TW201347446A - Method and apparatus for singular value decomposition of a channel matrix - Google Patents

Abstract

A method and apparatus for decomposing a channel matrix in a wireless communication system are disclosed. A channel matrix H is generated for channels between transmit antennas and receive antennas. A Hermitian matrix A=H<SP>H</SP>H or A=HH<SP>H</SP> is created. A Jacobi process is cyclically performed on the matrix A to obtain Q and DA matrixes such that A=QDAQ<SP>H</SP>. DA is a diagonal matrix obtained by singular value decomposition (SVD) on the A matrix. In each Jacobi transformation, real part diagonalization is performed to annihilate real parts of off-diagonal elements of the matrix and imaginary part diagonalization is performed to annihilate imaginary parts of off-diagonal elements of the matrix after the real part diagonalization. U, V and DH matrixes of H matrix are then calculated from the Q and DA matrices. DH is a diagonal matrix comprising singular values of the H matrix.

Description

Frequency rice matrix singular value decomposition method and device

The present invention relates to a wireless communication system. More specifically, the present invention relates to a method and apparatus for channel matrix singular value decomposition (SVD).

Orthogonal Frequency Division Multiplexing (OFDM) is a data transmission architecture in which data is partitioned into multiple smaller streams and each stream is transmitted using a subcarrier having a bandwidth less than one of the total available transmission bandwidth. The efficiency of OFDM depends on the selection of these subcarriers that are orthogonal to each other. These subcarriers do not interfere with each other while each carries a portion of all user data.

OFDM systems have advantages over other wireless communication systems. When the user data is split into streams carried by different subcarriers, the effective data transmission rate on each subcarrier is much smaller. Therefore, the symbol duration will be much longer. A large symbol duration can tolerate a large degree of delay. Therefore, it will not be seriously affected by multipath. Therefore, OFDM symbols can tolerate the degree of delay without a complex receiver design. However, traditional wireless systems require complex channel equalization architectures to combat multipath fading.

Another advantage of OFDM is that the generation of orthogonal subcarriers at the transmitter and receiver can be accomplished using an inverse fast Fourier transform (IFFT) and fast Fourier transform (FFT) engine. Since IFFT and FFT implementations are well known, OFDM can be easily implemented without the need for complex receivers.

Multiple Input Multiple Output (MIMO) refers to a wireless transmission and reception architecture in which a transmitter More than one antenna is used with one receiver. A MIMO system benefits from spatial diversity or spatial multiplexing processing and improves signal-to-noise ratio (SNR) and throughput.

In general, MIMO systems have two modes of operation: an open loop mode and a closed loop mode. The closed loop mode is used when channel state information (CSI) can be provided to the transmitter, and the open loop mode is used when no CSI is available at the transmitter. In closed loop mode, CSI is used to create a channel that is almost non-dependent by precoding at the transmitter and further antenna processing at the receiver to decompose and diagonalize the channel matrix. The CSI can be obtained at the transmitter by feedback from the receiver or by utilizing channel reciprocity.

A minimum mean square error (MMSE) receiver for open-loop MIMO must calculate the weight vector for data decoding and the convergence rate of these weight vectors is important. One of the correlation coefficient matrices, the direct vector inverse transform (DMI) technique, converges faster than a least mean square (LMS) or maximum SNR program. However, the complexity of the DMI program grows exponentially as the size of the matrix increases. A eigenbeamforming receiver for closed loop MIMO must perform SVD on the channel matrix. The complexity of the SVD program also grows exponentially as the size of the channel matrix increases.

The present invention relates to a method and apparatus for decomposing a channel matrix in a wireless communication system. The wireless communication system includes a transmitter having a plurality of transmit antennas and a receiver having a plurality of receive antennas. A channel matrix H is generated for the channel between the transmitting antenna and the receiving antenna. Create a Hermitian matrix A = H ^H H or A = HH ^H . A Jacobi program is executed cyclically on matrix A to obtain a Q and D _A matrix resulting in A = QD _A Q ^H , where D _A is a diagonal matrix obtained by SVD on the matrix A. In each Jacobi transformation, real diagonalization is performed to eliminate the real part of the non-diagonal elements of the matrix, and after the real part is diagonalized, imaginary diagonalization is performed to eliminate the off-diagonal elements of the matrix. The imaginary part. The U, V, and D _H matrices of the H matrix are then computed from the Q and D _A matrices, where D _H is a diagonal matrix containing the singular values of the matrix H.

103, 105, 107, 111, 207, 218‧‧‧ data streams

217, 215, D _A, D _H , Hermitian ‧ matrix

114, 202‧‧‧ antenna

203‧‧‧ Signal

209‧‧‧ output

CP‧‧‧ cyclic prefix

FFT‧‧‧fast Fourier transform

H ^H H, HH ^H ‧‧‧ Dimensions

IFFT‧‧‧Anti-fast Fourier transform

Jacobi‧‧ program

MIMO‧‧‧Multiple Input Multiple Output

OFDM‧‧ Orthogonal Frequency Division Multiplex

S/P‧‧‧Multiple serials

1 is a block diagram of an OFDM-MIMO system including a transmitter and a receiver for performing eigenbeamforming using SVD in accordance with the present invention.

Figure 2 is a flow diagram of a procedure for executing an SVD on the channel matrix H in accordance with the present invention.

Features of the invention may be incorporated into an integrated circuit (IC) or constructed within a circuit that includes multiple interconnect components.

The present invention proposes a channel evaluation component, a direct inverse transform of a channel correlation coefficient matrix in an MMSE receiver, and an SVD for a eigenbeamforming receiver.

1 is a block diagram of an OFDM-MIMO system 10 including a transmitter 100 and a receiver 200 for performing eigenbeamforming using SVD in accordance with the present invention. It should be understood that the system 10 shown in Figure 1 is merely an example and not a limitation, and that the present invention is applicable to any wireless communication system that requires a matrix decomposition using SVD. The transmitter 100 includes a channel encoder 102, a demultiplexer 104, a plurality of serial-to-serial (S/P) converters 106, a transmit beamformer 108, a plurality of IFFT units 110, and a cyclic prefix (CP) insertion unit. 112 and a plurality of transmitting antennas 114. Channel encoder 102 encode input data 101 by the demultiplexer 104 and the coded data stream 103 parsed into N _T data stream 105 a. N _T is the number of transmit antennas 114. OFDM processing is performed on each data stream 105. Each data stream 105 is converted to a plurality of data streams 107 by an S/P converter 106. The data stream 107 is then processed by the transmit beamformer 108. Transmit beamformer 108 performs a transmit precoding operation with a V matrix (see below) that is decomposed from a channel matrix and transmitted by receiver 200 as indicated by arrow 220. The IFFT unit 110 converts the data into a time domain data stream 111 and inserts a CP into each data stream 111 by each CP insertion unit 112 and issues it via a corresponding transmit antenna 114.

The receiver 200 includes a plurality of receiving antennas 202, a CP removing unit 204, an FFT unit 206, a receive beamformer 208, a multiplexer 210, a channel decoder 212, a channel evaluator 214, and a matrix decomposition and channel correlation. Coefficient matrix unit 216. The CP is removed from the received signal 203 by the CP removal unit 204 and processed by the FFT unit 206 to be converted into a frequency domain data stream 207. Receive beamformer 208 processes frequency domain data stream 207 with U and D matrices 217 decomposed from the channel matrix generated by matrix decomposition and channel correlation coefficient matrix unit 216. Each output 209 of the receive beamformer 208 is then multiplexed by the multiplexer 210 and decoded by the channel decoder 212, thus producing a decoded data stream 218. Channel estimator 214 preferably generates a channel matrix 215 from the training sequence transmitted by transmitter 100 via each transmit antenna 114. The matrix decomposition and channel correlation coefficient matrix unit 216 decomposes the channel matrix into U, V and D matrices and sends the V matrix 220 to the transmitter 100 and the U and D matrix 217 to the receiver beamformer 208, see below .

The present invention utilizes the Hermitian matrix and the diagonalization of the imaginary part to reduce the complexity of both the DMI and SVD programs. The present invention substantially surpasses conventional techniques to reduce complexity and provides benefits for asymmetric matrices that are far greater than the complexity that conventional techniques can provide.

The following definitions will be used throughout the invention.

Nt is the number of transmitting antennas.

Nr is the number of receiving antennas.

s(i) is the ith (Nt x 1) training vector of a subcarrier.

v(i) is the ith (Nr x 1) received noise vector, where v(i)~Nc(0,1).

y(i) is the ith (Nr x 1) received training vector of a subcarrier.

H is a (Nr x Nt) MIMO channel matrix, where h _ij represents the channel complex gain between the jth transmit antenna and the ith receive antenna.

The received signals corresponding to the training symbols are as follows:

The condition is T≧Nt MIMO training symbol. ρ is a total SNR that is independent of the number of transmit antennas.

By representing Y=[y(1), y(2),...y(T)], S=[s(1), s(2),...,s(T)] for a subcarrier and V= [v(1), v(2), ...v(T)], equation (1) can be rewritten as follows:

The most approximate estimate of the channel matrix H for a subcarrier is given by:

Wherein the superscript H represents Hermitian transposition and S is a training symbol sequence. Assume that the transmitted training symbol is a single power, E{|S _i | ² }=1.

As an alternative to the most probable channel evaluation, the Linear Minimum Mean Square Error (MMSE) channel estimate is given by:

Since S is known, SS ^H can be calculated offline. If the training symbol sequence S satisfies SS ^H =T. I _Nt , where I _Nt is an Nt×Nt identity matrix, then the training symbol sequence S is optimal. For example, the training symbol sequence for the subcarrier number (-26) according to the IEEE 802.11 specification for the 4-antenna HT-LTF pattern.

The input-output relationship of a MIMO system can be expressed as follows:

Where s=[s ₁ ,s ₂ ,...s _Nt ] ^T is the Nt×1 transmitted signal vector and s _i belongs to a finite constellation, v=[v ₁ ,v ₂ ,...v _Nt ] ^T is Nr×1 reception White Gaussian noise vector. H is an Nt×Nr MIMO channel matrix and h _ij represents a channel complex gain between the jth transmit antenna and the ith receive antenna. The MMSE-based data decoding procedure is then given by:

The matrix inverse conversion procedure of a 2 x 2 matrix is explained below.

Direct calculation of the inverse transformation of the 2×2 Hermitian matrix R.

One of the equations (6) Hermitian matrix R and its inverse matrix T are defined as and . The diagonal elements of the Hermitian matrix R (R ₁₁ and R ₂₂ ) are real numbers and their non-diagonal elements (R ₁₂ and R ₂₁ ) are conjugate symmetric. The inverse matrix T is also Hermitian. Since RT=I where I is a 2×2 identity matrix, the inverse matrix T is obtained by expanding the left side and using I to find the corresponding terms, as shown below:

The inverse matrix of the 2×2 Hermitian matrix R is calculated by eigenvalue decomposition.

One of the equations (6) Hermitian matrix R is defined as follows: R = QDQ ^H , where Q is positive and D is diagonal.

Wherein D ₁₁ and D ₂₂ are characteristic values of R.

The eigenvalues D ₁₁ and D ₂₂ are calculated as follows:

From RQ=QD, expand the left and right sides and find the corresponding terms to get the following equation: R ₁₁ Q ₁₁ + R ₁₂ Q ₂₁ = Q ₁₁ D ₁₁ Equation (10)

R ₁₁ Q ₁₂ + R ₁₂ Q ₂₂ = Q ₁₂ D ₂₂ Equation (11)

From Q ^H Q=I where I is a 2 × 2 unit matrix, expand the left and right sides and find the corresponding terms to get the following equation:

From equation (10),

Substituting equation (18) into equation (14),

Substituting equation (19) into equation (18) yields Q ₂₁ . From equation (13),

Substituting equation (20) into equation (17),

Substituting equation (21) into equation (20) yields Q ₂₂ . Then the inverse matrix is obtained by the following equation: R ^-1 = QD ^-1 Q ^H Equation (22)

A eigenbeamforming receiver using SVD will be described below, which is shown in Fig. 1. For the eigenbeamformed receiver, the channel matrix H for a subcarrier is decomposed into two beamformed positive matrices (U for transmission and V for reception) and a pair of angular matrices D by SVD.

H=UDV ^H equation (23)

Where U and V are positive matrices and D is a pair of angular matrices. U C ^nRxnR and V C ^nTxnT . For the transmitted symbol vector s, the transmit precoding is performed as follows: x = Vs Equation (24)

The received signal becomes as follows: y=HVs+n Equation (25)

Where n is the noise introduced into the channel. The receiver completes the decomposition by using a matched filter as follows: V ^H H ^H = V ^H VD ^H U ^H = D ^H U ^H Equation (26)

After normalizing the channel gain of the eigenbeam, the estimate of the transmitted symbol becomes as follows:

s is detected without the need for continuous interference cancellation by the MMSE type detector. D ^H D is a diagonal matrix composed of eigenvalues of H across the diagonal. U is a matrix of eigenvalues of HH ^H , V is a matrix of eigenvalues of H ^H H and D is a pair of angular matrices of singular values of ^H (the square root of the eigenvalues of HH ^H ).

An SVD program for an N x M channel matrix with N > 2 and M > 2.

The following SVD calculations (equations (28) through (52)) are based on the use of Givens rotation in a cyclic Jacobi program. The following describes the bilateral Jacobi program.

Step 1: Convert the complex data into real data.

Give the following A 2 × 2 complex matrix:

Step 1-1: Convert a _ii to a positive real number b _{11 as} follows: (a ₁₁ equals 1)

otherwise

among them .

Step 1-2: Triangulation. The matrix B is then converted to a triangular matrix W by multiplying a conversion matrix CSTriangle as follows: if (a _ij or a _{ji is} equal to zero and a _{jj is} equal to zero)

W =( CSTriangle )( B )= B equation (32)

otherwise

Where cosine parameter c is a real number, s is a complex number and c ² +| s | ² =1

Step 1-3: Phase cancellation. In order to convert the elements of the triangular matrix W into real numbers, the transformation matrix prePhC and postPhC are multiplied by the matrix W as follows: if (a _ij and a _{jj are} equal to zero and a _{ji is} not equal to zero)

Otherwise if (a _{ij is} not equal to zero) then

Where β = arg( w _ij ) and γ = arg( w _jj ), ie And if (a _{jj is} equal to zero)

otherwise

otherwise

Step 2: Symmetry - If the matrix realW is not a symmetric matrix then apply a symmetric rotation. Skip this step if the matrix realW is symmetrical.

If (a _{ji is} equal to zero and a _{jj is} equal to zero)

otherwise

Where s _ji = s _ij and c ² + s ² =1.

By expanding the left side and waiting for everything,

Step 3: Diagonalization - Apply a pair of angular rotations to eliminate the off-diagonal elements in the matrix symW (or realW).

If (a _{ij is} equal to zero and a _{jj is} equal to zero)

otherwise

Where c ² + s ² =1.

By expanding the left side and waiting for the corresponding off-diagonal terms,

Step 4: Rotate the matrices of the matrix to produce U and V matrices. The U and V matrices are obtained as follows: A = UDV ^H equation (50)

U =[ k ( diagM ) ^H ( symM ) ^H ( prePhC )( CSTriangle )] ^H equation (51)

V =( postPhC )( diagM ) Equation (52)

A cyclic generalized Jacobi program for a M x M square matrix.

In order to eliminate the non-diagonal elements of A (ie (i, j) and (j, i) elements), the above procedure is in a fixed order with a total of m = M (M - 1) / 2 different The index pair is applied to the M x M matrix A. This sequence of one-m conversion is referred to as a sweep. A sweep configuration can be cycled in columns or in rows. In either case, a new matrix A is obtained after each sweep to calculate the condition under j≠1 . If off(A) ≦ δ, the calculation stops. δ is a small number that depends on the accuracy of the calculation. Otherwise the calculation is repeated.

A cyclic generalized Jacobi program for an N x M rectangular matrix.

If the dimension N of the matrix A is greater than M, a square matrix is generated by adding zeros to the (N-M) rows of A. The augmented square matrix B=|A 0|. The procedure described above was then applied to B.

The expected factorization of the original data matrix A is obtained by the following equation: U ^T AV = diag ( λ ₁ , λ ₂ ,..., λ _M ) Equation (54)

If the dimension M of the matrix A is greater than N, a square matrix is generated by adding zeros to the (MN) column to A as follows:

The procedure described above was then applied to B.

The expected factorization of the original data matrix A is obtained by the following equation: U ^T AV = diag ( λ ₁ , λ ₂ ,..., λ _N ) Equation (57)

The SVD program according to the present invention will be explained below with reference to Fig. 2. Figure 2 is a flow chart of an SVD program in accordance with the present invention. The present invention proposes a method of executing an SVD program. A channel matrix H between the plurality of transmit antennas and the plurality of receive antennas is generated (step 202). A Hermitian matrix A is created for the resulting Nr x Nt channel matrix H (step 204). Matrix A is produced as A = H ^H H under Nr ≧ Nt and A = HH ^H under Nr < Nt conditions. The bilateral Jacobi program is then cyclically applied to the M x M matrix A to obtain a Q and D _A matrix resulting in A = QD _A Q ^H , where M = min(Nr, Nt), which will be explained below (step 206). D _A is a diagonal matrix containing the eigenvalues of the matrix H obtained by the SVD of the matrix A. Since the matrix A is Hermitian and symmetrical, the symmetry steps of the prior art are no longer needed and the procedure is greatly simplified. Once the SVD of _A is calculated, the U matrix, the V matrix, and the D _H matrix (H = UD _H V ^H ) of the _H matrix are calculated from the Q matrix and the D _A matrix (step 208).

The step 206 of performing SVD on the A matrix is explained below. Define a 2×2 Hermitian matrix symW from matrix A as follows:

Where a _ii , a _ij , a _ji , a _jj , b _ij and b _ji are real numbers and a _ij = a _ji and b _ij = b _ji . The matrix symW is generated from the matrix A for each Jacobi transformation as in the prior art method.

Real part diagonalization is performed on the matrix symW. The real part of the off-diagonal element of the matrix symW is eliminated by multiplying the matrix symW by the transformation matrix (diagRM) ^T and diagRM as follows:

Where r _ii and r _jj are real numbers, b _ij =b _ji and c ² +s ² =1.

The following equation is obtained by expanding the left side and waiting for the corresponding non-diagonal real terms.

Where t ² = 2 ζ t -1 = 0.

Equation (61); or

Then perform imaginary diagonalization. The imaginary part of the off-diagonal element is eliminated by multiplying the matrix of the diagonals of the real part by the transformation matrix (diagIM) ^T and diagIM as follows: Wherein c, s, r _ii , r _jj , b _ij , b _ji , d _ii and d _jj are real numbers, b _ij =b _ji and c ² +s ² =1.

The following equation is obtained by expanding the left side and waiting for the corresponding off-diagonal terms. y = cs ( γ _ii - γ _jj ) + (1-2 c ² ) b _ij equation (68)

If y> threshold value (for example, = 0.0001), then Equation (70).

The threshold is a small machine dependent number.

The transformation matrices for real triangulation and imaginary triangulation are then combined to calculate the U and V matrices as follows: A = UD _A V ^H equation (71)

U =[( diagIM ) ^H ( diagRM ) ^H ] ^H equation (72)

V = ( diagRM )( diagIM ) Equation (73)

In order to eliminate the non-diagonal elements of A (i.e., (i, j) and (j, i) elements), the above procedure is applied in a fixed order with a total of m = M (M - 1) / 2 different index pairs. In M × M matrix A, where M = min (Nr, Nt). After each step, a new matrix A is obtained, which is calculated under the condition of j≠1 . If off(A) ≦ δ where δ is a small machine dependent number, the calculation stops. Otherwise the calculation is repeated.

Once the SVD of matrix A is complete, the U matrix, V matrix, and D _H matrix of the H matrix are calculated from the Q matrix and the D _A matrix as follows: From equations (72) and (73), U = V and A matrix. Can be written as: A = QD _A Q ^H . When Nr≧Nt, since Q is equal to V, at H=UD _H V ^H and D _A =Q ^H AQ=Q ^H H ^H HQ=Q ^H VD _H U ^H UD _H V ^H Q=D _H U ^H UD _H = Under the condition of D _H D _H , D _A = D _H D _H (that is, D _H =sqrt(D _A )). The U, V and D _H matrices are then obtained as follows: U = HV(D _H ) ^-1 where V = Q and D _H = sqrt(D _A ).

When Nr>Nt, since Q is equal to V, at H=UD _H V ^H and D _A =Q ^H AQ=Q ^H HH ^H Q=Q ^H UD _H V ^H VD _H U ^H Q=D _H V ^H VD _H = Under the condition of D _H D _H , D _A = D _H D _H (that is, D _H =sqrt(D _A )). The U, V and D _H matrices are then obtained as follows: V = H ^H U(D _H ) ^-1 where U = Q and D _H = sqrt(D _A ).

Although the features and elements of the present invention have been described in terms of specific combinations in the preferred embodiments, each feature or element may be used alone (without other features and elements of the preferred embodiments) or with or without the invention. A combination of features and elements.

D _A, D _H, Hermitian‧‧ matrix

H ^H H, HH ^H ‧‧‧ Dimensions

Jacobi‧‧ program

Claims (4)

A method of integrating a circuit, comprising: a circuit configured to channel encode a data; a circuit configured to divide the channel encoded data into a first plurality of streams; configured to stream the first plurality of streams by the transmitter a circuit mapped to a second complex stream, wherein each of the second plurality of streams is associated with a respective transmit antenna; configured to perform an inverse fast Fourier transform (IFFT) for each of the second plurality of streams And a circuit for generating a complex IFFT stream, each of the complex IFFT streams having a plurality of subcarriers; a circuit configured to insert a cyclic prefix into each of the plurality of IFFT streams; configured to transmit using a corresponding transmit antenna a circuit for inserting the complex IFFT stream of the cyclic prefix, the corresponding transmit antenna using beamforming weights, the beamforming weights being derived using singular value decomposition (SVD), wherein the SVD uses a two-unit matrix and a singular value diagonal matrix The channel response matrix is decomposed. The integrated circuit of claim 1, wherein the beamforming weight is applied to the second plurality of streams before the IFFT. An integrated circuit comprising: a circuit configured to remove a cyclic prefix (CP) from a received signal; a circuit configured to convert the received signal into a complex frequency domain data stream; configured with U and D Processing a circuit of the complex frequency domain data stream, the U and D matrix being decomposed from a channel matrix; and configuring multiplexing and decoding the processed complex frequency domain data stream to generate a decoded data stream Circuit. The circuit as described in claim 3, further comprising: circuitry configured to generate a channel matrix from the training sequence, the training sequence being transmitted through an antenna by a transmitter; and configured to matrix the channel Decomposed into U, V, and D matrices and sent to the transmitter, and the U and D matrices are sent to a receive beamformer circuit.