of eigenvalues may be derived based on the Jacobi rotation matrices. For singular value decomposition, a fourth matrix with left singular vectors and a matrix of singular values may be derived based on the Jacobi rotation matrices.

Description

Eigenvalue decomposition and singular value decomposition of matrix using Jacobi rotation {EIGENVALUE DECOMPOSITION AND SINGULAR VALUE DECOMPOSITION OF MATRICES USING JACOBI ROTATION}

Claiming Priority Under 35 USC §119 This patent application claims priority to preliminary application No. 60/628,324 filed on November 15, 2004 under the name "Eigenvalue Decomposition and Singular Value Decomposition of Matrix Using Jacobi Rotation". , Which has been assigned to the assignee of this application and is hereby incorporated by reference.

TECHNICAL FIELD The present invention relates generally to communication, and more particularly, to a matrix decomposition technique.

A multiple input multiple output (MIMO) communication system uses multiple (T) transmit antennas at a transmitting station and multiple (R) receive antennas at a receiving station for data transmission. The MIMO channel formed by the T transmit antennas and R receive antennas can be decomposed into S spatial channels, where S≦min{T, R}. The S spatial channels can be used to transmit data in a manner that achieves higher overall throughput and/or greater reliability.

The MIMO channel response can be characterized by an R×T channel response matrix H , which includes the complex channel gains for both different transmit and receive antenna pairs. The channel response matrix H may be a diagonal matrix to obtain S eigen modes, which may be presented as an orthogonal spatial channel of the MIMO channel. Improved performance can be achieved by transmitting data in the native mode of the MIMO channel.

Channel response matrix H may be a diagonal matrix by performing singular value decomposition or eigenvalue decomposition of a correlation matrix of H for H. Singular value decomposition gives left and right singular vectors, and eigenvalue decomposition gives eigenvectors. The transmitting station transmits data in S eigen modes by using the right singular vector or the eigen vector. The receiving station receives data transmitted in the S eigenmodes using the left singular vector or the eigenvector.

Eigenvalue decomposition and singular value decomposition are computationally excessive. Therefore, there is a need for a technique to effectively decompose the matrix.

A technique for effectively decomposing a matrix using Jacobi rotation is described herein. This technique can be used to decompose the eigenvalues of the Hermitian matrix of complex values to obtain an eigenvector matrix and an eigenvalue matrix for the Hermitt matrix. The technique can also be used for singular value decomposition of an arbitrary matrix of complex values to obtain a left singular vector matrix, a right singular vector matrix, and a singular value matrix for an arbitrary matrix.

In an embodiment, multiple iterations of Jacobian rotation are performed on a first matrix of complex values using multiple Jacobian rotation matrices of complex values to make the non-diagonal component zero in the first matrix. The first matrix may be a channel response matrix H , a correlation matrix of R , H , or some other matrix. For each iteration, a partial matrix is formed and decomposed based on the first matrix to obtain an eigenvector for the partial matrix, and a Jacobian rotation matrix is formed as an eigen vector, which can be used to update the first matrix. A second matrix of complex values is derived based on the Jacobian rotation matrix. The second matrix includes an orthogonal vector and may be a right singular vector of H or a matrix V _i of an eigenvector of R.

가 유도될 수 있으며, 제 3 행렬 W _i 를 기초로 특이값들의 행렬

또한 유도될 수 있다. 제 2 SVD 실시예에 기반한 특이값 분해에 관하여, 야코비 회전 행렬을 기초로 직교 벡터를 가진 제 3 행렬 U _i 및 특이값의 행렬

가 유도될 수 있다.Regarding the eigenvalue decomposition, a third matrix D _i of eigenvalues may be derived based on the Jacobian rotation matrix. Regarding the singular value decomposition based on the first singular value decomposition (SVD) embodiment, a third matrix W _i of complex values may be derived based on the Jacobian rotation matrix, and a fourth matrix based on the third matrix W _i

Can be derived, and a matrix of singular values based on the third matrix W _i

It can also be derived. Regarding the singular value decomposition based on the second SVD embodiment, a third matrix U _i having an orthogonal vector based on a Jacobian rotation matrix and a matrix of singular values

Can be induced.

Various forms and embodiments of the present invention are described in more detail later.

The present invention is applicable to communications in general, and in particular to matrix decomposition techniques.

The word "exemplary" is used herein in the sense of "to serve as an example, illustration or illustration." No embodiment described herein as “exemplary” is to be construed as preferred or advantageous over other embodiments.

The matrix decomposition technique described herein includes a single carrier communication system having a single frequency subband, a multicarrier communication system having a plurality of subbands, a single carrier frequency division multiple access (SC-FDMA) system having a plurality of subbands, and other It can be used in various communication systems such as communication systems. Multiple subbands may be obtained by orthogonal frequency division multiplexing (OFDM), some other modulation technique, or some other configuration. OFDM divides the entire system bandwidth into multiple (K) orthogonal subbands, and these multiple orthogonal subbands are also called tones, subcarriers, bins, and the like. In OFDM, each subband is associated with each subcarrier that can be modulated with data. The SC-FDMA system transmits on subbands distributed across the system bandwidth using interleaved FDMA (IFDMA), or transmits on adjacent subbands of a block using localized FDMA (LFDMA), or enhanced It is also possible to transmit on adjacent subbands of multiple blocks using FDMA (EFDMA). In general, modulation symbols are transmitted in the frequency domain by OFDM and in the time domain by SC-FDMA. For brevity, much of the following description relates to a MIMO system with a single subband.

식(1) 여기서 성분 h _{i , j} (i = 1, … , R, j = 1, … , T)는 송신 안테나(j)와 수신 안테나(i) 간의 커플링 또는 복소 채널 이득을 나타낸다.The MIMO channel formed by multiple (T) transmit antennas and multiple (R) receive antennas can be characterized by an R×T channel response matrix H , which can be given as:

Equation (1) Here, the components h _{i , j} ( i = 1,…, R, j = 1,…, T) represent the coupling or complex channel gain between the transmit antenna j and the receive antenna i.

The channel response matrix H is a diagonal matrix, and multiple (S) eigen modes of H can be obtained, where S≤min{T, R}. For example, it is achieved by performing the diagonal upset the singular value decomposition H or eigenvalue decomposition of a correlation matrix of H.

The eigenvalue decomposition can be expressed as: R = H ^H · H = V · Λ · V ^H Equation (2) where R is the TxT correlation matrix of H; V is a TxT unitary matrix whose columns are the eigenvectors of R; Λ is a TxT diagonal matrix of the eigenvalues of R; " ^H " represents the conjugate transpose. The unitary matrix V is characterized by V ^H · V = I , where I is the identity matrix. The columns of the unitary matrix are orthogonal to each other, and each column has a unit power. The diagonal matrix Λ contains possible non-zero values along the diagonal, and zero elsewhere. The diagonal components of Λ are the eigenvalues of R. The eigenvalues are {λ ₁ , λ ₂ ,… , λ _S }, and represents the power gain for S eigenmodes. R is an Hermitian matrix whose components outside the diagonal have the following properties: r _i , _j = r ^* _j , _i ^{, where "*} " is the complex conjugate.

이다.The singular value decomposition can be expressed as: H = U · ∑ · V ^H Equation (3) where U is the R×R unitary matrix of the left singular vector of H; ∑ is an RxT diagonal matrix of singular values of H; V is a TxT unitary matrix of the right singular vector of H. U and V each contain a diagonal vector. Equations (2) and (3) indicate that the right singular vector of H is also the eigenvector of R. The diagonal components of ∑ are the singular values of H. The singular values are {σ ₁ , σ ₂ ,… , σ _S }, and represents the power gain for S eigenmodes. Since the singular values of H are also the square roots of the eigenvalues of R , i =1,… , About S

to be.

는 s 의 데이터 심벌의 추정치인 검출된 데이터 심벌을 S개까지 갖는 T×1 벡터이다. 표 1

The transmitting station can transmit data on the eigenmode of H using the right singular vector in V. Data transmission on the eigenmode typically provides better performance than simply transmitting data from the T transmit antennas without spatial processing. The receiving station can receive the data transmission transmitted on the eigenmode of H using the left singular vector in U or the eigen vector in V. Table 1 shows the spatial processing performed by the transmitting station, the received symbols at the receiving station, and the spatial processing performed by the receiving station. In Table 1, s is a T×1 vector having up to S data symbols to be transmitted, x is a T×1 vector having T transmission symbols to be transmitted from T transmission antennas, and r is R obtained from R reception antennas. Is an R×1 vector with four received symbols, n is an R×1 noise vector,

Is a Tx1 vector having up to S detected data symbols, which are estimates of the data symbols of s. Table 1

Eigenvalue decomposition and singular value decomposition of a complex matrix can be performed by an iterative process using Jacobian rotation, commonly referred to as Jacobian method and Jacobian transform. Jacobian rotation makes the non-diagonal component pair of a complex matrix zero by performing a horizontal rotation on the matrix. In a 2x2 complex Hermit matrix, only one iteration of the Jacobian rotation is required to obtain two eigenvectors and two eigenvalues for this 2x2 matrix. In a complex matrix with a dimension larger than 2×2, the iterative process performs several iterations of the Jacobian matrix to obtain the desired eigenvectors and eigenvalues, or singular vectors and singular values for the larger complex matrix. Each iteration of the Jacobian rotation for a larger complex matrix uses an eigenvector of a 2x2 submatrix, as described below.

식(4) 여기서 A, B, D는 임의의 실수값이고, θ _b 는 임의의 위상이다. _{The eigenvalue decomposition of the 2 ×2} Hermit matrix R 2 ×2 can be performed as follows. The Hermit matrix R _{2 ×2} can be expressed as:

Equation (4) where A, B, and D are arbitrary real values, and θ _b is an arbitrary phase.

식(5) 여기서 R _re는 실수값을 포함하고 (1, 2) 및 (2, 1) 위치에 대칭인 비-대각 성분을 갖는 대칭 실수 행렬이다. The first step in the eigenvalue decomposition of R _{2 ×2} is to apply a two-sided unitary transform as follows:

Equation (5) where R _re is a symmetric real matrix with non-diagonal components that contain real values and are symmetric at positions (1, 2) and (2, 1).

식(6) 여기서 각 φ는 다음과 같이 표현될 수 있다:

식(7)The symmetric real matrix R _re is a diagonal matrix using two-sided Jacobian rotation as follows:

Equation (6) where angle φ can be expressed as:

Equation (7)

식(8)2 × 2 unitary matrix V _{2 × 2} of eigenvectors of R _{2 × 2} may be derived as follows:

Equation (8)

식(9)The two eigenvalues λ ₁ and λ ₂ can be derived on the basis of Eq. (6) or Eq. Λ _2×2 = V ^H _{2 ×2} · R _{2 ×2} · V _{2×2 as follows:}

Equation (9)

In equation set (9), the order of the two eigenvalues is not constant, and λ ₁ may be larger or smaller than λ _2. However, if the angle φ is fixed at |2 φ |≤ π /2, then cos2 φ ≥0, if D> A and only in this case sin2 φ >0. Therefore, the order of the two eigenvalues can be determined by the relative sizes of A and D. When A> D, λ ₁ is a larger eigenvalue, and when D> A, λ ₂ is a larger eigenvalue. If A = D, then sin2 φ =1 and λ ₂ is the larger eigenvalue. If λ ₂ is a larger eigenvalue, two eigenvalues of Λ _2×2 are exchanged to maintain a predetermined order of the largest eigenvalue to the smallest eigenvalue, and the first and second columns of V _{2 ×2} It can also be exchanged accordingly. Thus V _{2 × 2} two Maintaining a predetermined order for the eigenvector to be greatest using the unique value V _{2 × 2} is arranged with the smallest eigenvalues from decomposing the size is more of a large matrix, the eigenvectors of and , This is desirable.

식(10) 식(10)은 R _{2 ×2}의 특성 방정식의 해이다. 식(10)에서, 우변의 두 번째 수에 대한 양의 부호로 λ₁이 구해지고, 우변의 두 번째 수에 대한 음의 부호로 λ₂가 구해지며, λ₁≥λ₂이다.The two eigenvalues λ ₁ and λ ₂ can also be calculated directly from the components of R _re as follows:

Equation (10) Equation (10) is the solution of the characteristic equation of R _{2 ×2. In equation (10), λ 1} is obtained as the positive sign for the second number on the _{right side, λ 2} is obtained as the negative sign for the second number on the right side, and λ ₁ ≥λ ₂ .

식(11a)

식(11b) c ₁ = r ₁·Re{r _{1 ,2}} = cos(∠r _{1 ,2}) 식(11c) s ₁ = -r ₁·Im{r _{1 ,2}} = sin(∠r _{1 ,2}) 식(11d) g ₁ = c ₁ + js ₁ 식(11e)

식(11f)

식(11g)

식(11h) τ < 0이라면, t < -t 식(11i)

식(11j) s = t·c 식(11k) τ < 0이라면,

, 그렇지 않으면

식(11l) 여기서 r _{1 ,1}, r _{1 ,2} 및 r _{2 ,2}는 R _{2 ×2}의 성분이고, r은 r _{1 ,2}의 크기이다. g ₁은 복소값이기 때문에 V _{2 ×2}는 두 번째 행에 복소값을 포함한다.Equation (8) requires the calculation of cos φ and sin φ to derive the components of V _{2 ×2.} The calculation of cos φ and sin φ is complex. The components of V _{2 ×2} can also be calculated directly from the components of R _{2 ×2 as follows:}

Equation (11a)

Equation (11b) c ₁ = r ₁ ·Re{ r _{1 ,2} } = cos(∠ r _{1 ,2} ) Equation (11c) s ₁ = -r ₁ ·Im{ r _{1 ,2} } = sin(∠ r _{1 ,2} ) Equation (11d) g ₁ = c ₁ + js ₁ equation (11e)

Equation (11f)

Formula (11g)

Equation (11h) If τ <0, then t < -t Equation (11i)

If equation (11j) s = t · c equation (11k) τ <0,

, Otherwise

Formula (11l) where r _{1 ,1} , r _{1 ,2} and r _{2 ,2} are components of R _{2 ×2} , and r is the size of r _{1 ,2.} Since g ₁ is a complex value, V _{2 ×2} contains the complex value in the second row.

*The equation set (11) is designed to reduce the amount of computation to derive V _{2 ×2} from R _{2 ×2.} For example, it is necessary to divide by r in equations (11c), (11d), and (11f). Instead, we take the inverse of r to obtain the r _1, is performed the product of r ₁ for the formula (11c), (11d), (11f). This reduces the number of division operations that are computationally more expensive than multiplication. In addition, instead of calculating the declination (phase) of the complex components r _{1 and 2} that require arctangent calculation and then calculating the cosine and sine of the phase value to obtain c ₁ and s ₁ , r ₁ is used only by the square root operation. Various trigonometric identities are used to solve c ₁ and s ₁ as functions of the real and imaginary parts of _,2. Moreover, instead of calculating the arctangent of equation (7) and the sine and cosine functions of equation (8), other trigonometric identities are used to solve c and s as functions of the components of R _{2 × 2.}

Equation set (11) calculates V _{2 × 2} by performing a complex Jacobian rotation on R _{2 ×2.} The set of calculations in equation set (11) is designed to reduce the number of multiplication, square root, and reciprocal operations required to derive V _{2 ×2.} This can significantly reduce the computational complexity for larger-sized matrix decomposition using V _{2 ×2.}

식(12a) z = x·r 식(12b) λ ₁ = y + z 식(12c) λ ₂ = y - z 식(12d) 1. 고유값 분해 The eigenvalues of R _{2 ×2} can be calculated as follows:

Equation (12a) z = x · r Equation (12b) λ ₁ = y + z equation (12c) λ ₂ = y -z equation (12d) 1. Eigenvalue decomposition

As shown in equation (2), the eigenvalue decomposition of an N×N Hermit matrix larger than 2×2 can be performed in an iterative process. This iterative process uses Jacobian rotation repeatedly to zero non-diagonal components in the NxN Hermit matrix. During the iterative process, an N×N unitary transform matrix is formed based on the 2×2 Hermit submatrices of the N×N Hermit matrix and applied repeatedly to make the N×N Hermit matrix a diagonal matrix. Each unitary transformation matrix contains four non-trivial components (i.e., components other than 0 or 1) derived from the components of the corresponding 2x2 Hermit submatrix. The transformation matrix is also referred to as the Jacobian rotation matrix. After completing all Jacobian rotations, the resulting diagonal matrix contains the real eigenvalues of the N×N Hermit matrix, and the product of all unitary transformation matrices is the N×N matrix of the eigenvectors for the N×N Hermit matrix. to be.

In the following description, the index i represents the number of repetitions and is initialized to i = 0. R is the N×N Hermit matrix to be decomposed, and N> 2. The N×N matrix D _i is an approximation of the diagonal matrix Λ of the eigenvalues of R , and is initialized to D ₀ = R. The N×N matrix V _i is an approximation of the matrix V of the eigenvectors of R , and is initialized to V ₀ = I.

식(13) 여기서 d _{p ,q} 는 D _i 에서 (p, q) 위치의 성분이고; p ∈ {1, … , N}, q ∈ {1, … , N}이며, p ≠ q이다. D _pq 는 D _i 의 2×2 부분 행렬이고, D _pq 의 4개의 성분은 D _i 에서 (p, p), (p, q), (q, p), (q, q) 위치의 4개의 성분이다. 인덱스 p 및 q에 대한 값은 후술하는 바와 같이 다양한 방식으로 선택된다.One iteration of the Jacobian rotation to update the matrices D _i and V _{i may be performed as follows.} First, based on the current D _i , a 2×2 Hermitian matrix D _pq is formed as follows:

Formula (13) where d _{p ,q} is a component at the ( p , q ) position in D _i; p ∈ {1,… , N}, q ∈ {1,… , N}, and p ≠ q . D _pq is a four position and the 2 × 2 sub-matrix of D _i, the four components of the D _pq is in _{D i (p, p),} (p, q), (q, p), (q, q) It is an ingredient. The values for indices p and q are selected in various ways, as described below.

The eigenvalue decomposition of D _pq is performed as shown in the examples to obtain the 2 × 2 unitary matrix of eigenvectors example expression set of D _pq (11). For _{the eigenvalue decomposition of D pq} , R _{2 × 2} in equation (4) is replaced by D _pq , and V _{2 × 2} from equation (11l) is given as V _pq.

식(14) 여기서 v _{1 ,1}, v _{1 ,2}, v _{2 ,1}, v _{2 ,2}는 V _pq 의 4개의 성분이다. T _pq 의 다른 비-대각 성분들은 모두 0이다. 식(11l)은 T _pq 가 v _{2 ,1} 및 v _{2 ,2}에 대한 복소값을 포함하는 복소 행렬임을 지시한다. T _pq 는 야코비 회전을 수행하는 변환 행렬이라고도 한다.Then, an N×N complex Jacobian rotation matrix T _pq is formed by the matrix V _pq. T _pq is (p , p ), ( p , q ), ( q , p ) replaced by (1, 1), (1, 2), (2, 1), (2, 2) of V _{pq, respectively} , ( q , q ) is an identity matrix with four components of the position. T _pq has the form:

Equation (14) where v _{1 ,1} , v _{1 ,2} , v _{2 ,1} , v _{2 ,2} are the four components of V _pq. All other non-diagonal components of T _{pq are zero.} Equation (11l) indicates that T _pq is a complex matrix including complex values for v _{2 ,1} and v _{2 ,2.} T _pq is also referred to as a transformation matrix that performs Jacobian rotation.

The matrix D _i is updated as follows: D _{i +1} = T ^H _pq · D _i · T _pq Equation (15) Equation (15) is at positions ( p , q ) and ( q , p ) in D _{i, respectively.} Zero the two non-diagonal components d _{p ,q} and d _{q ,p.} The above calculation may change the value of other non-diagonal components of D _i.

The matrix V _i is updated as follows: V _{i +1} = V _i · T _pq Equation (16) V _i may be presented as a cumulative transformation matrix containing all Jacobian rotation matrices T _pq used for D _i .

Each iteration of the Jacobian rotation zeroes the two non-diagonal components of D _i. Multiple iterations of Jacobian rotation can be performed for different values of indices p and q to make all non-diagonal components of D _{i zero.} The indices p and q can be selected in a predetermined manner by sweeping over all possible values.

One sweep for all possible values for indices p and q can be performed as follows. The index p can be stepped by 1 from 1 to N-1. For each value of p, the index q can be stepped by 1 from p +1 to N. Iteration of Jacobian rotation to update matrices D _i and V _i can be performed for different combinations of values for p and q. At each iteration, D _pq is formed based on the current values of D _i and p and q for that iteration, V _pq is formed for D _pq as shown in equation set (11), and equation (14) T _pq is formed by V _pq as shown in, D _i is updated as shown in equation (15), and V _i is updated as shown in equation (16). at a given combination of values for p and q, the ratio of the position of the D _i (q, p) - if the size of the diagonal elements is less than a predetermined threshold Jacobi rotation to update D _i and V _i will be omitted May be.

The sweep consists of N·(N-1)/2 iterations of Jacobian rotation to update D _i and V _i for all possible values of indices p and q. Each iteration of the Jacobian rotation makes the two non-diagonal components of D _i zero, but it is also possible to change other components that became zero earlier. Effect of the sweep of the index p and q have any ratio of D _i - to be reduced in size to a diagonal elements D _i is close to a diagonal matrix Λ. V _i contains the accumulation of all Jacobian rotation matrices collectively giving D _i. Therefore, V _i is closer to the V according to D _i is closer to Λ.

Any number of sweeps can be performed to get an increasingly accurate approximation of V and Λ. Computer simulations have shown that 4 sweeps are sufficient to reduce the non-diagonal component of D _i to negligible levels, and 3 sweeps are sufficient for most applications. A predetermined number of sweeps (eg, 3 or 4 sweeps) may be performed. Alternatively, the ratio of D _i - diagonal components may choose whether to be checked after each sweep D _i is accurate enough. For example, after each sweep, the total error (e.g., the power of all non-diagonal components of D _i ) may be calculated and compared to the error threshold, or if the total error is less than the error threshold, the iterative process may end. have. Other conditions or criteria may be used to terminate the iterative process.

The values for indices p and q may be chosen in a deterministic manner. For instance, in each iteration (i), the largest ratio of D _i - the diagonal components are identified can be represented by d _{p, q.} Jacobian rotation can be performed with D _pq containing three other components at positions ( p , p ), ( q , p ), ( q , q ) of this maximum non-diagonal component d _{p ,q} and D _i have. The iterative process can be performed until the termination condition is met. The termination condition may be, for example, termination of a predetermined number of repetitions, satisfaction of the aforementioned error criterion, or any other condition or criterion.

At the end of the iterative process, the final V _i is a good approximation of V , and the final D _i is a good approximation of Λ. The columns of V _i can be provided as the eigenvectors of R , and the diagonal components of D _i can be provided as the eigenvalues of R. The eigenvalues of the final D _i are arranged from the largest value to the smallest value because the eigenvectors of V _{pq for each iteration are arranged.} The eigenvectors of the final V _i are also arranged based on the associated eigenvalue of D _i.

FIG. 1 shows an iterative process 100 for performing eigenvalue decomposition of an N×N Hermit matrix R using Jacobian rotation when N>2. The matrices V _i and D _i are initialized with V ₀ = I and D ₀ = R and the index i is initialized with i = 1 (block 110).

In iteration ( i ), either in a predetermined manner (e.g., by stepping over all possible values for these indices) or in a deterministic manner (e.g., by selecting the index values for the largest non-diagonal components). ) Values for indices p and q are selected (block 112). Then, a 2x2 matrix D _pq having four components of the matrix D _i is formed at the position determined by the indices p and q (block 114). Then, for example, eigenvalue decomposition of D _pq as shown in equation set (11) is performed to obtain a 2 × 2 matrix of eigenvectors of V _{pq pq} D (block 116). As shown in equation (14), an N×N complex Jacobian rotation matrix T _pq is formed based on the matrix V _pq (block 118). The matrix D _i is then updated based on T _pq as represented by equation (15) (block 120). Further, the matrix V _i is updated based on T _pq as represented by equation (16) (block 122).

A determination is made as to whether to end the eigenvalue decomposition of R (block 124). The termination criterion may be based on the number of iterations or sweeps already performed, an error criterion, and the like. If the answer to block 124 is no, then index i is incremented (block 126), and the process returns to block 112 for the next iteration. Otherwise, at the end, D _i is provided as an approximation of the diagonal matrix Λ , and V _i is provided as an approximation of the eigenvector matrix V of R (block 128).

In a MIMO system having multiple subbands (eg, a MIMO system using OFDM), multiple channel response matrices H ( k ) can be obtained for different subbands. The iterative process is performed for each of the channel response matrices H ( k ) to obtain the matrices D _i ( k ) and V _i ( k ), each of which is unique to R ( k ) = H ^H ( k ) · H ( k) It is an approximation of the diagonal matrix of vectors Λ ( k ) and the matrix V ( k).

There is usually a high degree of correlation between adjacent subbands in a MIMO channel. This correlation can be utilized by an iterative process to reduce the amount of computation to derive D _i ( k ) and V _i ( k ) for the subbands of interest. For example, an iterative process may be performed for one subband at a time, starting at one end of the system bandwidth and across to the other end of the system bandwidth. For each subband ( k ) excluding the first subband, the final solution V _i ( k -1) obtained for the previous subband ( k -1) may be used as the initial solution for the current subband ( k). Initialization for each subband ( k ) can be given as V ₀ ( k ) = V _i ( k -1) and D ₀ ( k ) = V ₀ ^H ( k ) R ( k ) V ₀ ( k ) have. The iterative process acts on the initial solution of D ₀ ( k ) and V ₀ ( k ) for the subband (k ) until the termination condition is met.

The above-described concept may be used with respect to time. For each time interval ( t ), the final solution V _i ( t -1) obtained for the previous time interval ( t -1) may be used as the initial solution to the current time interval ( t ). Initialization for each time interval ( t ) can be given as V ₀ ( t ) = V _i ( t -1) and D ₀ ( t ) = V ₀ ^H ( t ) R ( t ) V ₀ ( t ) Where R ( t ) = H ^H ( t ) · H ( t ) and H ( t ) is the channel response matrix for the time interval ( t ). Then, the iterative process acts on the initial solutions of D ₀ ( t ) and V ₀ ( t ) over the time interval (t ) until the termination condition is met. The concept can also be used for both frequency and time. The above concept can be used for both frequency and time. For each subband in each time interval, the last solution obtained for the previous subband and/or the last solution obtained for the previous time interval may be used as the initial solution for the current subband and the time interval. 2. Singular value decomposition

는 모두 에르미트 행렬이다. 둘째, V 의 열인 H 의 우측 특이 벡터들은 R 의 고유 벡터이기도 하다. 이에 대응하여, U 의 열인 H 의 좌측 특이 벡터들은

의 고유 벡터이기도 하다. 셋째, R 의 0이 아닌 고유값들은

의 0이 아닌 고유값들과 동일하며, H 의 해당 특이값들의 제곱이다.An iterative process may be used for singular value decomposition of any complex matrix H greater than 2×2. Specific value decomposition of H is given as ^{H = U · Σ · V H} . The following observations can be made with respect to H. First, the matrix R = H ^H H and

Are all Hermitian matrices. Second, the right singular vectors of H , the row of V , are also the eigenvectors of R. Correspondingly, the left singular vectors of H , the column of U, are

It is also the eigenvector of. Third, the non-zero eigenvalues of R

It is equal to the non-zero eigenvalues of and is the square of the corresponding singular values of H.

식(17) 여기서 h ₁은 H _{2 ×2}의 제 1 열의 성분들을 가진 2×1 벡터이고; h ₂는 H _{2 ×2}의 제 2 열의 성분들을 가진 2×1 벡터이다.A 2×2 matrix of complex values H _{2 ×2} can be expressed as:

Equation (17) where h ₁ is a 2×1 vector with the components of the first column of H _{2 ×2;} h ₂ is a 2×1 vector with the components of the second column of H _{2 ×2.}

식(18a)

식(18b)

식(18c)

식(18d) 에르미트 행렬 R _{2 ×2}에서, r _{2 ,1}= r ^* _1,2이므로 r _{2 ,1}은 계산될 필요가 없다. 식 집합(11)이 R _{2 ×2}에 적용되어 행렬 V _{2 ×2}를 구할 수도 있다. V _{2 ×2}는 H _{2 ×2}의 우측 특이 벡터이기도 한 R _{2 ×2}의 고유 벡터들을 포함한다.Right singular vectors of H _{2 × 2} is a specific ^{_{H H 2 × 2 · H 2}} × 2 vector, can be calculated by using the eigenvalue decomposition described in Equation Set (11). The 2×2 Hermit matrix R _{2 ×2} is defined as R _{2 ×2} = H ^H _{2 ×2} · H _{2 ×2} , and the components of R _{2 ×2} are calculated based on the components of H _{2 ×2 as follows:} Can be:

Equation (18a)

Equation (18b)

Equation (18c)

Equation (18d) In the Hermit matrix R _{2 ×2} , r _{2 ,1} = r ^* _1,2, so r _{2 ,1} need not be calculated. Equation set (11) is applied to the R _{2 × 2} can obtain the matrix V _{2 × 2.} V _{2 ×2} contains the eigenvectors of R _{2 ×2, which} is also the right singular vector of H _{2 ×2.}

는

= H _2×2· H ^H _{2 ×2}로 정의되고,

의 성분들은 다음과 같이 H _{2 ×2}의 성분들을 기초로 계산될 수 있다:

식(19a)

식(19b)

식(19c)

식(19d) 식 집합(11)이

에 적용되어 행렬

를 구할 수도 있다.

는 H _{2 ×2}의 좌측 특이 벡터이기도 한

의 고유 벡터들을 포함한다.Left singular vectors of H _{2 × 2 2 × 2} is H, and H · ^H _{2 × 2} own vector may be computed by using the eigenvalue decomposition described in Equation Set (11). 2x2 Hermit matrix

Is

= H _{2 × 2} · H ^H _{2 × 2} is defined as,

The components of can be calculated based on the components of H _{2 ×2 as follows:}

Equation (19a)

Equation (19b)

Equation (19c)

Equation (19d) Equation set (11) is

Applied to the matrix

You can also get

Is also the left singular vector of H _{2 ×2}

Contains the eigenvectors of.

The iterative process described above for the eigenvalue decomposition of the N×N Hermit matrix R may be used for the singular value decomposition of any complex matrix H larger than 2×2. H has a dimension of R×T, where R is the number of rows and T is the number of columns. The iterative process for singular value decomposition (SVD) of H can be performed in several ways.

In the first SVD embodiment, the iterative process derives an approximation of the right singular vector of V and the scaled left singular vector of U · Σ. In this embodiment, T × T matrix V _i is initialized to an approximate value of V, and V ₀ = I. The R×T matrix W _i is an approximation of U · ∑ and initialized with W ₀ = H.

식(20a)

식(20b)

식(20c)

식(20d) 여기서 w _p 는 W _i 의 열(p)이고, w _q 는 W _i 의 열(q)이며, w _{l ,p} 는 W _i 에서 (ㅣ, p) 위치의 성분이다. 인덱스 p 및 q는 p ∈ {1, … , T}, q ∈ {1, … , T}이며, p ≠ q이다. 인덱스 p 및 q에 대한 값은 후술하는 바와 같이 다양한 방식으로 선택된다.In the first SVD embodiment, one iteration of Jacobian rotation to update matrices V _i and W _{i may be performed as follows.} First, a 2x2 Hermitian matrix M _pq is formed based on the current W _i. M _pq is a 2 × 2 sub-matrix of W _i ^H · W _i, in the ^{_{W i H · W i (p}} , p), (p, q), (q, p), (q, q) 4 position Contains dog ingredients. The components of M _pq can be calculated as follows:

Equation (20a)

Equation (20b)

Equation (20c)

Equation (20d) wherein w _p is a column (p) of W _i, w _q is the heat (q) of W _i, w _{l, p} is a component of the position (l, p) from the W _i. The indices p and q are p ∈ {1,… , T}, q ∈ {1,… , T}, and p ≠ q . The values for indices p and q are selected in various ways, as described below.

The eigenvalue decomposition of M _pq is performed as shown in _pq M 2 × 2 unitary matrix V _pq example expression set for example to save 11 of eigenvectors of. For this eigenvalue decomposition, R _{2 ×2} is replaced by M _pq and V _{2 ×2} is given as V _pq.

Then, a _TxT complex Jacobian rotation matrix T _pq is formed by the matrix V pq. T _pq is (p , p ), ( p , q ), ( q , p ) replaced by the (1, 1), (1, 2), (2, 1), (2, 2) components of V _pq , ( q , q ) is an identity matrix with four components of the position. T _pq has the form shown in equation (14).

The matrix V _i is updated as follows: V _{i +1} = V _i · T _pq equation (21)

Also, the matrix W _i is updated as follows: W _{i +1} = W _i · T _pq equation (22)

In the first SVD embodiment, the iterative process repeatedly zeros the non-diagonal components of W _i ^H · W _i without explicitly calculating H ^H · H. The indices p and q can be swept by stepping p from 1 to T-1 and q from p+1 to T. Alternatively, | The values of p and q with the largest W _p ^H · W _q | can be selected for each iteration. The iterative process is performed until the termination condition is met, and the termination condition may be a predetermined number of sweeps, a predetermined number of repetitions, and an error criterion satisfaction.

= ( W _i ^H · W _i )^1/2로 주어질 수 있다. 비-제곱 대각 행렬에서,

의 0이 아닌 대각 성분들은 W _i ^H · W _i 의 대각 성분들의 제곱근으로 주어진다. U 의 최종 해는

= W _i ·

^-1로 주어질 수 있다.At the end of the iterative process, the final V _i is a good approximation of V , and the final W _i is a good approximation of U · Σ. When converged, W _i ^H · W _i = ∑ ^T · ∑ and U = W _i · ∑ ^-1 , where " ^T " represents the transpose. In the squared diagonal matrix, the final solution of ∑ is

= ( W _i ^H · W _i ) can be given as ^1/2. In a non-squared diagonal matrix,

The nonzero diagonal components of W _i ^H · W _i are given as the square root of the diagonal components of W i H · W i. The final solution of U is

= W _i ·

Can be given as ^-1.

2 shows an iterative process 200 for performing singular value decomposition of any complex matrix H greater than 2×2 using Jacobian rotation according to the first SVD embodiment. The matrices V _i and W _i are initialized with V ₀ = I and W ₀ = H and the index i is initialized with i = 1 (block 210).

At iteration i , values for indices p and q are selected in a predetermined or deterministic manner (block 212). Then, a 2x2 matrix M _pq is formed with the four components of the matrix W _{i at} the positions determined by the indices p and q as indicated by the equation set 20 (block 214). Then, for the M _pq as shown in example equation set (11), eigenvalue decomposition is performed to obtain a 2 × 2 matrix V _{pq pq} M of specific vectors (block 216). As shown in equation (14), a _TxT complex Jacobian rotation matrix T _pq is formed based on the matrix V pq (block 218). The matrix V _i is then updated based on T _pq as shown in equation (21) (block 220). Further, the matrix W _i is updated based on T _pq as represented by equation (22) (block 222).

및

를 구한다(블록 228). H 의 우측 특이 벡터 행렬 V 의 근사값으로서 V _i 가 제공되고, H 의 좌측 특이 벡터 행렬 U 의 근사값으로서

가 제공되며, H 의 특이값 행렬 ∑ 의 근사값으로서

가 제공된다(블록 230).A determination is made as to whether to end singular value decomposition of H (block 224). The termination criterion may be based on the number of iterations or sweeps already performed, an error criterion, and the like. If the answer to block 224 is no, index i is incremented (block 226), and the process returns to block 212 for the next iteration. Otherwise, when the end is reached, post-processing is performed on W _i

And

Find (block 228). As an approximation of the right singular vectors of the matrix V H being provided with a V _i, a of the left singular vectors of H matrix U approximation

Is provided, and as an approximation of the singular value matrix ∑ of H

Is provided (block 230).

It left singular vectors of H are Claim 1 SVD embodiment performs, loosen the scaled left singular vector H · V = U · Σ can be found by the following normalization. Left singular vectors of H may be obtained by performing the iteration process for the eigenvalue decomposition of H · H ^H.

이고, 여기서 및

는 H ^H _2×2의 2개의 열이고 H _{2 ×2}의 행들의 복소 켤레이기도 하다. H _{2 ×2}의 좌측 특이 벡터들은 H ^H _{2 ×2}의 우측 특이 벡터이기도 하다. H _{2 ×2}의 우측 특이 벡터들은 식 집합(18)에 대해 상술한 바와 같이 야코비 회전을 이용하여 계산될 수 있다. H _{2 ×2}의 좌측 특이 벡터들은 식 집합(19)에 대해 상술한 바와 같이 야코비 회전을 이용하여 H ^H _{2 ×2}의 우측 특이 벡터들을 계산함으로써 구해질 수 있다.In the second SVD embodiment, the iterative process directly derives an approximation of the right singular vector of V and the left singular vector of U. In this SVD example, the Jacobian rotation is applied to both sides to obtain the left and right singular vectors at the same time. For any complex 2x2 matrix H _{2 ×2} = [ h ₁ h ₂ ], the conjugate transpose matrix of this matrix is

Is, where And

Two columns of H ^H _{2 × 2,} and is also the line of the complex conjugate of H _{2 × 2.} Left singular vectors of H _{2 × 2} are also the right singular vectors of ^H H _{2 × 2.} The right singular vectors of H _{2 ×2} can be calculated using Jacobian rotation as described above for equation set (18). The left side of the H _{2 × 2} specific vectors can be found by calculating the right singular vector of the I using the Jacobi rotation H ^H _{2 × 2} as described above for equation set (19).

The second SVD for example, T × T matrix V _i is an approximation of V is initialized to V ₀ = I. R × R matrix U _i is initialized to an approximate value of U ₀ = I U. The RxT matrix D _i is an approximation of ∑ and initialized with D ₀ = H.

식(23a)

식(23b)

식(23c)

식(23d) 여기서 d _{p 1}은 D _i 의 열(p ₁)이고, d _{q 1}은 D _i 의 열(q ₁)이며, d _{l , p 1}은 D _i 에서 (ㅣ, p ₁) 위치의 성분이다. 인덱스 p ₁ 및 q ₁은 p ₁ ∈ {1, … , T}, q ₁ ∈ {1, … , T}이며, p ₁ ≠ q ₁이다. 인덱스 p ₁ 및 q ₁은 후술하는 바와 같이 다양한 방식으로 선택될 수 있다.For the second SVD embodiment, one iteration of Jacobian rotation for updating matrices V _i , U _i and D _{i may be performed as follows.} First, a 2x2 Hermit matrix X _{p 1 q 1} is formed based on the current D _i. X _{p 1 q 1} is D _i ^H · D and _i 2 × 2 sub-matrix _{of, (p 1, p 1)} , (p 1, q 1), (q 1, p 1) in the D _i ^H · D _i , ( q ₁ , q ₁ ) contains four components of the position. The _{four components of X p 1 q 1} can be calculated as follows:

Equation (23a)

Equation (23b)

Equation (23c)

Equation (23d) where d _{p 1} is the column of D _i (p ₁ ), d _{q 1} is the column of D _i (q ₁ ), and d _{l , p 1} is the position of ( ㅣ , p ₁ ) in D _i It is an ingredient. The indices p ₁ and q ₁ are p ₁ ∈ {1,… , T}, q ₁ ∈ {1,… , T}, and p ₁ ≠ q ₁ . The indices p ₁ and q ₁ may be selected in various ways as described later.

The eigenvalue decomposition of X _p X _{p 1, q 1} is performed as shown in the example ₁ to obtain the 2 × 2 matrix V _{p 1 q 1} of eigenvectors example expression set of _{q 1} (11). For this eigenvalue decomposition, R _{2 ×2} is replaced by X _{p 1 q 1} , and V _{2 × 2} is given as V _{p 1 q 1} . Then, the T×T complex Jacobian rotation matrix T _{p 1 q 1} is formed by the matrix V _{p 1 q 1} , (p ₁ , p ₁ ), ( p ₁ , q ₁ ), ( q ₁ , p ₁ ), ( q ₁ , q ₁ ) contains four components of V _{p 1 q 1.} T _{p 1 q 1} has the form shown in equation (14).

식(24a)

식(24b)

식(24c)

식(24d) 여기서

는 D _i 의 행(p ₂)이고,

는 D _i 의 행(q ₂)이며, d _{p 2 , l} 은 D _i 에서 (p ₂, ㅣ) 위치의 성분이다. 인덱스 p ₂ 및 q ₂는 p ₂ ∈ {1, … , R}, q ₂ ∈ {1, … , R}이며, p ₂ ≠ q ₂이다. 인덱스 p ₂ 및 q ₂는 후술하는 바와 같이 다양한 방식으로 선택될 수도 있다.Another 2×2 Hermit matrix Y _{p 2 q 2 is} also formed based on the current D _i. Y _{p 2 q 2} is 2 × 2 sub-matrix of D _i · D _i ^H, D _i · In ^{_{D i H (p 2, p}} 2), (p 2, q 2), (q 2, p 2) , ( q ₂ , q ₂ ) contains the component of the position. The components of Y _{p 2 q 2} can be calculated as follows:

Equation (24a)

Equation (24b)

Equation (24c)

Equation (24d) where

Is the row of D _i (p ₂ ),

Is a row (q _2), the D _i, d _{p 2, l} is in the D _i (p _2, l) is the component of the position. The indices p ₂ and q ₂ are p ₂ ∈ {1,… , R}, q ₂ ∈ {1,… , R}, and p ₂ ≠ q ₂ . The indices p ₂ and q ₂ may be selected in various ways as described later.

Y _{p 2} The eigenvalue decomposition of Y _{p 2 q 2} is performed as shown in the examples to obtain the 2 × 2 matrix U _{p 2 q 2} of eigenvectors example expression set of _{q 2} (11). For this eigenvalue decomposition, R _{2 ×2} is replaced by Y _{p 2 q 2} and V _{2 × 2} is given as U _{p 2 q 2} . Then, the R×R complex Jacobian rotation matrix S _{p 2 q 2} is formed by the matrix U _{p 2 q 2} , (p ₂ , p ₂ ), ( p ₂ , q ₂ ), ( q ₂ , p ₂ ), ( q ₂ , q ₂ ) contains four components of U _{p 2 q 2.} S _{p 2 q 2} has the form shown in equation (14).

The matrix V _i is updated as follows: V _{i +1} = V _i · T _{p 1 q 1} Equation (25)

The matrix U _i is updated as follows: U _{i +1} = U _i · S _{p 2 q 2} Equation (26)

The matrix D _i is updated as follows: D _{i +1} = S ^H _{p 2 q 2} · D _i · T _{p 1 q 1} Equation (27)

가 가장 큰 p ₂ 및 q ₂의 값이 선택될 수 있다. 종료 조건과 만날 때 까지 반복 프로세스가 수행되며, 종료 조건은 미리 결정된 회수의 스위프, 미리 결정된 회수의 반복, 오차 기준 만족 등일 수 있다.The second for the SVD embodiment, the iterative process is the alternative (1) Specific having an index p ₁ and q ₁ In H ^H · H - In the Jacobi rotation and (2) H · H ^H them to zero diagonal elements A Jacobian rotation with non-diagonal components with indices p ₂ and q _{2 equal to zero is obtained.} Index p ₁ and q ₁ can be swept by the stepping from one value for each p ₁ T-1 to the stepping p _1, q ₁ and to the p ₁ from +1 to T. The index p ₂ and q ₂ may be a stepping p ₂ p ₂ each from 1 to the value R-1, and sweep by stepping up the q ₂ from p ₂ R +1. As an example, for the squared matrix H , the indices may be set to p ₁ = p ₂ and q ₁ = q _2. As another example, for a squared or non-squared matrix H , a set of p ₁ and q ₁ is selected, then a set of p ₂ and q ₂ is selected, a new set of p ₁ and q ₁ is selected, and then p ₂ and q the new values are selected alternately for such a new set ₂ is selected, the index p ₁ and q ₁ and the index p ₂ and q _2. Alternatively, at every iteration | The values of p ₁ and q ₁ with the largest d ^H _{p 1} · d _{p 1} | can be selected,

The largest values of p ₂ and q ₂ can be selected. The iterative process is performed until the termination condition is met, and the termination condition may be a predetermined number of sweeps, a predetermined number of repetitions, and an error criterion satisfaction.

의 양호한 근사값이고, 최종 U _i 가 U 의 양호한 근사값이고, 최종 D _i 가

의 양호한 근사값이며,

및

는 각각 V 및 ∑ 의 회전된 버전일 수 있다. 상술한 계산은 좌측 및 우측 특이 벡터 해를 충분히 포함하지 않아 최종 D _i 의 대각 성분은 양의 실수값이다. 최종 D _i 의 성분들은 크기가 H 의 특이값과 동일한 복소값일 수도 있다. V _i 및 D _i 는 다음과 같이 회전되지 않을 수도 있다:

식(28a)

식(28b) 여기서 P 는 단위 크기 및 D _i 의 해당 대각 성분들의 음의 위상인 위상을 갖는 대각 성분들을 갖는 T×T 대각 행렬이다.

및

는 각각 ∑ 및 V 의 최종 근사값이다.At the end of the iterative process, the final V _i is

Is a good approximation of, the final U _i is a good approximation of U , and the final D _i is

Is a good approximation of

And

May be the rotated versions of V and Σ , respectively. Since the above calculation does not sufficiently include the left and right singular vector solutions, the diagonal component of the final D _i is a positive real value. Components of the final D _i may be complex values whose magnitude is equal to the singular value of H. V _i and D _i may not be rotated as follows:

Equation (28a)

Equation (28b) where P is a TxT diagonal matrix with diagonal components having a unit size and a phase that is the negative phase of the corresponding diagonal components of D _i.

And

Is the final approximation of ∑ and V, respectively.

3 shows an iterative process 300 for performing singular value decomposition of an arbitrary complex matrix H greater than 2×2 using Jacobian rotation according to a second SVD embodiment. The matrices V _i , U _i and D _i are initialized to V ₀ = I , U ₀ = I , D ₀ = H and the index i is initialized to i = 1 (block 310).

In iteration ( i ), values for indices p ₁ , q ₁ , p ₂ , q ₂ are selected in a predetermined or deterministic manner (block 312). As indicated by equation set (23), a 2x2 matrix X _{p 1 q 1} is formed from the four components of the matrix D _{i at} the positions determined by the indices p ₁ and q ₁ (block 314). Then, for example, as shown in equation set (11) the eigenvalue decomposition of X _{p 1 q 1} is performed obtain a 2 × 2 matrix V _{p 1 q 1} of eigenvectors of the X _{p 1 q 1} (block 316 ). Then, a TxT complex Jacobian rotation matrix T _{p 1 q 1} is formed based on the matrix V _{p 1 q 1} (block 318). Further, as shown in the equation set 24, a 2x2 matrix Y _{p 2 q 2} is formed from the four components of the matrix D _{i at} the positions determined by the indices p ₂ and q ₂ (block 324). Then, for example, eigenvalue decomposition of Y _{p 2 q 2} as shown in equation set (11) is performed to Y _{p 2 q 2} of obtains a 2 × 2 matrix U _{p 2 q 2} of eigenvectors (block 326 ). Then, the RxR complex Jacobian rotation matrix S _{p 2 q 2} is formed based on the matrix U _{p 2 q 2} (block 328).

The matrix V _i is then updated based on T _{p 1 q 1} as shown in equation (25) (block 330). The matrix U _i is updated based on S _{p 2 q 2} as shown in equation (26) (block 332). The matrix D _i is updated based on T _{p 1 q 1} and S _{p 2 q 2} as shown in equation (27) (block 334).

및

를 구한다(블록 340). V 의 근사값으로서 행렬

가 제공되고, U 의 근사값으로서 행렬 U _i 가 제공되며, ∑ 의 근사값으로서

가 제공된다(블록 342).A determination is made as to whether to end singular value decomposition of H (block 336). The termination criterion may be based on the number of iterations or sweeps already performed, an error criterion, and the like. If the answer to block 336 is no, then index i is incremented (block 338), and the process returns to block 312 for the next iteration. Otherwise, when the end is reached, post-processing is performed on D _i and V _i

And

Find (block 340). Matrix as an approximation of V

The service is, and the matrix U _i provided as the approximate value of U, as an approximate value of Σ

Is provided (block 342).

의 좌측 특이 벡터는 가장 큰 특이값에서부터 가장 작은 특이값으로 배열된다.In both the first and second embodiments , the eigenvectors of V _pq (for the first SVD embodiment) and the eigenvectors of V _{p 1 q 1} and U _{p 2 q 2} (for the second SVD embodiment) for each iteration Are arranged, so the right singular vector of the final V _i and the final U _i or

The left singular vector of is arranged from the largest singular value to the smallest singular value.

In a MIMO system having multiple subbands, an iterative process is performed for each channel response matrix H ( k ) to obtain matrices V _i ( k ), U _i ( k ) and D _i ( k ), each of which corresponds to H right singular vector matrix V for a (k) (k), is an approximation of diagonal matrix Σ (k) of left singular vector matrix U (k) and the singular values. Starting at one end of the system bandwidth and across to the other end of the system bandwidth, an iterative process can be performed for one subband at a time. When the first SVD embodiment, for the first sub-band of each sub-band (k) except, prior to sub-band (k -1) to the final V _i (k -1), the current sub-band (k) obtained for the It can be used as an initial solution for V ₀ ( k ) = V _i ( k -1) and W ₀ ( k ) = H ( k ) · V ₀ ( k ). If SVD second embodiment, first, for each sub-band (k) except the first sub-band, the final solution obtained for a previous sub-band _{(k -1) V i (k} -1) and U _i (k -1) Can be used as an initial solution to the current subband ( k ), so V ₀ ( k ) = V _i ( k -1), U ₀ ( k ) = U _i ( k -1), D ₀ ( k ) = U ₀ ^H ( k ) · H ( k ) · V ₀ ( k ). In both embodiments, the iterative process acts on the initial solution for the subband k until an end condition is met for the subband. This concept may be used over time or both over frequency and time, as described above.

4 shows a process 400 of decomposing a matrix using Jacobian rotation. Multiple iterations of Jacobian rotation are performed on the first matrix of complex values using multiple Jacobian rotation matrices (block 412). The first matrix may be a channel response matrix H , a correlation matrix of H , R , or some other matrix. The Jacobian rotation matrix may be T _pq , T _{p 1 q 1} , S _{p 2 q 2} , and/or any other matrix. At each iteration, a partial matrix is formed and decomposed based on the first matrix to obtain an eigenvector for this partial matrix, and a Jacobian rotation matrix is formed using the eigenvectors, which can be used to update the first matrix. A second matrix of complex values is derived based on a number of Jacobian rotation matrices (block 414). The second matrix includes orthogonal vectors, and may be a right singular vector of H or a matrix V _i of eigenvectors of R.

가 유도될 수 있으며, 제 3 행렬 W _i 를 기초로 특이값들의 행렬

가 유도될 수도 있다(블록 422). 제 2 SVD 실시예에 기반한 특이값 분해의 경우, 다수의 야코비 회전 행렬들을 기초로 직교 벡터들을 갖는 제 3 행렬 U _i 및 특이값들의 행렬

가 유도될 수도 있다(블록 424). For eigenvalue decomposition, a third matrix D _i of eigenvalues may be derived based on a number of Jacobian rotation matrices, as determined at block 416 (block 420). In the case of singular value decomposition based on the first SVD embodiment or method, a third matrix W _i of complex values may be derived based on a plurality of Jacobian rotation matrices, and orthogonal vectors based on the third matrix W _i 4th procession

Can be derived, and a matrix of singular values based on the third matrix W _i

May be derived (block 422). In the case of singular value decomposition based on the second SVD embodiment, a third matrix U _i having orthogonal vectors based on a plurality of Jacobian rotation matrices and a matrix of singular values

May be derived (block 424).

5 shows an apparatus 500 for decomposing a matrix using Jacobian rotation. The apparatus 500 includes means for performing multiple iterations of Jacobian rotation on a first matrix of complex values with a plurality of Jacobian rotation matrices (block 512) and of complex values based on the plurality of Jacobian rotation matrices. Means for deriving a second matrix V _i (block 514).

를, 그리고 제 3 행렬을 기초로 특이값들의 행렬

를 유도하는 수단(블록 522)을 더 포함한다. 제 2 SVD 실시예에 기반한 특이값 분해의 경우, 장치(500)는 다수의 야코비 회전 행렬들을 기초로 직교 벡터들을 갖는 제 3 행렬 U _i 및 특이값들의 행렬

를 유도하는 수단(블록 524)을 더 포함한다. 3. 시스템 For eigenvalue decomposition, apparatus 500 further comprises means (block 520) for deriving a third matrix D _i of eigenvalues based on the plurality of Jacobian rotation matrices. In the case of singular value decomposition based on the first SVD embodiment, the apparatus 500 provides a third matrix W _i of complex values based on a plurality of Jacobian rotation matrices, and a fourth matrix having orthogonal vectors based on the third matrix.

And the matrix of singular values based on the third matrix

And means for deriving (block 522). In the case of singular value decomposition based on the second SVD embodiment, the apparatus 500 includes a third matrix U _i having orthogonal vectors based on a plurality of Jacobian rotation matrices and a matrix of singular values.

And means for deriving (block 524). 3. System

6 shows a block diagram of an embodiment of an access point 610 and a user terminal 650 in a MIMO system 600. The access point 610 includes a plurality of antennas _{(N ap} ) that can be used for data transmission and reception. The user terminal 650 is provided with the antenna of the multiple (N _ut) which can be used for data transmission and reception. For simplicity, the following description MIMO system 600, the time-division duplex downlink channel response matrix H _dn (k) for the communication (TDD) in use, and each sub-band (k) is the uplink for that sub-band It is assumed that it is relative to the channel response matrix H _up ( k ) or that H _dn ( k ) = H ( k ) and H _up ( k ) = H ^T ( k ).

On the downlink, a transmit (TX) data processor 614 at access point 610 receives traffic data from data source 612 and other data from controller/processor 630. TX data processor 614 formats, encodes, interleaves, and modulates the received data to generate data symbols, which are modulation symbols for the data. A TX spatial processor 620 multiplexes the received data symbols with the pilot symbols, if applicable, to a processing between the ball eigenvectors or specific right vector N _ap transmitters (TMTR) (622a ~ 622ap) N ap It provides three transmission symbol streams. Each transmitter 622 processes its respective transmission symbol stream to generate a downlink modulated signal. _{N ap} downlink modulated signals from the transmitters 622a to 622ap are transmitted from the antennas 624a to 624ap, respectively.

Receiving the N _ut antennas (652a ~ 652ut) is transmitted downlink modulated symbols from the user terminal 650, and each antenna 652 provides a received signal to a respective receiver (RCVR) (654). Each receiver 654 performs processing complementary to the processing performed by the antenna 622 to provide received symbols. The receive (RX) spatial processor 660 performs spatial matching filtering on received symbols from all receivers 654a to 654ut to provide the detected data symbols, and these data symbols are transmitted by the access point 610. It is an estimate of the data symbol. The RX data processor 670 further processes the detected data symbols (e.g., symbol demapping, deinterleaving and decoding) to provide the decoded data to the data sink 672 and/or the controller/processor 680. .

를 제공한다. 프로세서(678 및/또는 680)는 여기서 설명한 기술을 이용하여 각 행렬

를 분해하여, 다운링크 채널 응답 행렬 H (k)에 대한 V (k) 및 ∑ (k)의 추정치인

및

를 구할 수 있다. 프로세서(678 및/또는 680)는 표 1에 나타낸 바와 같이,

를 기초로 관심 있는 각 부대역에 대한 다운링크 공간 필터 행렬 M _dn(k)를 유도할 수 있다. 프로세서(680)는 다운링크 매칭 필터링을 위해 RX 공간 프로세서(660)에 M _dn(k)를 그리고/또는 업링크 공간 처리를 위해 TX 공간 프로세서(690)에

를 제공할 수 있다.The channel processor 678 processes the received pilot symbols to provide an estimate of the downlink channel response for each subband of interest.

Provides. Processors 678 and/or 680 are each matrix

Decomposing the downlink channel response matrix H ( k ), which is the estimate of V ( k ) and ∑ ( k)

And

Can be obtained. Processors 678 and/or 680, as shown in Table 1,

The downlink spatial filter matrix M _dn ( k ) for each subband of interest can be derived based on. _{The processor 680 draws} M dn ( k ) to the RX spatial processor 660 for downlink matching filtering and/or to the TX spatial processor 690 for uplink spatial processing.

Can provide.

로 추가 공간 처리된다. TX 공간 프로세서(690)로부터의 송신 심벌들은 송신기(654a~654ut)에 의해 추가 처리되어 N_ut개의 업링크 변조 신호를 생성하고, 이 업링크 변조 신호는 안테나(652a~652ut)에 의해 전송된다.The processing for the uplink may be the same as or different from the processing for the downlink. Traffic data from data source 686 and other data from controller/processor 680 are processed (e.g., encoded, interleaved and modulated) by TX data processor 688 to be multiplexed with pilot symbols, and TX spatial For each subband of interest by processor 690

It is treated with additional space. The transmission symbols from TX spatial processor 690 are further processed by transmitters 654a-654ut _{to generate N ut} uplink modulated signals, which are transmitted by antennas 652a-652ut.

At the access point 610, uplink modulated signals are received by antennas 624a-624ap and processed by receivers 622a-622ap to generate received symbols for uplink transmission. The RX spatial processor 640 provides the detected data symbols by performing spatial matching filtering on the received data symbols. RX data processor 642 processes the detected data symbols and provides decoded data to data sink 644 and/or controller/processor 630.

를 분해하여

를 구할 수 있다. 프로세서(628 및/또는 630)는 또한

를 기초로 관심 있는 각 부대역에 대한 업링크 공간 필터 행렬 M _up(k)를 유도할 수도 있다. 프로세서(630)는 다운링크 매칭 필터링을 위해 RX 공간 프로세서(640)에 M _up(k)를 그리고/또는 다운링크 공간 처리를 위해 TX 공간 프로세서(620)에

를 제공할 수 있다.The channel processor 628 processes the received pilot symbols and provides an estimate of H ^H ( k ) or U ( k ) for each subband of interest depending on how the uplink pilot is transmitted. Processors 628 and/or 630 can use the techniques described herein to

By disassembling

Can be obtained. Processors 628 and/or 630 are also

It is also possible to derive an uplink spatial filter matrix M _up ( k ) for each subband of interest based on. The processor 630 transmits M _up ( k ) to the RX spatial processor 640 for downlink matching filtering and/or to the TX spatial processor 620 for downlink spatial processing.

Can provide.

The controllers/processors 630 and 680 control operations at the access point 610 and the user terminal 650, respectively. The memories 632 and 682 store data and program codes for the access point 610 and the user terminal 650, respectively. Processors 628, 630, 678, 680 and/or other processors may perform eigenvalue decomposition and/or singular value decomposition of the channel response matrix.

The matrix decomposition technique described herein can be implemented by various means. For example, these technologies may be implemented in hardware, firmware, software, or a combination thereof. For hardware implementations, the processing unit used to perform the matrix decomposition is one or more application specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processing devices (DSPDs), programmable logic devices (PLDs), field programmable It may be implemented in a gate array (FPGA), a processor designed to perform the functions described herein, a controller, a microcontroller, a microprocessor, other units, or a combination thereof.

In the case of firmware and/or software implementation, the matrix decomposition technique may be implemented with modules (eg, procedures, functions, etc.) that perform the functions described herein. The software code may be stored in a memory (eg, memory 632 or 682 of FIG. 6) and executed by a processor (eg, processor 630 or 680 ). The memory unit may be implemented inside the processor or outside the processor.

Headings are included here to help locate references and specific sections. These headings do not limit the scope of the concepts described therein, and these concepts may have applicability in other sections throughout the entire specification.

The above description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the present invention. Various modifications to these embodiments are readily apparent to those skilled in the art, and the general principles defined herein can be applied to other embodiments without departing from the spirit or scope of the present invention. Accordingly, the present invention is not limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

1 shows a process for performing eigenvalue decomposition using Jacobian rotation.

2 shows a process of performing singular value decomposition using Jacobian rotation according to the first SVD embodiment.

3 shows a process of performing singular value decomposition using Jacobi rotation according to the second SVD embodiment.

4 shows a process for decomposing a matrix using Jacobian rotation.

5 shows an apparatus for decomposing a matrix using Jacobian rotation.

6 shows a block diagram of an access point and a user terminal.

Claims (12)

An apparatus for wireless communication, Perform multiple iterations of Jacobian rotation on the first matrix using Jacobi rotation matrices of complex values to zero out non-diagonal components in the first matrix of complex values, Derive a second matrix of complex values based on the Jacobian rotation matrices, the second matrix comprising orthogonal vectors; and A processor configured to transmit data via a plurality of antennas using the orthogonal vectors; and A memory coupled to the processor, Device for wireless communication. The method of claim 1, The processor is configured to transmit data via the plurality of antennas using right singular vectors of the second matrix, Device for wireless communication. An apparatus for wireless communication, Perform multiple iterations of Jacobian rotation on the first matrix using Jacobe rotation matrices of complex values to zero out non-diagonal components in the first matrix of complex values, Derive a second matrix of complex values based on the Jacobian rotation matrices, the second matrix comprising orthogonal vectors; and A processor configured to receive data via a plurality of antennas using the orthogonal vectors, and A memory coupled to the processor, Device for wireless communication. The method of claim 3, The processor is configured to receive data via the plurality of antennas using left singular vectors of the second matrix, Device for wireless communication. An apparatus for wireless communication, Means for performing multiple iterations of Jacobian rotation on the first matrix using Jacobi rotation matrices of complex values to zero out non-diagonal components in the first matrix of complex values ; Means for deriving a second matrix of complex values based on the Jacobian rotation matrices, the second matrix comprising orthogonal vectors; And Means for transmitting data over a plurality of antennas using the orthogonal vectors; Device for wireless communication. The method of claim 5, The means for transmitting is configured to transmit data via the plurality of antennas using right singular vectors of the second matrix, Device for wireless communication. An apparatus for wireless communication, Means for performing multiple iterations of Jacobian rotation on the first matrix using Jacobi rotation matrices of complex values to zero out non-diagonal components in the first matrix of complex values ; Means for deriving a second matrix of complex values based on the Jacobian rotation matrices, the second matrix comprising orthogonal vectors; And Means for receiving data via a plurality of antennas using the orthogonal vectors; Device for wireless communication. The method of claim 7, wherein The means for receiving is configured to receive data via the plurality of antennas using right singular vectors of the second matrix, Device for wireless communication. A method for wireless communication, Performing multiple iterations of Jacobian rotation on the first matrix using Jacobi rotation matrices of the complex values to zero out non-diagonal components in the first matrix of complex values; Deriving a second matrix of complex values based on the Jacobian rotation matrices, the second matrix comprising orthogonal vectors; And Transmitting data via a plurality of antennas using the orthogonal vectors; Method for wireless communication. The method of claim 9, The transmitting step includes transmitting data over the plurality of antennas using right singular vectors of the second matrix, Method for wireless communication. A method for wireless communication, Performing multiple iterations of Jacobian rotation on the first matrix using Jacobi rotation matrices of complex values to zero out non-diagonal components in a first matrix of complex values; Deriving a second matrix of complex values based on the Jacobian rotation matrices, the second matrix comprising orthogonal vectors; And Receiving data via a plurality of antennas using the orthogonal vectors; Method for wireless communication. The method of claim 11, The receiving step includes receiving data via the plurality of antennas using right singular vectors of the second matrix, Method for wireless communication.

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