EIGENVALUE DECOMPOSITION AND SINGULAR VALUE DECOMPOSITION OF MATRICES USING JACOBI ROTATION
I. Claim of Priority under 35 U.S.C. §119
[0001] The present Application for Patent claims priority to Provisional Application
Serial No. 60/628,324, entitled "Eigenvalue Decomposition and Singular Value Decomposition of Matrices Using Jacobi Rotation," filed November 15, 2004, assigned to the assignee hereof, and hereby expressly incorporated by reference herein.
BACKGROUND
I. Field
[0002] The present invention relates generally to communication, and more specifically to techniques for decomposing matrices.
II. Background
[0003] A multiple-input multiple-output (MIMO) communication system employs multiple (T) transmit antennas at a transmitting station and multiple (R) receive antennas at a receiving station for data transmission. A MIMO channel formed by the T transmit antennas and the R receive antennas may be decomposed into S spatial channels, where S < min {T, R} . The S spatial channels may be used to transmit data in a manner to achieve higher overall throughput and/or greater reliability.
[0004] A MIMO channel response may be characterized by an R xT channel response matrix H , which contains complex channel gains for all of the different pairs of transmit and receive antennas. The channel response matrix H may be diagonalized to obtain S eigenmodes, which may be viewed as orthogonal spatial channels of the MEvIO channel. Improved performance may be achieved by transmitting data on the eigenmodes of the MIMO channel.
[0005] The channel response matrix H may be diagonalized by performing either singular value decomposition of H or eigenvalue decomposition of a correlation matrix of H . The singular value decomposition provides left and right singular vectors, and the eigenvalue decomposition provides eigenvectors. The transmitting station uses the right singular vectors or the eigenvectors to transmit data on the S eigenmodes. The receiving station uses the left singular vectors or the eigenvectors to receive data transmitted on the S eigenmodes.
[0006] Eigenvalue decomposition and singular value decomposition are very computationally intensive. There is therefore a need in the art for techniques to efficiently decompose matrices.
SUMMARY
[0007] Techniques for efficiently decomposing matrices using Jacobi rotation are described herein. These techniques may be used for eigenvalue decomposition of a Hermitian matrix of complex values to obtain a matrix of eigenvectors and a matrix of eigenvalues for the Hermitian matrix. The techniques may also be used for singular value decomposition of an arbitrary matrix of complex values to obtain a matrix of left singular vectors, a matrix of right singular vectors, and a matrix of singular values for the arbitrary matrix.
[0008] In an embodiment, 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. The first matrix may be a channel response matrix a correlation matrix of which is or some other 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 is derived based on the Jacobi rotation matrices. The second matrix contains orthogonal vectors and may be a matrix of right singular vectors o r eigenvectors of
[0009] For eigenvalue decomposition, a third matri f eigenvalues may be derived based on the Jacobi rotation matrices. For singular value decomposition (SVD) based on a first SVD embodiment, a third matrix of complex values may be derived based on the Jacobi rotation matrices, a fourth matri ith orthogonal vectors may be derived based on the third matrix , and a matrix of singular values may also be derived based on the third matrix For singular value decomposition based on a second SVD embodiment, a third matrix with orthogonal vectors and a matri of singular values may be derived based on the Jacobi rotation matrices. [0010] Various aspects and embodiments of the invention are described in further detail below.
BRIEF DESCRIPTION OF THE DRAWINGS
[0011] FIG. 1 shows a process for performing eigenvalue decomposition using Jacobi rotation. [0012] FIG. 2 shows a process for performing singular value decomposition using
Jacobi rotation in accordance with the first SVD embodiment. [0013] FIG. 3 shows a process for performing singular value decomposition using
Jacobi rotation in accordance with the second SVD embodiment. [0014] FIG. 4 shows a process for decomposing a matrix using Jacobi rotation.
[0015] FIG. 5 shows an apparatus for decomposing a matrix using Jacobi rotation.
[0016] FIG. 6 shows a block diagram of an access point and a user terminal.
DETAILED DESCRIPTION
[0017] The word "exemplary" is used herein to mean "serving as an example, instance, or illustration." Any embodiment described herein as "exemplary" is not necessarily to be construed as preferred or advantageous over other embodiments.
[0018] The matrix decomposition techniques described herein may be used for various communication systems such as a single-carrier communication system with a single frequency subband, a multi-carrier communication system with multiple subbands, a single-carrier frequency division multiple access (SC-FDMA) system with multiple subbands, and other communication systems. Multiple subbands may be obtained with orthogonal frequency division multiplexing (OFDM), some other modulation techniques, or some other construct. OFDM partitions the overall system bandwidth into multiple (K) orthogonal subbands, which are also called tones, subcarriers, bins, and so on. With OFDM, each subband is associated with a respective subcarrier that may be modulated with data. An SC-FDMA system may utilize interleaved FDMA (IFDMA) to transmit on subbands that are distributed across the system bandwidth, localized FDMA (LFDMA) to transmit on a block of adjacent subbands, or enhanced FDMA (EFDMA) to transmit on multiple blocks of adjacent subbands. In general, modulation symbols are sent in the frequency domain with OFDM and in the time domain with SC-FDMA. For clarity, much of the following description is for a MIMO system with a single subband. [0019] A MIMO channel formed by multiple (T) transmit antennas and multiple (R) receive antennas may be characterized by an R xT channel response matrix , which may be given as:
where entry htJ , for ι = l, ..., R and j = l, ...,T , denotes the coupling or complex channel gain between transmit antenna j and receive antenna i. [0020] The channel response matri may be diagonalized to obtain multiple (S) eigenmodes o where S ≤ min {T, R} . The diagonalization may be achieved by, for example, performing either singular value decomposition of or eigenvalue decomposition of a correlation matrix o [0021] The eigenvalue decomposition may be expressed as:
where is a TxT correlation matrix of s a TxT unitary matrix whose co umns are eigenvectors of s a T x T diagonal matrix of eigenvalues o ; and "H" denotes a conjugate transpose.
The unitary matrix is characterized by the property where s the identity matrix. The columns of the unitary matrix are orthogonal to one another, and each column has unit power. The diagonal matri contains possible non-zero values along the diagonal and zeros elsewhere. The diagonal elements o are eigenvalues of . These eigenvalues are denoted as and represent the power gains for the S eigenmodes. is a Hermitian matrix whose off-diagonal elements have the following property: where " * " denotes a complex conjugate. [0022] The singular value decomposition may be expressed as: where s an R xR unitary matrix of left singular vectors of s an R xT diagonal matrix of singular values o and s a T x T unitary matrix of right singular vectors o
ach contain orthogonal vectors. Equations (2) and (3) indicate that the right singular vectors o are also the eigenvectors of The diagonal elements of Σ are the singular values of These singular values are denoted as and represent the channel gains for the S eigenmodes. The singular values o are also the square roots of the eigenvalues o , so tha
[0023] A transmitting station may use the right singular vectors in to transmit data on the eigenmodes of Transmitting data on eigenmodes typically provides better performance than simply transmitting data from the T transmit antennas without any spatial processing. A receiving station may use the left singular vectors in r the eigenvectors i to receive the data transmission sent on the eigenmodes of 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 TxI vector with up to S data symbols to be transmitted, x is a TxI vector with T transmit symbols to be sent from the T transmit antennas, r is an R xI vector with R received symbols obtained from the R receive antennas, n is an R xI noise vector, and s is a TxI vector with up to S detected data symbols, which are estimates of the data symbols in s .
Table 1
[0024] Eigenvalue decomposition and singular value decomposition of a complex matrix may be performed with an iterative process that uses Jacobi rotation, which is also commonly referred to as Jacobi method and Jacobi transformation. The Jacobi rotation zeros out a pair of off-diagonal elements of the complex matrix by performing a plane rotation on the matrix. For a 2x2 complex Hermitian matrix, only one iteration of the Jacobi rotation is needed to obtain the two eigenvectors and two eigenvalues for this 2x2 matrix. For a larger complex matrix with dimension greater than 2x2 , the iterative process performs multiple iterations of the Jacobi rotation to obtain the desired eigenvectors and eigenvalues, or singular vectors and singular values, for the larger complex matrix. Each iteration of the Jacobi rotation on the larger complex matrix uses the eigenvectors of a 2x2 submatrix, as described below.
[0025] Eigenvalue decomposition of a 2x2 Hermitian matri may be performed as follows. The Hermitian matri may be expressed as:
where A, B, and D are arbitrary real values, and θb is an arbitrary phase. [0026] The first step of the eigenvalue decomposition of s to apply a two-sided unitary transformation, as follows:
where is a symmetric real matrix containing real values and having symmetric off- diagonal elements at locations (1, 2) and (2, 1).
[0027] The symmetric real matri s then diagonalized using a two-sided Jacobi rotation, as follows:
where angle φ may be expressed as:
[0028] A 2x2 unitary matrix of eigenvectors of may be derived as:
[0029] The two eigenvalues A1 and A2 may be derived based on equation (6), or based on the equatioas follows:
[0030] In equation set (9), the ordering of the two eigenvalues is not fixed, and A1 may be larger or smaller than However, if angle φ is constrained such that then co , and sin if and only if D > A . Thus, the ordering of the two eigenvalues may be determined by the relative magnitudes of A and D. is the larger eigenvalue if A > D , and is the larger eigenvalue if D > A . If A = D , then si an is the larger eigenvalue. If is the larger eigenvalue, then the two eigenvalues i may be swapped to maintain a predetermined ordering of largest to smallest eigenvalues, and the first and second columns of may also be swapped correspondingly. Maintaining this predetermined ordering for the two eigenvectors in results in the eigenvectors of a larger size matrix decomposed using to be ordered from largest to smallest eigenvalues, which is desirable.
[0031] The two eigenvalues may also be computed directly from the elements of as follows:
Equation (10) is the solution to a characteristic equation of In equation (10), is obtained with the plus sign for the second quantity on the right hand side, an is obtained with the minus sign for the second quantity, where [0032] Equation (8) requires the computation of cos φ and sin φ to derive the elements of The computation of cos φ and sin φ is complex. The elements of may be computed directly from the elements o , as follows:
where rlil3 rlj2 and r2>1 are elements o and r is the magnitude of r1;2. Since ^1 is a complex value contains complex values in the second row.
[0033] Equation set (11) is designed to reduce the amount of computation to derive fro For example, in equations (lie), (lid), and (Hf), division by r is required. Instead, r is inverted to obtain r\, and multiplication by r\ is performed for equations (lie), (lid), and (Hf). This reduces the number of divide operations, which are computationally more expensive than multiplies. Also, instead of computing the argument (phase) of the complex element r\ ;2, which requires an arctangent operation, and then computing the cosine and sine of this phase value to obtain C1 and s\, various trigonometric identities are used to solve for C1 and S1 as a function of the real and imaginary parts of r1>2 and using only a square root operation. Furthermore, instead of computing the arctangent in equation (7) and the sine and cosine functions in equation (8), other trigonometric identities are used to solve for c and s as functions of the elements o
[0034] Equation set (11) performs a complex Jacobi rotation on to obtain
The set of computations in equation set (11) is designed to reduce the number of multiply, square root, and invert operations required to deri This can greatly reduce computational complexity for decomposition of a larger size matrix using
[0035] The eigenvalues o may be computed as follows:
1. Eigenvalue Decomposition
[0036] Eigenvalue decomposition of an NxN Hermitian matrix that is larger than
2x2 , as shown in equation (2), may be performed with an iterative process. This iterative process uses the Jacobi rotation repeatedly to zero out the off-diagonal elements in the NxN Hermitian matrix. For the iterative process, NxN unitary transformation matrices are formed based on 2 x 2 Hermitian submatrices of the NxN Hermitian matrix and are repeatedly applied to diagonalize the NxN Hermitian matrix. Each unitary transformation matrix contains four non-trivial elements (i.e., elements other than 0 or 1) that are derived from elements of a corresponding 2x2 Hermitian submatrix. The transformation matrices are also called Jacobi rotation matrices. After completing all of the Jacobi rotation, the resulting diagonal matrix contains the real eigenvalues of the NxN Hermitian matrix, and the product of all of the unitary transformation matrices is an N xN matrix of eigenvectors for the- NxN
Hermitian matrix. [0037] In the following description, index / denotes the iteration number and is initialized as i = 0. an NxN Hermitian matrix to be decomposed, where N > 2.
An N x N matri is an approximation of the diagonal matrix of eigenvalues of and is initialized as An N X N matrix is an approximation of the matrix of eigenvectors o and is initialized as [0038] A single iteration of the Jacobi rotation to update matrices an may be performed as follows. First, a 2x2 Hermitian matrix is formed based on the curren as follows:
where is the element at location (p,q) i and
is a 2x2 submatrix of and the four elements of re four elements at location in The values for indices p and q may be selected in various manners, as described below. [0039] Eigenvalue decomposition of is then performed, e.g., as shown in equation set (11), to obtain a 2x2 unitary matrix of eigenvectors of For the eigenvalue decomposition om equation (4) is replaced with an from equation (111) is provided as
[0040] An N x N complex Jacobi rotation matrix s then formed with matrix
T_pq is an identity matrix with the four elements at locations (p, p) , (p,q) , (q,p) and (q,q) replaced with the (1, 1), (1, 2), (2, 1) and (2, 2) elements, respectively, of has the following form:
where v1;1, v1>2, v2jl and v2,2 are the four elements of . AU of the other off-diagonal elements o are zeros. Equation (111) indicates that is a complex matrix containing complex values for v2jl and τ>2>2. is also called a transformation matrix that performs the Jacobi rotation. [0041] Matrix s then updated as follows:
Equation (15) zeros out two off-diagonal elements dPΛ and dqφ at locations (p,q) and (q, p) , respectively, in The computation may alter the values of the other off- diagonal elements in [0042] Matrix is also updated as follows:
may be viewed as a cumulative transformation matrix that contains all of the Jacobi rotation matrices sed on
[0043] Each iteration of the Jacobi rotation zeros out two off -diagonal elements of
Multiple iterations of the Jacobi rotation may be performed for different value of indices p and q to zero out all of the off-diagonal elements of The indices p and q may be selected in a predetermined manner by sweeping through all possible values.
[0044] A single sweep across all possible values for indices p and q may be performed as follows. The index p may be stepped from 1 through N -I in increments of one. For each value of p, the index q may be stepped from p + 1 through N in increments of one. An iteration of the Jacobi rotation to updat nd ay be performed for each different combination of values for p and q. For each iteration, is formed based on the values of p and q and the curren for that iteration, s computed for as shown in equation set (11), s formed with as shown in equation (14), s updated as shown in equation (15), an s updated as shown in equation (16). For a given combination of values for p and q, the Jacobi rotation to update nd may be skipped if the magnitude of the off-diagonal elements at locations (p,q) and (q, p) i is below a predetermined threshold.
[0045] A sweep consists of N - (N -I)/ 2 iterations of the Jacobi rotation to update an or all possible values of p and q. Each iteration of the Jacobi rotation zeros ou two off-diagonal elements o but may alter other elements that might have been zeroed out earlier. The effect of sweeping through indices p and q is to reduce the magnitude of all off-diagonal elements of so that approaches the diagonal matri contains an accumulation of all Jacobi rotation matrices that collectively giv Thus, approaches approaches A .
[0046] Any number of sweeps may be performed to obtain more and more accurate approximations o and Computer simulations have shown that four sweeps should be sufficient to reduce the off -diagonal elements o o a negligible level, and three sweeps should be sufficient for most applications. A predetermined number of sweeps (e.g., three or four sweeps) may be performed. Alternatively, the off-diagonal elements of may be checked after each sweep to determine whether is sufficiently accurate. For example, the total error (e.g., the power in all off-diagonal elements of may be computed after each sweep and compared against an error threshold, and the iterative process may be terminated if the total error is below the error threshold. Other conditions or criteria may also be used to terminate the iterative process.
[0047] The values for indices p and q may also be selected in a deterministic manner.
As an example, for each iteration i, the largest off-diagonal element of may be identified and denoted as Jacobi rotation may then be performed with containing this largest off-diagonal element dPA and three other elements at locations (p, p) , (q, p) , and {q,q) in The iterative process may be performed until a termination condition is encountered. The termination condition may be, for example, completion of a predetermined number of iterations, satisfaction of the error criterion described above, or some other condition or criterion.
[0048] Upon termination of the iterative process, the final s a good approximation of and the final s a good approximation of The columns of ay be provided as the eigenvectors o and the diagonal elements o ay be provided as the eigenvalues of . The eigenvalues in the final re ordered from largest to smallest because the eigenvectors in for each iteration are ordered. The eigenvectors in the fin are also ordered based on their associated eigenvalues in
[0049] FIG. 1 shows an iterative process 100 for performing eigenvalue decomposition of an NxN Hermitian matrix where N > 2 , using Jacobi rotation. Matrices and are initialized a and and index i is initialized as i = 1 (block
110).
[0050] For iteration i, the values for indices p and q are selected in a predetermined manner (e.g., by stepping through all possible values for these indices) or a deterministic manner (e.g., by selecting the index values for the largest off-diagonal element) (block 112). A 2x2 matri is then formed with four elements of matrix at the locations determined by indices p and q (block 114). Eigenvalue decomposition o is then performed, e.g., as shown in equation set (11), to obtain a 2x2 matrix of eigenvectors of block 116). An NxN complex Jacobi rotation matri is then formed based on matrix as shown in equation (14) (block 118). Matrix is then updated based on s shown in equation (15) (block 120). Matri is also updated based on s shown in equation (16) (block 122).
[0051] A determination is then made whether to terminate the eigenvalue decomposition of block 124). The termination criterion may be based on the number of iterations or sweeps already performed, an error criterion, and so on. If the answer is 'No' for block 124, then index / is incremented (block 126), and the process returns to block 112 for the next iteration. Otherwise, if termination is reached, then matrix is provided as an approximation of diagonal matrix and matri is provided as an approximation of matrix of eigenvectors o (block 128). [0052] For a MIMO system with multiple subbands (e.g., a MIMO system that utilizes
OFDM), multiple channel response matrices may be obtained for different subbands. The iterative process may be performed for each channel response matrix to obtain matrices and , which are approximations of diagonal matrix nd matri f eigenvectors, respectively, o
[0053] A high degree of correlation typically exists between adjacent subbands in a
MBvIO channel. This correlation may be exploited by the iterative process to reduce the amount of computation to derive and for the subbands of interest. For example, the iterative process may be performed for one subband at a time, starting from one end of the system bandwidth and traversing toward the other end of the system bandwidth. For each subband k except for the first subband, the final solutio obtained for the prior subband Jc -I may be used as an initial solution for the current subband Jc. The initialization for each subband Jc may be given as: and The iterative process then operates on the initial solutions of and for subband Jc until a termination condition is encountered.
[0054] The concept described above may also be used across time. For each time interval t, the final solution obtained for a prior time interval t - 1 may be used as an initial solution for the current time interval t. The initialization for each time and where esponse matrix for time interval t. The iterative process then operates on the initial solutions of and for time interval t until a termination condition is encountered. The concept may also be used across both frequency and time. For each subband in each time interval, the final solution obtained for a prior subband and/or the final solution obtained for a prior time interval may be used as an initial solution for the current subband and time interval. 2. Singular Value Decomposition
[0055] The iterative process may also be used for singular value decomposition of an arbitrary complex matri that is larger than 2x2. The singular value decomposition o is given as . The following observations may be made regarding First, matrix and matri are both Hermitian matrices. Second, right singular vectors o which are the columns of are also eigenvectors o Correspondingly, left singular vectors of which are the columns of are also eigenvectors of Third, the non-zero eigenvalues o re equal to the non-zero eigenvalues o and are the square of corresponding singular values of [0056] A 2x2 matrix f complex values may be expressed as:
where is a 2x1 vector with the elements in the first column o and s a 2x1 vector with the elements in the second column of
[0057] The right singular vectors o are the eigenvectors of and may be computed using the eigenvalue decomposition described above in equation set (11). A 2 x 2 Hermitian matrix is defined as and the elements of may be computed based on the elements o , as follows:
For Hermitian matrix does not need to be computed since Equation set (11) may be applied t to obtain a matri ontains the eigenvectors o which are also the right singular vectors of
[0058] The left singular vectors o are the eigenvectors of and may also be computed using the eigenvalue decomposition described above in equation set (11). A 2x2 Hermitian matrix is defined as and the elements of may be computed based on the elements o as follows:
Equation set (11) may be applied to o obtain a matrix contains the eigenvectors of which are also the left singular vectors o
[0059] The iterative process described above for eigenvalue decomposition of an
N xN Hermitian matrix ay be used for singular value decomposition of an arbitrary complex matrix larger than 2x2. as a dimension of R xT , where R is the number of rows and T is the number of columns. The iterative process for singular value decomposition (SVD) o may be performed in several manners.
[0060] In a first SVD embodiment, the iterative process derives approximations of the right singular vectors in and the scaled left singular vectors in For this embodiment, a T xT matrix is an approximation of and is initialized as An R x T matrix is an approximation o and is initialized as
[0061] For the first SVD embodiment, a single iteration of the Jacobi rotation to update matrices and ay be performed as follows. First, a 2x2 Hermitian matrix
is formed based on the current is a 2x2 submatrix of and contains four elements at locations (p,p) , (p,q) , (q, p) and (q,q) in The elements o may be computed as follows:
where s column p o is column q of and is the element at location (£,p) in Indices p and g are such that p& {1,...,T } , qe {1,...,T }, and p ≠ q . The values for indices p and q may be selected in various manners, as described below. [0062] Eigenvalue decomposition of is then performed, e.g., as shown in equation set (11), to obtain a 2x2 unitary matrix of eigenvectors of For this eigenvalue decomposition, is replaced wit an is provided a
[0063] A T xT complex Jacobi rotation matrix is then formed with matrix is an identity matrix with the four elements at locations (p, p) , (p,q) , (q, p) and (q, q) replaced with the (1, 1), (1, 2), (2, 1) and (2, 2) elements, respectively, of has the form shown in equation (14).
[0064] Matri is then updated as follows:
[0065] Matrix is also updated as follows:
[0066] For the first SVD embodiment, the iterative process repeatedly zeros out off- diagonal elements of without explicitly computin The indices p and q may be swept by stepping p from 1 through T - I and, for each value of p, stepping q from p + 1 through T. Alternatively, the values of p and q for which is largest may be selected for each iteration. The iterative process is performed until a termination condition is encountered, which may be a predetermined number of sweeps, a predetermined number of iterations, satisfaction of an error criterion, and so on. [0067] Upon termination of the iterative process, the fina is a good approximation of and the final is a good approximation of When converged,
where "τ " denotes a transpose. For a square diagonal matrix, the final solution o may be given as For a non- square diagonal matrix, the non-zero diagonal elements of re given by the square roots of the diagonal elements of The final solution of ay be given as:
[0068] . 2 shows an iterative process 200 for performing singular value decomposition of an arbitrary complex matrix hat is larger than 2x2 using Jacobi rotation, in accordance with the first SVD embodiment. Matrices and re initialized and index i is initialized as i = 1 (block 210).
[0069] For iteration i, the values for indices p and q are selected in a predetermined or deterministic manner (block 212). A 2x2 matrix s then formed with four elements of matrix1 at the locations determined by indices p and q as shown in equation set (20) (block 214). Eigenvalue decomposition o is then performed, e.g., as shown in equation set (11), to obtain a 2x2 matrix of eigenvectors of (block 216). A TxT complex Jacobi rotation matri formed based on matrix , as shown in equation (14) (block 218). Matrix is then updated based o as shown in equation (21) (block 220). Matrix s also updated based on as shown in equation (22) (block 222).
[0070] A determination is then made whether to terminate the singular value decomposition of lock 224). The termination criterion may be based on the number of iterations or sweeps already performed, an error criterion, and so on. If the answer is 'No' for block 224, then index i is incremented (block 226), and the process returns to block 212 for the next iteration. Otherwise, if termination is reached, then post processing is performed on to obtain d (block 228). Matrix is provided as an approximation of matrix of right singular vectors o , matrix is provided as an approximation of matri of left singular vectors of nd matrix is provided as an approximation of matri f singular values o (block 230)
[0071] The left singular vectors of may be obtained by performing the first SVD embodiment and solving for scaled left singular vectors and then normalizing. The left singular vectors of may also be obtained by performing the iterative process for eigenvalue decomposition o
[0072] In a second SVD embodiment, the iterative process directly derives approximations of the right singular vectors in and the left singular vectors in . This SVD embodiment applies the Jacobi rotation on a two-sided basis to simultaneously solve for the left and right singular vectors. For an arbitrary complex
2x2 matri he conjugate transpose of this matrix i where an e the two columns o and are also the complex conjugates of the rows of . The left singular vectors o re also the right singular vectors of The right singular vectors of may be computed using Jacobi rotation, as described above for equation set (18). The left singular vectors of ay be obtained by computing the right singular vectors of sing Jacobi rotation, as described above for equation set (19). [0073] For the second SVD embodiment, a T xT matri is an approximation of and is initialized as An R X R matrix s an approximation of n is initialized a An R xT matri is an approximation of Σ and is initialized as [0074] the second SVD embodiment, a single iteration of the Jacobi rotation to update matrices and may be performed as follows. First, a 2x2 Hermitian matrix s formed based on the current s a 2x2 submatrix of and contains four elements at locations (P1^1) , (^1, #1)> (^v P\)an<^ (#i>#i)m
The four elements of may be computed as follows:
where is colum is colum and s the element at location Indices p\ and q\ are such that and px ≠ qx . Indices p\ and ^1 may be selected in various manners, as described below.
[0075] Eigenvalue decomposition of is then performed, e.g., as shown in equation set (11), to obtain a 2x2 matrix of eigenvectors of For this eigenvalue decomposition, is replaced with and is provided as A TxT complex Jacobi rotation matrix is then formed with matrix and contains the four elements of at location and has the form shown in equation (14).
[0076] Another 2x2 Hermitian matrix is also formed based on the current is a 2x2 submatrix of and contains elements at locations (p2,p2) ,anc*m The elements o may be computed as follows:
wher is row /?2 o is row q2 o and is the element at location (p2,£) in Indices p% and g2 are such that , and P2 ≠ q2. Indices p2 and qt may also be selected in various manners, as described below. [0077] Eigenvalue decomposition of is then performed, e.g., as shown in equation set (11), to obtain a 2x2 matrix HP2 q2 of eigenvecto For this eigenvalue decomposition, s replaced with and vided as . An R xR complex Jacor bi rotation matri is then formed with matrix and contains the four elements o at locations and has the form shown in equation (14). [0078] Matrix is then updated as follows:
[0079] Matrix is updated as follows:
[0080] Matri s updated as follows:
[0081] For the second SVD embodiment, the iterative process alternately finds (1) the
Jacobi rotation that zeros out the off-diagonal elements with indices p\ and q\ i and (2) the Jacobi rotation that zeros out the off-diagonal elements with indices p2 and ^2 i The indices p\ and q\ may be swept by stepping p\ from 1 through T- I and, for each value of p\, stepping q\ from P1 + 1 through T. The indices p2 and <j2 may also be swept by stepping ^2 from 1 through R -I and, for each value of p2, stepping qt from p2 + 1 through R. As an example, for a square matrix the indices may be set as P1 = p2 and qx - q2 . As another example, for a square or non-square matrix a set of p\ and q\ may be selected, then a set of p2 and g2 may be selected, then a new set of pi and ^r1 may be select, then a new set of /?2 and ^2 may be selected, and so on, so that new values are alternately selected for indices p\ and q\ and indices /?2 and ^2.
Alternatively, for each iteration, the values Of ^1 and q\ for whic is largest may be selected, and the values of /?2 and g2 for which is largest may be selected. The iterative process is performed until a termination condition is encountered, which may be a predetermined number of sweeps, a predetermined number of iterations, satisfaction of an error criterion, and so on. [0082] Upon termination of the iterative process, the final is a good approximation o the final is a good approximation of and the final is a good approximation o where and may be rotated versions of and respectively. The computation described above does not sufficiently constrain the left and right singular vector solutions so that the diagonal elements of the final are positive real values. The elements of the final may be complex values whose magnitudes are equal to the singular values o an may be unrotated as follows:
where is a TxT diagonal matrix with diagonal elements having unit magnitude and phases that are the negative of the phases of the corresponding diagonal elements of re the final approximations of , respectively.
[0083] FIG. 3 shows an iterative process 300 for performing singular value decomposition of an arbitrary complex matrix that is larger than 2x2 using Jacobi rotation, in accordance with the second SVD embodiment. Matrices and are initialized a and index i is initialized as i = 1 (block 310). [0084] For iteration i, the values for indices p\, q\, p2 and q2 are selected in a predetermined or deterministic manner (block 312). A 2x2 matrix is formed with four elements of matrix at the locations determined by indices p\ and q\, as shown in equation set (23) (block 314). Eigenvalue decomposition of is then performed, e.g., as shown in equation set (11), to obtain a 2x2 matrix of eigenvectors o (block 316). A TxT complex Jacobi rotation matrix is then formed based on matri (vblock 318) '. A 2x2 matrix s also formed with four elements of matrix at the locations determined by indices p2 and #2, as shown in equation set (24) (block 324). Eigenvalue decomposition of is then performed, e.g., as shown in equation set (11), to obtain a 2x2 matrix of eigenvectors of block 326). An R xR complex Jacobi rotation matrix s then formed based on matri (block 328).
[0085] Matrix is then updated based on as shown in equation (25) (block
330). Matrix is updated based on as shown in equation (26) (block 332). Matrix is updated based on and as shown in equation (27) (block 334).
[0086] A determination is then made whether to terminate the singular value decomposition of (block 336). The termination criterion may be based on the number of iterations or sweeps already performed, an error criterion, and so on. If the answer is 'No' for block 336, then index i is incremented (block 338), and the process returns to block 312 for the next iteration. Otherwise, if termination is reached, then post processing is performed on nd obtai nd block 340). Matrix is provided as an approximation o , matrix is provided as an approximation , and matrix s provided as an approximation of block 342)
[0087] For both the first and second SVD embodiments, the right singular vectors in the final nd the left singular vectors in the final or are ordered from the largest to smallest singular values because the eigenvectors in (for the first SVD embodiment) and the eigenvectors in and (for the second SVD embodiment) for each iteration are ordered. [0088] For a MIMO system with multiple subbands, the iterative process may be performed for each channel response matrix to obtain matrices an which are approximations of the matrix of right singular vectors, the matrix of left singular vectors, and the diagonal matrix f singular values, respectively, for tha The iterative process may be performed for one subband at a time, starting from one end of the system bandwidth and traversing toward the other end of the system bandwidth. For the first SVD embodiment, for each subband k except for the first subband, the final solutio obtained for the prior subband k - 1 may be used as an initial solution for the current subband k, so that and For the second SVD embodiment, for each subband k except for the first subband, the final solution nd btained for the prior subband k — 1 may be used as initial solutions for the current subband k, so that For both embodiments, the iterative process operates on the initial solutions for subband k until a termination condition is encountered for the subband. The concept may also be used across time or both frequency and time, as described above.
[0089] FIG. 4 shows a process 400 for decomposing a matrix using Jacobi rotation.
Multiple iterations of Jacobi rotation are performed on a first matrix of complex values with multiple Jacobi rotation matrices of complex values (block 412). The first matrix may be a channel response matri , a correlation or some other matrix. The Jacobi rotation matrices may b r some other matrices. 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 is derived based on the multiple Jacobi rotation matrices (block 414). The second matrix contains orthogonal vectors and may be matrix of right singular vectors o r eigenvectors o
[0090] For eigenvalue decomposition, as determined in block 416, a third matrix f eigenvalues may be derived based on the multiple Jacobi rotation matrices (block 420). For singular value decomposition based on the first SVD embodiment or scheme, a third matri of complex values may be derived based on the multiple Jacobi rotation matrices, a fourth matri with orthogonal vectors may be derived based on the third matrix , and a matri of singular values may also be derived based on the third matri (block 422). For singular value decomposition based on the second SVD embodiment, a third matrix with orthogonal vectors and a matrix of singular values may be derived based on the multiple Jacobi rotation matrices (block 424).
[0091] FIG. 5 shows an apparatus 500 for decomposing a matrix using Jacobi rotation.
Apparatus 500 includes means for performing multiple iterations of Jacobi rotation on a first matrix of complex values with multiple Jacobi rotation matrices of complex values (block 512) and means for deriving a second matri of complex values based on the multiple Jacobi rotation matrices (block 514).
[0092] For eigenvalue decomposition, apparatus 500 further includes means for deriving a third matri f eigenvalues based on the multiple Jacobi rotation matrices (block 520). For singular value decomposition based on the first SVD embodiment, apparatus 500 further includes means for deriving a third matrix of complex values based on the multiple Jacobi rotation matrices, a fourth matrix with orthogonal vectors based on the third matrix, and a matrix of singular values based on the third matrix (block 522). For singular value decomposition based on the second SVD embodiment, apparatus 500 further includes means for deriving a third matri ith orthogonal vectors and a matrix of singular values based on the multiple Jacobi rotation matrices (block 524).
3. System
[0093] FIG. 6 shows a block diagram of an embodiment of an access point 610 and a user terminal 650 in a MJJVIO system 600. Access point 610 is equipped with multiple (Nap) antennas that may be used for data transmission and reception. User terminal 650 is equipped with multiple (Nut) antennas that may be used for data transmission and reception. For simplicity, the following description assumes that MDVIO system 600 uses time division duplexing (TDD), and the downlink channel response matrix for each subband k is reciprocal of the uplink channel response matrix for that subband, o [0094] On the downlink, at access point 610, a transmit (TX) data processor 614 receives traffic data from a data source 612 and other data from a controller/processor 630. TX data processor 614 formats, encodes, interleaves, and modulates the received data and generates data symbols, which are modulation symbols for data. A TX spatial processor 620 receives and multiplexes the data symbols with pilot symbols, performs spatial processing with eigenvectors or right singular vectors if applicable, and provides Nap streams of transmit symbols to Nap transmitters (TMTR) 622a through 622ap. Each transmitter 622 processes its transmit symbol stream and generates a downlink modulated signal. Nap downlink modulated signals from transmitters 622a through 622ap are transmitted from antennas 624a through 624ap, respectively.
[0095] At user terminal 650, Nut antennas 652a through 652ut receive the transmitted downlink modulated signals, 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 transmitters 622 and provides received symbols. A receive (RX) spatial processor 660 performs spatial matched filtering on the received symbols from all receivers 654a through 654ut and provides detected data symbols, which are estimates of the data symbols transmitted by access point 610. An RX data processor 670 further processes (e.g., symbol demaps, deinterleaves, and decodes) the detected data symbols and provides decoded data to a data sink 672 and/or a controller/processor 680.
[0096] A channel processor 678 processes received pilot symbols and provides an estimate of the downlink channel response, for each subband of interest. Processor 678 and/or 680 may decompose each matrix using the techniques described herein to obtai an hich are estimates of nd or the downlink channel response matrix Processor 678 and/or 680 may derive a downlink spatial filter matrix for each subband of interest based on as shown in Table 1. Processor 680 may provid RX spatial processor 660 for downlink matched filtering and/o to a TX spatial processor 690 for uplink spatial processing.
[0097] The processing for the uplink may be the same or different from the processing for the downlink. Traffic data from a data source 686 and other data from controller/ processor 680 are processed (e.g., encoded, interleaved, and modulated) by a TX data processor 688, multiplexed with pilot symbols, and further spatially processed by a TX spatial processor 690 with for each subband of interest. The transmit symbols from TX spatial processor 690 are further processed by transmitters 654a through 654ut to generate Nut uplink modulated signals, which are transmitted via antennas 652a through 652ut.
[0098] At access point 610, the uplink modulated signals are received by antennas 624a through 624ap and processed by receivers 622a through 622ap to generate received symbols for the uplink transmission. An RX spatial processor 640 performs spatial matched filtering on the received data symbols and provides detected data symbols. An RX data processor 642 further processes the detected data symbols and provides decoded data to a data sink 644 and/or controller/processor 630.
[0099] A channel processor 628 processes received pilot symbols and provides an estimate of either or for each subband of interest, depending on the manner in which the uplink pilot is transmitted. Processor 628 and/or 630 may decompose each matrix using the techniques described herein to obtain Processor 628 and/or 630 may also derive an uplink spatial filter matrix for each subband of interest based on Processor 680 may provide to RX spatial processor 640 for uplink spatial matched filtering and/or to TX spatial processor 620 for downlink spatial processing.
[00100] Controllers/processors 630 and 680 control the operation at access point 610 and user terminal 650, respectively. Memories 632 and 682 store data and program codes for access point 610 and 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 matrices.
[00101] The matrix decomposition techniques described herein may be implemented by various means. For example, these techniques may be implemented in hardware, firmware, software, or a combination thereof. For a hardware implementation, the processing units used to perform matrix decomposition may be implemented within one or more application specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processing devices (DSPDs), programmable logic devices (PLDs), field programmable gate arrays (FPGAs), processors, controllers, micro- controllers, microprocessors, other electronic units designed to perform the functions described herein, or a combination thereof.
[00102] For a firmware and/or software implementation, the matrix decomposition techniques may be implemented with modules (e.g., procedures, functions, and so on) that perform the functions described herein. The software codes may be stored in a memory (e.g., memory 632 or 682 in FIG. 6 and executed by a processor (e.g., processor 630 or 680). The memory unit may be implemented within the processor or external to the processor.
[00103] Headings are included herein for reference and to aid in locating certain sections. These headings are not intended to limit the scope of the concepts described therein under, and these concepts may have applicability in other sections throughout the entire specification.
[00104] The previous 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 will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other embodiments without departing from the spirit or scope of the invention. Thus, the present invention is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.
[00105] WHAT IS CLAIMED IS: