WO2006053340A2 - 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 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 h_tJ , 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 θ_b 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 equatio ^as 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 r_lil3 r_lj2 and r_2>1 are elements o and r is the magnitude of r_1;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 r_1>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 o ^m 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 v_1;1, v_1>2, v_2jl 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 v_2jl 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 d_PΛ and d_qφ 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 d_PA 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 matrix ₁ 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 p_x ≠ q_x . 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 q_t 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 H_{P2 q2} of eigenvecto For this eigenvalue decomposition, s replaced with and vided as

. An R xR complex Jaco

^r 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 q_t from p₂ + 1 through R. As an example, for a square matrix

the indices may be set as P₁ = p₂ and q_x - 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 (N_ap) antennas that may be used for data transmission and reception. User terminal 650 is equipped with multiple (N_ut) 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 N_ap streams of transmit symbols to N_ap transmitters (TMTR) 622a through 622ap. Each transmitter 622 processes its transmit symbol stream and generates a downlink modulated signal. N_ap downlink modulated signals from transmitters 622a through 622ap are transmitted from antennas 624a through 624ap, respectively.

[0095] At user terminal 650, N_ut 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 N_ut 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:

Claims

1. An apparatus comprising: at least one processor configured to perform a plurality of iterations of Jacobi rotation on a first matrix of complex values with a plurality of Jacobi rotation matrices of complex values, and to derive a second matrix of complex values based on the plurality of Jacobi rotation matrices, the second matrix comprising orthogonal vectors; and a memory coupled to the at least one processor.

2. The apparatus of claim 1, wherein for each of the plurality of iterations the at least one processor is configured to form a submatrix based on the first matrix, to decompose the submatrix to obtain eigenvectors for the submatrix, to form a Jacobi rotation matrix with the eigenvectors, and to update the first matrix with the Jacobi rotation matrix.

3. The apparatus of claim 2, wherein for each of the plurality of iterations, the at least one processor is configured to order the eigenvectors for the submatrix based on eigenvalues for the submatrix.

4. The apparatus of claim 1, wherein the at least one processor is configured to derive a third matrix of eigenvalues based on the plurality of Jacobi rotation matrices.

5. The apparatus of claim 1, wherein the at least one processor is configured to derive a third matrix of complex values based on the plurality of Jacobi rotation matrices; and to derive a fourth matrix with orthogonal vectors based on the third matrix.

6. The apparatus of claim 5, wherein the at least one processor is configured to derive a matrix of singular values based on the third matrix.

7. The apparatus of claim 1, wherein the at least one processor is configured to derive a third matrix with orthogonal vectors based on the plurality of Jacobi rotation matrices.

8. The apparatus of claim 7, wherein the at least one processor is configured to derive a matrix of singular values based on the plurality of Jacobi rotation matrices.

9. The apparatus of claim 1, wherein the at least one processor is configured to select different values for row and column indices of the first matrix for the plurality of iterations of the Jacobi rotation.

10. The apparatus of claim 1, wherein for each of the plurality of iterations the at least one processor is configured to identify a largest off-diagonal element in the first matrix, and to perform the Jacobi rotation based on the largest off-diagonal element.

11. The apparatus of claim 1, wherein the at least one processor is configured to terminate the Jacobi rotation on the first matrix after a predetermined number of iterations.

12. The apparatus of claim 1, wherein the at least one processor is configured to determine whether an error criterion is satisfied, and to terminate the plurality of iterations of the Jacobi rotation upon satisfaction of the error criterion.

13. The apparatus of claim 1, wherein the first matrix has a dimension larger than 2x2.

14. A method comprising: performing a plurality of iterations of Jacobi rotation on a first matrix of complex values with a plurality of Jacobi rotation matrices of complex values; and deriving a second matrix of complex values based on the plurality of Jacobi rotation matrices, the second matrix comprising orthogonal vectors.

15. The method of claim 14, wherein the performing the plurality of iterations of Jacobi rotation on the first matrix comprises, for each iteration, forming a submatrix based on the first matrix, decomposing the submatrix to obtain eigenvectors for the submatrix, forming a Jacobi rotation matrix with the eigenvectors, and updating the first matrix with the Jacobi rotation matrix.

16. The method of claim 14, further comprising: deriving a third matrix of complex values based on the plurality of Jacobi rotation matrices; and deriving a fourth matrix with orthogonal vectors based on the third matrix.

17. The method of claim 14, further comprising: deriving a third matrix with orthogonal vectors based on the plurality of Jacobi rotation matrices.

18. An apparatus comprising: means for performing a plurality of iterations of Jacobi rotation on a first matrix of complex values with a plurality of Jacobi rotation matrices of complex values; and means for deriving a second matrix of complex values based on the plurality of Jacobi rotation matrices, the second matrix comprising orthogonal vectors.

19. The apparatus of claim 18, wherein the means for performing the plurality of iterations of Jacobi rotation on the first matrix comprises, for each iteration, means for forming a submatrix based on the first matrix, means for decomposing the submatrix to obtain eigenvectors for the submatrix, means for forming a Jacobi rotation matrix with the eigenvectors, and means for updating the first matrix with the Jacobi rotation matrix.

20. The apparatus of claim 18, further comprising: means for deriving a third matrix of complex values based on the plurality of Jacobi rotation matrices; and means for deriving a fourth matrix with orthogonal vectors based on the third matrix.

21. The apparatus of claim 18, further comprising: means for deriving a third matrix with orthogonal vectors based on the plurality of Jacobi rotation matrices.

22. An apparatus comprising: at least one processor configured to initialize a first matrix to an identity matrix, to initialize a second matrix to a Hermitian matrix of complex values, to perform a plurality of iterations of Jacobi rotation on the second matrix by forming a Jacobi rotation matrix of complex values for each iteration based on the second matrix, and updating the first and second matrices for each iteration based on the

Jacobi rotation matrix for the iteration, to provide the first matrix as a matrix of eigenvectors, and to provide the second matrix as a matrix of eigenvalues; and a memory coupled to the at least one processor.

23. The apparatus of claim 22, wherein for each of the plurality of iterations the at least one processor is configured to form a submatrix based on the second matrix, to decompose the submatrix to obtain eigenvectors for the submatrix, and to form the Jacobi rotation matrix with the eigenvectors for the submatrix.

24. An apparatus comprising: means for initializing a first matrix to an identity matrix; means for initializing a second matrix to a Hermitian matrix of complex values; means for performing a plurality of iterations of Jacobi rotation on the second matrix, comprising means for forming a Jacobi rotation matrix of complex values for each iteration based on the second matrix, and means for updating the first and second matrices for each iteration based on the Jacobi rotation matrix for the iteration; means for providing the first matrix as a matrix of eigenvectors; and means for providing the second matrix as a matrix of eigenvalues.

25. The apparatus of claim 24, wherein the means for forming the Jacobi rotation matrix of complex values for each iteration comprises means for forming a submatrix based on the second matrix, means for decomposing the submatrix to obtain eigenvectors for the submatrix, and means for forming the Jacobi rotation matrix with the eigenvectors for the submatrix.

26. An apparatus comprising: at least one processor is configured to initialize a first matrix to an identity matrix, to initialize a second matrix to a matrix of complex values, to perform a plurality of iterations of Jacobi rotation on the second matrix by forming a Jacobi rotation matrix for each iteration based on the second matrix, and updating the first and second matrices for each iteration based on the Jacobi rotation matrix for the iteration, and to provide the first matrix as a matrix of right singular vectors; and a memory coupled to the at least one processor.

27. The apparatus of claim 26, wherein for each of the plurality of iterations the at least one processor is configured to form a submatrix based on the second matrix, to decompose the submatrix to obtain eigenvectors for the submatrix, and to form the Jacobi rotation matrix with the eigenvectors.

28. The apparatus of claim 26, wherein the at least one processor is configured to derive a matrix of singular values based on the second matrix.

29. The apparatus of claim 26, wherein the at least one processor is configured to derive a matrix of left singular vectors based on second matrix.

30. An apparatus comprising: means for initializing a first matrix to an identity matrix; means for initializing a second matrix to a matrix of complex values; means for performing a plurality of iterations of Jacobi rotation on the second matrix, comprising means for forming a Jacobi rotation matrix for each iteration based on the second matrix, and means for updating the first and second matrices for each iteration based on the Jacobi rotation matrix for the iteration; and means for provide the first matrix as a matrix of right singular vectors.

31. The apparatus of claim 30, wherein the means for forming the Jacobi rotation matrix for each iteration comprises means for forming a submatrix based on the second matrix, means for decomposing the submatrix to obtain eigenvectors for the submatrix, and means for forming the Jacobi rotation matrix with the eigenvectors.

32. An apparatus comprising: at least one processor is configured to initialize a first matrix to an identity matrix, to initialize a second matrix to the identity matrix, to initialize a third matrix to a matrix of complex values, to perform a plurality of iterations of Jacobi rotation on the third matrix by, for each iteration, forming a first Jacobi rotation matrix based on the third matrix, forming a second Jacobi rotation matrix based on the third matrix, updating the first matrix based on the first Jacobi rotation matrix, updating the second matrix based on the second Jacobi rotation matrix, and updating the third matrix based on the first and second Jacobi rotation matrices, and to provide the second matrix as a matrix of left singular vectors; and a memory coupled to the at least one processor.

33. The apparatus of claim 32, wherein for each of the plurality of iterations the at least one processor is configured to form a first submatrix based on the third matrix, to decompose the first submatrix to obtain eigenvectors for the first submatrix, and to form the first Jacobi rotation matrix with the eigenvectors for the first submatrix.

34. The apparatus of claim 33, wherein for each of the plurality of iterations the at least one processor is configured to form a second submatrix based on the third matrix, to decompose the second submatrix to obtain eigenvectors for the second submatrix, and to form the second Jacobi rotation matrix with the eigenvectors for the second submatrix.

35. The apparatus of claim 32, wherein the at least one processor is configured to derive a matrix of right singular vectors based on the first matrix.

36. The apparatus of claim 32, wherein the at least one processor is configured to derive a matrix of singular values based on the third matrix.

37. An apparatus comprising: means for initializing a first matrix to an identity matrix; means for initializing a second matrix to the identity matrix; means for initializing a third matrix to a matrix of complex values, means for performing a plurality of iterations of Jacobi rotation on the third matrix comprising, for each iteration, means for forming a first Jacobi rotation matrix based on the third matrix, means for forming a second Jacobi rotation matrix based on the third matrix, means for updating the first matrix based on the first Jacobi rotation matrix, means for updating the second matrix based on the second Jacobi rotation matrix, and means for updating the third matrix based on the first and second Jacobi rotation matrices; and means for providing the second matrix as a matrix of left singular vectors.

38. The apparatus of claim 37, wherein the means for forming the first Jacobi rotation matrix comprises means for forming a first submatrix based on the third matrix, means for decomposing the first submatrix to obtain eigenvectors for the first submatrix, and means for forming the first Jacobi rotation matrix with the eigenvectors for the first submatrix.

39. The apparatus of claim 38, wherein the means for forming the second Jacobi rotation matrix comprises means for forming a second submatrix based on the third matrix, means for decomposing the second submatrix to obtain eigenvectors for the second submatrix, and means for forming the second Jacobi rotation matrix with the eigenvectors for the second submatrix.

40. An apparatus comprising: at least one processor is configured to perform a first plurality of iterations of Jacobi rotation on a first matrix of complex values to obtain a first unitary matrix with orthogonal vectors, and to perform a second plurality of iterations of the Jacobi rotation on a second matrix of complex values to obtain a second unitary matrix with orthogonal vectors, wherein the first unitary matrix is used as an initial solution for the second unitary matrix; and a memory coupled to the at least one processor.

41. The apparatus of claim 40, wherein the at least one processor is configured to perform a third plurality of iterations of the Jacobi rotation on a third matrix of complex values to obtain a third unitary matrix with orthogonal vectors, wherein the second unitary matrix is used as an initial solution for the third unitary matrix.

42. The apparatus of claim 40, wherein the first and second matrices of complex values are channel response matrices for two frequency subbands.

43. The apparatus of claim 40, wherein the first and second matrices of complex values are channel response matrices for two time intervals.

44. An apparatus comprising: means for performing a first plurality of iterations of Jacobi rotation on a first matrix of complex values to obtain a first unitary matrix with orthogonal vectors; and means for performing a second plurality of iterations of the Jacobi rotation on a second matrix of complex values to obtain a second unitary matrix with orthogonal vectors, wherein the first unitary matrix is used as an initial solution for the second unitary matrix.

45. The apparatus of claim 44, wherein the first and second matrices of complex values are channel response matrices for two frequency subbands.