WO2019141353A1

WO2019141353A1 - Signal decoder and method for determining the qr decomposition of an m×n input matrix

Info

Publication number: WO2019141353A1
Application number: PCT/EP2018/051048
Authority: WO
Inventors: Alon SIDI
Original assignee: Huawei Technologies Co., Ltd.
Priority date: 2018-01-17
Filing date: 2018-01-17
Publication date: 2019-07-25

Abstract

The present invention provides signal decoder 100 for determining the QR decomposition of an M x N input matrix (H), with M≥ N, with a product of an M x N orthonormal matrix (Q) and an N x N upper triangular matrix (R). The signal decoder 100 is configured to calculate a conjugated transposed matrix (H^H) of the input matrix (H) and to calculate a Hermitian matrix (G) by multiplying the conjugated transposed matrix (H^H) of the input matrix (H) and the input matrix (H) in the form G = H^HH. Further, the signal decoder 100 is configured to provide an augmented N x (M + N) matrix (K₀) having as a left matrix the matrix (I) and as a right matrix the matrix (II), and to multiply the augmented matrix (K₀) from the left with a series of lower triangular matrices (A_k) where k ranges from 0 to (N— 1).

Description

Signal decoder and method for determining the QR decomposition of an M*N input matrix

TECHNICAL FIELD

The present invention relates to a signal decoder for determining the QR decomposition of an M x N input matrix H, a method for determining the QR decomposition of an M x N input matrix H and a computer program.

BACKGROUND

In mobile communication systems such as an OFDMA (Orthogonal frequency-division multiple access)/MIMO (Multiple Input Multiple Output) communication system, QR decomposition (QRD) is widely used. The QR decomposition or QR factorization of a matrix is a decomposition of a matrix into a product of an orthogonal matrix Q and an upper triangular matrix R.

A very popular application of QR decomposition is the usage of the Q and R matrices by the Sphere decoder or the K-Best Sphere decoder, which are near maximum-likelihood channel equalizers in the MIMO environment.

Many wireless communication standards including the 802.11 family of standards use QR decomposition. Recent versions of 802.11 , including 802.11 ax, support high-order MIMO schemes, up to 8x8, with a very large number of frequency tones. This requires the development of up to 8x8 QR decomposition with reasonable complexity in terms of latency and required chip area.

The most common implementation of the QR decomposition is based on iterative Givens Rotations which zero elements below the diagonal of the input matrix H. As an illustration, by augmenting H and I from the right and multiplying both by an appropriate orthonormal Givens Rotation matrix G. In order to differentiate between scalar values and matrices and vectors, matrices and vectors are presented in the formulas in bold font.

With

Then the following matrix is received

That is, the lower leftmost element of H is zeroed while keeping the right side matrix orthonormal. By performing successive operations as such, an upper triangular matrix R is produced on the left, while since the Givens Rotation matrices are orthonormal, an

orthonormal matrix Q^H is produced on the right and naturally, H = QR. The Givens Rotations are traditionally performed via CORDIC machines, which require a very large area.

SUMMARY

In view of the above-mentioned problems and disadvantages, the present invention aims to improve the QR decomposition. The present invention has thereby the object to provide a signal decoder for determining the QR decomposition and a method of determining the QR decomposition, which operate with better performance compared to the corresponding solutions known in the art. The object of the present invention is achieved by the solution provided in the enclosed independent claims. Advantageous implementations of the present invention are further defined in the dependent claims.

In particular, the present invention proposes to use the Gauss Jordan Elimination on a

Hermitian matrix G in the form G = H ^H H and a conjugated transposed matrix H ^H of the input matrix H with a unique scaling. The process is halted when the N x N matrix

which equals the upper triangular matrix (R) and the N x M matrix which equals the

conjugated transposed matrix (Q^H) is received and all the elements below the diagonal of the matrix are zeroed. The process is performed in a column by column manner starting

from the leftmost side of G. The conjugated transposed matrix of an M-by-N matrix A with complex entries is the N-by-M matrix A^* obtained from A by taking the transpose and then taking the complex conjugate of each matrix element. A Hermitian matrix or self-adjoint matrix is a complex square matrix that is equal to its own conjugated transposed matrix. This implies that the element in the z-th row and j- th column is equal to the complex conjugate of the element in the j- th row and z-th column, for all indices z and j.

A first aspect of the present invention provides a Signal decoder for determining the QR decomposition of an M x N input matrix H, with M ³ N, with a product of an M x N orthonormal matrix Q and an N x N upper triangular matrix R, wherein the signal decoder is configured to calculate a conjugated transposed matrix of the input matrix H and to calculate a Hermitian matrix G by multiplying the conjugated transposed matrix H ^H of the input matrix H and the input matrix H in the form G = H ^H H; to provide an augmented (iV x M) + N matrix K₀ having as a left matrix the matrix which is initialized as the Hermitian

matrix G and as a right matrix the matrix

which is initialized as the conjugated transposed matrix H ^H of the input matrix H; to multiply the augmented matrix K₀ from the left with a series of lower triangular matrices A where k ranges from 0 to N— 1 which is equivalent to multiplying K₀ with the inverse lower triangular Cholesky decomposition matrix L^-1 ; to provide the upper triangular matrix R as the result of the product of the Hermitian matrix G with a series of lower triangular matrices A_k where k ranges from 0 to N— 1 from the left which is equivalent to multiplying G with the inverse lower triangular Cholesky

decomposition matrix L^-1 from the left; and to provide a conjugated transposed matrix Q^H of the orthonormal matrix Q as the result of the product of the conjugated transposed matrix H ^H of the input matrix H with a series of lower triangular matrices A_k where k ranges from 0 to N— 1 from the left which is equivalent to multiplying H ^H with the inverse lower triangular Cholesky decomposition matrix L^-1 from the left.

This present invention proposes for a given complex or real input matrix H to use a modification of the Gauss Jordan Elimination (GJE) procedure on the matrix G = H^H H. That is, initializing a 'left matrix' as G and augmenting to it a 'right matrix' from the right which is initialized as H ^H and performing on both of these matrices the same row operations which result in zeroing of all the elements below the diagonal elements of the 'left matrix' in a column by column manner starting from the leftmost side of the 'left matrix'. However, in the proposed solution and in contrast to the traditional GJE procedure, when all the elements below a certain diagonal element of the 'left matrix', which is currently being processed have been zeroed, the row with the same row index as of this diagonal element in both the 'left matrix' and the 'right matrix' may be scaled by the reciprocal of the square root of this diagonal element value itself. The process may be named QRD via Gauss Jordan Elimination (QRD-GJE).

The present invention offers advantages in its implementation simplicity, high accuracy, reduced silicon area and easy scalability. Instead of complex matrix calculations, the present invention relies in part on simple vector by scalar multiplications and simple vector by scalar multiplication and accumulation (MAC). For a single pipeline stage only multipliers are needed and for the whole scheme only a single divider and a single inv-sqrt (inverted square root) unit is needed. In addition to design simplification, the proposed solution leads to area reduction compared to a conventional implementation.

The invention herein describes a hardware, which calculates the QRD based on GJE with the following major differences. The hardware performs GJE on the augmented matrix [G|H^H] in contrast to traditional matrix inversion where [G|I] is processed. In contrast to the traditional GJE where all the elements above and below the diagonal of the N x N‘left matrix’ of the processed augmented matrix are zeroed, the process herein is halted when only all the elements below the diagonal of the N x N‘left matrix’ are zeroed.

In contrast to GJE, the pivot rows, which are the final outputs, may be uniquely scaled by the reciprocal of the square root of the pivot itself. At the end of the process [G|H^H] transform into [R|Q^H] In an implementation form of the first aspect, the signal decoder is configured to provide the orthonormal matrix Q from the conjugated transposed matrix Q^H of the orthonormal matrix Q. Hence, the orthonormal matrix Q can be calculated easily from its conjugated transposed matrix Q^H. In an implementation form of the first aspect, the signal decoder is configured to perform a Gauss Jordan Elimination on the augmented matrix K₀, wherein the Gauss Jordan Elimination is terminated when the N x (M + N) augmented matrix K_N having as a left matrix the N x N matrix which equals the upper triangular matrix R and as a right matrix the N x M matrix which equals the conjugated transposed matrix Q^H is received and all the elements below the diagonal of the matrix are zeroed. Such provision is easy to implement and allows

increased performance.

In a further implementation form of the first aspect, the signal decoder is configured to scale each of the pivot rows by the reciprocal of the square root of the row’s pivot itself. Such provision is easy to implement and allows increased performance. In an implementation form of the first aspect, the signal decoder is configured to perform recursions indexed by k that ranges from 0 to N— 1 on the augmented matrix K₀, with the (N— k) x (N— k ) Hermitian matrices C_k

Hermitian matrices D_fc, and the N x N matrices A_k

and the recursions on the N x N matrices and the N x N matrices being

The A multiplications are implemented as simple row multiply or multiply and accumulate (MAC) operations. This proceeding increases the performance.

In a further implementation form of the first aspect, the input matrix G is the product H^HH where the M x N channel matrix H is a transmission channel of received data signals. The input matrix G can be easily provided based on the channel matrix H. The data signals may include typical mobile communication signals for example phase-shift keying (PSK) signals or quadrature amplitude modulation (QAM) signals.

In an implementation form of the first aspect, the signal decoder is a part of a MIMO decoder and/or a MIMO precoder. Due to increasing complexity of MIMO systems, the proposed solution is especially suitable for such systems as parallel execution of the filtering operations is possible.

A second aspect of the present invention provides a method for determining the QR decomposition of an M x N input matrix H, with M ³ N, with a product of an M x N orthonormal matrix Q and an N x N upper triangular matrix R, comprising calculating a conjugated transposed matrix H^H of the input matrix H and to calculate a Hermitian matrix G by multiplying the conjugated transposed matrix H^H of the input matrix H and the input matrix H in the form G = H^H H; providing an N x (M + N) augmented matrix K₀ having as a left matrix the matrix which is initialized as the Hermitian matrix G and as a right matrix

the matrix which is initialized as the conjugated transposed matrix H^H of the input matrix H; multiplying the augmented matrix K₀ from the left with a series of lower triangular matrices A_k where k ranges from 0 to N— 1 which is equivalent to multiplying K₀ with the inverse lower triangular Cholesky decomposition matrix L^-1; providing the upper triangular matrix R as the result of the product of the Hermitian matrix G with a series of lower triangular matrices A_k where k ranges from 0 to N— 1 from the left which is equivalent to multiplying G with the inverse lower triangular Cholesky decomposition matrix L^-1 from the left; and providing a conjugated transposed matrix Q^H of the orthogonal matrix Q as the result of the product of the conjugated transposed matrix H^H of the input matrix H with a series of lower triangular matrices A where k ranges from 0 to N— 1 from the left which is equivalent to multiplying H^H with the inverse lower triangular Cholesky decomposition matrix L^-1 from the left. The same advantages and modifications as described above apply. A third aspect of the present invention provides a computer program with a program code for performing the method as described above when the computer program runs on a computer or on the signal decoder as described above. The same advantages and modifications as described above apply. It has to be noted that all devices, elements, units and means described in the present application could be implemented in the software or hardware elements or any kind of combination thereof. All steps which are performed by the various entities described in the present application as well as the functionalities described to be performed by the various entities are intended to mean that the respective entity is adapted to or configured to perform the respective steps and functionalities. Even if, in the following description of specific embodiments, a specific functionality or step to be performed by external entities is not reflected in the description of a specific detailed element of that entity which performs that specific step or functionality, it should be clear for a skilled person that these methods and functionalities can be implemented in respective software or hardware elements, or any kind of combination thereof.

BRIEF DESCRIPTION OF DRAWINGS

The above described aspects and implementation forms of the present invention will be explained in the following description of specific embodiments in relation to the enclosed drawings, in which

Fig. 1 shows an example of a signal decoder for determining the QR decomposition of an input matrix H.

Fig. 2 shows a flowchart of an exemplary implementation of a method for determining the QR decomposition of an input matrix H.

DETAILED DESCRIPTION OF EMBODIMENTS

The signal decoder 100 is for example part of a transmission system like a wireless communication system. The signal decoder 100 may be implemented as a system on a chip, a microprocessor, a DSP (Digital Signal Processor) or the like. The signal decoder 100 is located at endpoints of a communication channel 120, for example at base stations, user equipment or the like and may be part of a MIMO decoder.

The signal decoder 100 is configured to determine the QR decomposition of an M x N input matrix H, with M ³ N, with a product of an M x N orthonormal matrix Q and an N x N upper triangular matrix R.

The channel matrix H is a transmission channel of received data signals. The input matrix G can be easily provided based on the channel matrix H. The data signals may include typical mobile communication signals for example phase-shift keying (PSK) signals or quadrature amplitude modulation (QAM) signals.

In a first block 120 of the signal decoder 100, the signal decoder 100 calculates a conjugated transposed matrix of the input matrix H. In a second block 130 of the signal decoder 100, the signal decoder 100 calculates a Hermitian matrix G by multiplying the conjugated transposed matrix of the input matrix H and the input matrix H in the form G = H. The conjugated transposed matrix and the Hermitian matrix G are provided to a third block 140, where the remaining steps of the QR decomposition are performed. The process can be named QRD via Gauss Jordan Elimination (QRD-GJE). The blocks 120, 130 and 140 can be implemented in hardware and/or software.

Figure 2 shows steps of an exemplary QR decomposition via Gauss Jordan Elimination (QRD-GJE) method, which may be performed by the signal decoder 100.

The method determines the QR decomposition of an M x N input matrix H, with M ³ N, with a product of an M x N orthonormal matrix Q and an N x N upper triangular matrix R.

In a first step 200 a conjugated transposed matrix of the input matrix H and a Hermitian matrix G by multiplying the conjugated transposed matrix H ^H of the input matrix H and the input matrix H in the form G = H ^H H are calculated.

In a second step 210 an N x (M + N) augmented matrix K₀ having as a left matrix the matrix which is initialized as the Hermitian matrix G and as a right matrix the matrix which is

initialized as the conjugated transposed matrix H ^H of the input matrix H is provided. In a third step 220 the augmented matrix K₀ is multiplied from the left with a series of lower triangular matrices A where k ranges from 0 to N— 1 which is equivalent to multiplying K₀ with the inverse lower triangular Cholesky decomposition matrix L ¹.

In a fourth step 230 the upper triangular matrix R is provided as the result of the product of the Hermitian matrix G with a series of lower triangular matrices A_k where k ranges from 0 to N— 1 from the left which is equivalent to multiplying G with the inverse lower triangular Cholesky decomposition matrix L^-1 from the left.

In a fifth step 240 a conjugated transposed matrix Q^H of the orthogonal matrix Q is provided as the result of the product of the conjugated transposed matrix H ^H of the input matrix H with a series of lower triangular matrices A_k where k ranges from 0 to N— 1 from the left which is equivalent to multiplying H ^H with the inverse lower triangular Cholesky decomposition matrix L^-1 from the left.

Such procedure reduces calculation complexity and the resulting signal decoder 100 is less complex and needs less space.

In the following, a more detailed description of the QR decomposition via Gauss Jordan Elimination (QRD-GJE) method is given.

Given a matrix H and the defined matrix G

H^H H, it is easy to show that

G = H^HH = LL^H = R^H Q^H QR = R^H R, (1) where L is the lower triangular Cholesky decomposition of G, H QR and obviously L^H =

R. Therefore, if an augmented matrix K₀ is defined as

K₀ [G|H^H], (2) then by multiplying K₀ from the left by L^-1, the following is received

L^-1 K₀ = [L^-1 L^-1H¹¹] = [R|Q^H] (3)

That is, the QRD of the matrix H is produced at equation (3). The following is based on performing row operations on matrices and which are initialized as B_Q = G = H^HH and

Define the (N — k) x (N — k) lower right sub matrix of as

and define the N x N matrix A k

Where, I_k is a k x k identity matrix, I_{N-k -1} is a (N — k— 1) x (N — k— 1) identity matrix, then with the definitions (4-5), the recursions are finally stated as

The practical interpretation of equations (6-7) is that the A_k multiplications are implemented as simple row multiply or multiply and add (MAC) operations. Specifically, for the pivot rows:

for the rows below the pivot lines, where was defined

For indexing inside a matrix, for example the matrix [k + 1,0: N— 1], the value k is

incremented by 1 for each step of the recursion. The value k starts with 0 and reaches to N— 1.

At the end of the process the following is received

R = (14)

Q^H = (15)

Note that equations (4-15) simply explain the steps of the invention. A mathematical justification of the algorithm is given below.

It is easy to notice that equations (8-9) and (12-13) imply a vector by scalar multiplication, whereas equations (10-11) imply a vector by scalar multiplication accumulation (MAC). Per a single pipeline stage only multipliers may be needed and for the whole scheme only a single divider and a single inverse-square root unit may be needed. The following diagrams illustrate a processing of a 4x4 real matrix according to the above described scheme. The counter k for the number of recursions ranges from 0 to 3, with N = 4 and k from 0 to N - 1.

Multiply by and store as a 1x7 intermediate result r^T , wherein r is a result vector.

Multiply by and store in-place.

Subtract from this row the multiplication and store in-place.

Subtract from this row the multiplication and store in-place.

Subtract from this row the multiplication and store in-place.

Now, after the last row has been reached, k is incremented to 1. Multiply by and store

as a 1x6 intermediate result r^T .

Multiply by and store in-place.

Subtract from this row the multiplication and store in-place.

Subtract from this row the multiplication and store in-place.

Now, after the last row has been reached, k is incremented to 2. Multiply by and store

as a 1x5 intermediate result r^T.

Multiply by and store in-place.

Subtract from this row the multiplication and store in-place.

Now, after the last row has been reached, k is incremented to 3. Multiply by and store

in-place. As the limit of k is reached with the value 3, the recursion is finished.

In the following, a mathematical justification of the QRD-GJE process is given.

Using equations (4-5), one can write

in that case

where is also a Hermitian matrix. Using equation (A3) as an example, the generalization

can be achieved . Further, from equation (1) it can be stated L^-1GL ^-H = I, (A5) hence

L ¹— A_n-1 A-_LA_Q . (A6)

And since the A_k matrices are lower triangular, then L^_1and L are also lower triangular and consequently it can be stated that L is actually the Cholesky decomposition of G which is unique.

Now, since recursively multiplying the A_k, k = 0: N— 1 matrices from the left of any matrix is equivalent to multiplying it by L^-1 as stated in equation (A6). Then, if the same operations on G and H^H are performed then R and Q^H are produced respectively as stated in equation (3).

The present invention has been described in conjunction with various embodiments as examples as well as implementations. However, other variations can be understood and effected by those persons skilled in the art and practicing the claimed invention, from the studies of the drawings, this disclosure and the independent claims. In the claims as well as in the description the word“comprising” does not exclude other elements or steps and the indefinite article“a” or“an” does not exclude a plurality. A single element or other unit may fulfill the functions of several entities or items recited in the claims. The mere fact that certain measures are recited in the mutual different dependent claims does not indicate that a combination of these measures cannot be used in an advantageous implementation.

Claims

1. Signal decoder (100) for determining the QR decomposition of an M x N input matrix (H), with M ³ N, with a product of an M x N orthonormal matrix (Q) and an N x N upper triangular matrix (R), wherein the signal decoder (100) is configured to

calculate a conjugated transposed matrix (H^H) of the input matrix (H) and to calculate a Hermitian matrix (G) by multiplying the conjugated transposed matrix (H^H) of the input matrix (H) and the input matrix (H) in the form G = H;

provide an augmented N x (M + N) matrix (K₀) having as a left matrix the matrix

which is initialized as the Hermitian matrix (G) and as a right matrix the matrix

which is initialized as the conjugated transposed matrix (H^H) of the input matrix (H);

multiply the augmented matrix (K₀) from the left with a series of lower triangular matrices (A_k) where k ranges from 0 to (N— 1) which is equivalent to multiplying (K₀) with the inverse lower triangular Cholesky decomposition matrix (L^-1);

provide the upper triangular matrix (R) as the result of the product of the Hermitian matrix (G) with a series of lower triangular matrices (A_k) where k ranges from 0 to (N— 1) from the left which is equivalent to multiplying (G) with the inverse lower triangular Cholesky decomposition matrix (L ¹) from the left; and

provide a conjugated transposed matrix (Q^H) of the orthonormal matrix (Q) as the result of the product of the conjugated transposed matrix (H^H) of the input matrix (H) with a series of lower triangular matrices (A ) where k ranges from 0 to (N— 1) from the left which is equivalent to multiplying (H^H) with the inverse lower triangular Cholesky decomposition matrix (L - ) from the left.

2. Signal decoder (100) according to claim 1, wherein

the signal decoder (100) is configured to provide the orthonormal matrix (Q) from the conjugated transposed matrix (Q^H) of the orthonormal matrix (Q).

3. Signal decoder (100) according to claim 1 or 2, wherein

the signal decoder (100) is configured to perform a Gauss Jordan Elimination on the augmented matrix (K₀), wherein the Gauss Jordan Elimination is terminated when the N x (M + N ) augmented matrix (K_w) having as a left matrix the N x N matrix which equals

the upper triangular matrix (R) and as a right matrix the N x M matrix which equals the

conjugated transposed matrix (Q^H) is received and all the elements below the diagonal of the matrix are zeroed.

4. Signal decoder (100) according to claim 3, wherein

the signal decoder (100) is configured to scale each of the pivot rows by the reciprocal of the square root of the row’s pivot itself.

5. Signal decoder (100) according to one of the claims 1 to 4, wherein

the signal decoder (100) is configured to perform recursions indexed by k that ranges from 0 to N— 1 on the augmented matrix (K₀), with (N— k) x (N— k ) Hermitian matrices

(C_k)

Hermitian matrices D_k, and N x N matrices A

and the recursions on the N x N matrices

and the N x M matrices being

6. Signal decoder (100) according to one of the claims 1 to 5, wherein

the input matrix (H) is a channel matrix for a transmission channel of received data signals.

7. Signal decoder (100) according to one of the claims 1 to 6, wherein

the signal decoder (100) is part of a MIMO decoder.

8. Method for determining the QR decomposition of an M x N input matrix (H), with M ³ N , with a product of an M x N orthonormal matrix (Q) and an N x N upper triangular matrix (R), comprising calculating a conjugated transposed matrix (H ^/) of the input matrix (H) and to calculate a Hermitian matrix (G) by multiplying the conjugated transposed matrix (H^H) of the input matrix (H) and the input matrix (H) in the form G = H^HH;

providing an N x (M + N) augmented matrix (K₀) having as a left matrix the matrix (B_Q) which is initialized as the Hermitian matrix (G) and as a right matrix the matrix

multiplying the augmented matrix (K₀) from the left with a series of lower triangular matrices (A_k) where k ranges from 0 to (N— 1) which is equivalent to multiplying (K₀) with the inverse lower triangular Cholesky decomposition matrix (L^-1);

providing the upper triangular matrix (R) as the result of the product of the Hermitian matrix (G) with a series of lower triangular matrices (A_k) where k ranges from 0 to (N— 1) from the left which is equivalent to multiplying (G) with the inverse lower triangular Cholesky decomposition matrix (L ¹) from the left; and

providing a conjugated transposed matrix (Q^H) of the orthogonal matrix (Q) as the result of the product of the conjugated transposed matrix (H^H) of the input matrix (H) with a series of lower triangular matrices (A_k ) where k ranges from 0 to (N— 1) from the left which is equivalent to multiplying (H^H) with the inverse lower triangular Cholesky decomposition matrix (L ¹) from the left.

9. A computer program with a program code for performing the method according to claim 8 when the computer program runs on a computer or on the signal decoder (100) according to one of the claims 1 to 7.