WO2019141352A1

WO2019141352A1 - Signal decoder and method for performing hermitian matrix inversion

Info

Publication number: WO2019141352A1
Application number: PCT/EP2018/051047
Authority: WO
Inventors: Alon SIDI; Doron Ezri; Shimon SHILO; Yaron Ben-Arie
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 a signal decoder (100) for performing matrix inversion of an N x N Hermitian input matrix (G), wherein the signal decoder (100) is configured to provide an N x 2N augmented matrix (K₀) having as a left matrix the matrix (B^L _O) and as a right matrix the matrix (B^R _O); perform a recursive sequence of operations indexed by k that ranges from 0 to (N — 1) each of which includes a row and column permutation operation followed by a multiplication from the left of a lower triangular matrix (A_K) on the matrix (B^L _O) and the matrix (B^R _O); provide a lower triangular matrix (L); provide a conjugated transposed upper triangular matrix (L^H) of the lower triangular matrix (L); calculate a permuted, inverted input matrix (G^_1) by multiplying the conjugated transposed upper triangular matrix (L^H) and the lower triangular matrix (L); and provide the inverse input matrix (G^_1).

Description

Signal decoder and method for performing Hermitian matrix inversion

TECHNICAL FIELD

The present invention relates to a signal decoder for performing an inversion of a Hermitian matrix, a method for performing an inversion of a Hermitian matrix 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 it is often required to perform an inversion of Hermitian (or symmetrical) matrices. For instance, in a ZF/MMSE (Minimum Mean Square Error) equalizer on the receiver side, or a ZF (Zero- forcing) pre-coder on the transmitter side.

Hermitian Matrix Inversion (HMI) is widely used in wireless communication. Hermitian Matrix Inversion is the basic building block of linear MIMO equalizers such as the Zero Forcing (ZF) equalizer and the Minimum Mean Square Error (MMSE) equalizer. Further, Hermitian Matrix Inversion is also heavily used in MIMO for null-steering, for instance, the ZF pre-coder.

A Hermitian matrix or self-adjoint matrix is a complex square matrix that is equal to its own conjugate transpose. This implies that the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j.

Many wireless communication standards including the 802.11 family of standards use Hermitian Matrix Inversion. Recent versions of 802.11, including 802.1 lax, support high- order MIMO schemes, up to 4x4 (and even higher dimensions), with a very large number of frequency tones. This requires the development of up to 4 x 4 (or higher dimensions) HMI units with reasonable complexity in terms of latency and required chip area.

The most common implementation of the Hermitian Matrix Inversion is based on the Cholesky decomposition of a Hermitian matrix G, i.e. G = LL^H such that L is a lower triangular matrix with real values on the diagonal. After L is calculated, its inverse L^-1 is calculated and is followed by the computation of G^-1 via the multiplication G^-1 = L^-HL^-1, where L^-H is the conjugated transposed upper triangular matrix of the lower triangular matrix L^-1. In order to differentiate between scalar values and matrices and vectors, matrices and vectors are presented in the formulas in bold font.

The Cholesky decomposition is known to be the most efficient method that calculates L when a scalar processor is used. However, in modem applications, it is desired to speed up the calculation via parallelism, i.e. a number of multipliers that is as the size of the matrix, and in this case the Cholesky decomposition becomes inefficient and very cumbersome in terms of hardware implementation.

Furthermore, for some applications, it may be desired to perform a pivoted calculation in order to achieve higher accuracy and in this case also the Cholesky decomposition becomes very cumbersome when parallelism is required.

SUMMARY

In view of the above-mentioned problems and disadvantages, the present invention aims to improve for performing matrix inversion of an N x N Hermitian input matrix. The present invention has thereby the object to provide signal decoder configured for performing matrix inversion and a method for performing matrix inversion, which operates with better performance compared to the corresponding solutions known in the art.

The objective 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.

A first aspect of the present invention provides a signal decoder for performing matrix inversion of an N x N Hermitian input matrix G, wherein the signal decoder is configured to provide an N x 2 N augmented matrix K₀ having as a left matrix the matrix which is

initialized as the input matrix G and as a right matrix the matrix

which is initialized as an N x N identity matrix I_N; to perform a recursive sequence of operations indexed by k that ranges from 0 to N— 1 each of which includes a row and column permutation operation followed by a multiplication from the left of a lower triangular matrix A_k on the matrix and the matrix ; to provide a lower triangular matrix

corresponding to the matrix to

provide a conjugated transposed upper triangular matrix

of the lower triangular matrix L; to calculate a permuted, inverted input matrix by multiplying the conjugated transposed upper triangular matrix

and the lower triangular matrix

in the form

and to provide the inverse input matrix G^-1 by permuting the permuted, inverted input matrix

.

The present invention has the advantage of providing an innovative solution, which has relatively low complexity while keeping high accuracy. Due to the nature of the process, it is very easily pipelined. For comparison, a Cholesky decomposition based solution would require complicated addressing and additional memories in order to achieve a much lower utilization of its multipliers and would be much harder to pipeline.

The fundamental idea behind the proposed solution is to use the Gauss Jordan Elimination (GJE) on an input matrix G with a unique scaling of the pivot rows and stop the process when all the elements below its diagonal are zeroed. The process is performed in a column by column manner starting from the leftmost side of G and prior to the processing of each column, a pivoting step is performed which greatly increases the accuracy of the computation. In order to receive G^-1, another permutation step is performed. This process is dubbed ‘Pivoted Gauss Jordan Elimination Inversion of a Hermitian Matrix’ (PGJEIH) and its advantages are in its implementation simplicity, full utilization of its hardware (compared to prior art solutions), reduced silicon area and ease of scalability.

The invention herein describes a hardware, which performs signal processing. The signal processing may be defined by two steps.

In step one a process which is based on the GJE is performed with the following major differences. 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, are scaled by the reciprocal of the square root of the pivot itself. Further, in contrast to GJE, in this step a pivoting step is performed prior to zeroing the elements of every processed column, which greatly increases the stability and accuracy of the process. At the end of the process [G|I_W] transform into [l"|L] where L is the inverse of the Cholesky decomposition of G which is a permuted version of G. In step two

is calculated and then permutated in order to receive G - .

In an implementation form of the first aspect, the signal decoder is configured to provide an equivalent permutation matrix

of all recursive permutation matrices of the row and column permutation operations, and is further configured to provide the inverse input matrix G^-1 by multiplying the permuted, inverted input matrix from the left with the equivalent

permutation matrix

and from the right with the transposed equivalent permutation matrix

In an implementation form of the first aspect, the signal decoder is configured to perform a pivoted Gauss Jordan Elimination process on the augmented matrix K₀, wherein the Gauss Jordan Elimination is terminated when the N x 2 N augmented matrix K_N having as a left matrix the N x N matrix

which equals the upper triangular matrix

P and as a right matrix the N x N matrix

which equals the lower triangular matrix

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 range from 0 to N— 1 on the augmented matrix K₀, with the (N— k) x (N— k ) Hermitian matrix C_k

the (N— k) x lvector d_k

the scalar value q_k

the permutation matrix P_fe which is the l_N matrix with its q_k’s row interchanged by its k’s row in the case pivoting is required or when pivoting is not required P_k is simply the I_N matrix, the N x N matrices

and

^and

the (N— k) x (N— k) Hermitian matrix

the N x N matrix A_k

the N x N matrix

and the recursions on the N x N matrices

and the N x N matrices

being

Such recursions have the advantage that the permutations are performed by multiplexing registers and not by real multiplications. The A_k 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 may be a transmission channel of received data signals. The input matrix G can be easily provided based on the 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 performing Hermitian matrix inversion of an N x N Hermitian input matrix G, comprising providing an N x 2 N augmented matrix K₀ having as a left matrix the matrix

which is initialized as the input matrix G and as a right matrix the matrix

which is initialized as an N x N identity matrix I_N; performing a recursive sequence of operations indexed by k that ranges from 0 to N— 1 each of which includes a row and column permutation operation followed by multiplication from the left of a lower triangular matrix A_k on the matrix

and the matrix providing a

lower triangular matrix

corresponding to the matrix

providing a conjugated transposed upper triangular matrix

of the lower triangular matrix

calculating a permuted, inverted input matrix by multiplying the conjugated transposed upper triangular matrix

and the lower triangular matrix L in the form

and providing the inverse input matrix G^-1 by permuting the permuted, inverted input matrix

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 performing matrix inversion of an N x N Hermitian input matrix G.

Fig. 2 shows a flowchart of an exemplary implementation of a method for performing matrix inversion of an N x N Hermitian input matrix G.

DETAIFED 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 110, for example at base stations, user equipment or the like and may be part of a MIMO decoder. The signal decoder 100 is configured for performing matrix inversion of an N x N Hermitian input matrix G. The Hermitian input matrix G may be a channel matrix or may be derived from a channel matrix H being a transmission channel of received data signals. 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 performs a Pivoted Gauss Jordan Elimination on the Hermitian input matrix G. Thereby, a lower triangular matrix

corresponding to the matrix and a permutation matrix are provided. Details of the Pivoted

Gauss Jordan Elimination are explained later. In a second block 130 of the signal decoder 100, the signal decoder 100 calculates a conjugated transposed upper triangular matrix

of the lower triangular matrix

In a third block 140 of the signal decoder 100, the signal decoder 100 calculates a permuted, inverted input matrix

by multiplying the conjugated transposed upper triangular matrix

and the lower triangular matrix

in the form

In a fourth block 150 of the signal decoder 100, the signal decoder 100 provides the inverse input matrix G^-1 by permuting the permuted, inverted input matrix

in the form G^-1 =

The process can be named Pivoted Gauss Jordan Elimination Inversion of a Hermitian Matrix (PGJEIH). The blocks 120, 130, 140 and 150 can be implemented in hardware and/or software.

Figure 2 shows steps of an exemplary Pivoted Gauss Jordan Elimination Inversion of a Hermitian Matrix (PGJEIH) method, which may be performed by the signal decoder 100.

In a first step 200 an N x 2 N augmented matrix K₀ is provided having as a left matrix the matrix which is initialized as the input matrix G and as a right matrix the matrix which

is initialized as an N x N identity matrix I_N.

In a second step 210 a recursive sequence of operations indexed by k that ranges from 0 to N — 1 is performed each of which includes a row and column permutation operation followed by multiplication from the left of a lower triangular matrix A_k on the matrix and the

matrix

In a third step 220 a lower triangular matrix L corresponding to the matrix is provided.

In a fourth step 230 a conjugated transposed upper triangular matrix

of the lower triangular matrix is provided.

In a fifth step 240 a permuted, inverted input matrix

is calculated by multiplying the conjugated transposed upper triangular matrix

and the lower triangular matrix

in the form

In a sixth step 250 the inverse input matrix G^-1 is provided by permuting the permuted, inverted input matrix

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

The proposed invention is based on performing row and column permutation operations on matrices and which are initialized as and where G is the N x N input

matrix to be inverted and l_N is the N x N identity matrix. This permutation can be

mathematically expressed by a permutation matrix P_k, which multiplies both B_k and B_k from the left and from the right.

In the following, a more detailed description of the Pivoted Gauss Jordan Elimination

Inversion of a Hermitian Matrix (PGJEIH) method is given. In a first step of the processing the matrix C_k is defined as

which is the (N— k) x (N— k ) lower right sub matrix of B_k and define

Then the permutation matrix P_k is simply the I_w matrix with its q_k’ s row interchanged by its k’ s row in the case pivoting is required or when pivoting is not required P_k is simply the l_N matrix for k that ranges from 0 to N— 1, hence P_k = P_k .

Define the permutated versions of B_k and B_k as

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

And define the N x N matrix A_{k -}

Then with the definitions (1-7), the recursion is finally stated as

The practical interpretation of equations (8) and (9) is that the permutations are performed by multiplexing registers and not by real multiplications. The A_k multiplications are implemented as simple row multiply or multiply and accumulate operations (both are implemented by parallel multiply accumulate units). Specifically, for the pivot rows:

and -

for the rows below the pivot lines, where we 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.

In a second step, the signal decoding or precoding is completed. The second step may be termed as the computation of PGJEIH from PGJEH.

In order to receive G ¹ it is defined

and compute via matrix multiplication

where is an equivalent permutation matrix of all the recursive permutation matrices.

Using equations (17-19) it is finally stated

That is, once has been calculated, G^-1 is achieved by simply permuting according

to On the other hand if pivoting is not required then by setting P_k = l_N for k that ranges from 0 to N— 1 no permutations induced by P_k are required during the PGJEH process and also the permutation in (20) can be discarded since G^-1 would equal

Note that equations (1-20) simply explain the required steps of the proposed solution, a mathematical justification of the algorithm is included below.

It is easy to notice that equations (10-11) and equations (14-15) imply a vector by scalar multiplication, whereas (12-13) imply a vector by scalar multiplication and accumulation (MAC). Since the same coefficients, i.e. multiply rows on both and , it

means that for a N x N matrix, and a set of N parallel MACs, this can be used to perform a multiplication of only the relevant elements of the same row of

and in a single cycle

and thus achieve 100% utilization of the MACs. The following diagrams illustrate a processing of a 4 x 4 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. Note that the markings‘f and‘c’ in the diagrams below stand for final value and current value respectively.

Permute both G and I by P₀ from the left and right.

Multiply by and store as a 1 x 4 intermediate result r^T .

Multiply by and store in-place. Note that multiplying by '1' is actually a placement

and not a multiplication.

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. Permute both and

by P₁ from the left and right.

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

as a 1 x 4 intermediate result r^T .

Multiply by and store in-place.

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

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

by P₂ from the left and right.

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

as a 1 x 4 intermediate result r^T .

Multiply by and store in-place.

Subtract from this row the multiplication Bf [3,2] r^T and store in-place. Permute both B| and B ? by P₃ from the left and right.

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

in-place. Note that no permutation was required here.

As the limit of k is reached with the value 3, the recursion is finished. Please note that the upper triangular matrix directly corresponds to and can be extracted from the matrix

and that the lower triangular matrix L directly corresponds to and can be extracted from the matrix.

In the following, a mathematical justification of the PGJEH process is given.

Using equations (1-7), one can write

(Al)

(A2)

(A3)

In that case

(A4)

where (h is also a Hermitian matrix.

Using this step as an example, one can therefore generalize the PGJEH procedure by the following recursion:

Define the upper triangular matrix

(A5)

and the lower triangular matrix L

(A6)

Where it was defined:

(A7)

In view of the above it is easy to see that

(A8)

From equation (A8) one can derive

(A9)

Where it was defined

(A10) Finally, from equations (A8-A10) one can state

(Al l) i.e. is actually the Cholesky decomposition of However, if on the other hand, one choose to not perform any pivoting, i.e., then L is the Cholesky

decomposition of G. Therefore, it is obvious from equations (A5-A6) that the PGJEH algorithm operates

from the left and from the right on G and I simultaneously and produces and respectively.

From equations (A8), (A10) and (Al 1) it is obvious that (A 12)

Finally, the computation of G^-1 can be finished by (A13)

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 performing matrix inversion of an N x N Hermitian input matrix (G), wherein the signal decoder (100) is configured to

provide an N x 2 N augmented matrix (K₀) having as a left matrix the matrix

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

as an N x N identity matrix (I_N);

perform a recursive sequence of operations indexed by k that ranges from 0 to (N— 1) each of which includes a row and column permutation operation followed by a multiplication from the left of a lower triangular matrix (A_k ) on the matrix and the matrix

provide a lower triangular matrix corresponding to the matrix (

provide a conjugated transposed upper triangular matrix of the lower triangular

matrix

calculate a permuted, inverted input matrix by multiplying the conjugated

transposed upper triangular matrix

and the lower triangular matrix in the form

and

provide the inverse input matrix (G^-1) by permuting the permuted, inverted input matrix

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

the signal decoder (100) is configured to provide an equivalent permutation matrix

of all recursive permutation matrices of the row and column permutation operations, and

is further configured to provide the inverse input matrix (G^-1) by multiplying the permuted, inverted input matrix from the left with the equivalent permutation matrix

and from the right with the transposed equivalent permutation matrix

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

the signal decoder (100) is configured to perform a pivoted Gauss Jordan Elimination process on the augmented matrix (K₀), wherein the Gauss Jordan Elimination is terminated when the N x 2 N augmented matrix (K_w) having as a left matrix the N x N matrix

which equals the upper triangular matrix and as a right matrix the N x N matrix which

equals the lower triangular matrix

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 the (N— k) x (N— k ) Hermitian matrix

(C_fc)

the (N— k) x lvector (d_k)

the scalar value (q_k)

the permutation matrix (P_k) which is the I_w matrix with its q_k’s row interchanged by its k’s row in the case pivoting is required or when pivoting is not required P_k is simply the I_w matrix for k that ranges from 0 to N— 1, the N x N matrices

and

the N x N matrix (A_k)

the N x N matrix

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

B

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

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.

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

the signal decoder (100) is a part of a MIMO decoder and/or a MIMO precoder.

8. Method for performing matrix inversion of an N x N Hermitian input matrix (G), comprising

providing an N x 2 N augmented matrix (K₀) having as a left matrix the matrix

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

which is initialized as an N x N identity matrix (I_N);

performing a recursive sequence of operations indexed by k that ranges from 0 to (N—

1) each of which includes a row and column permutation operation followed by multiplication from the left of a lower triangular matrix (A_k ) on the matrix (B_Q) and the matrix

providing a lower triangular matrix (L) corresponding to the matrix

providing a conjugated transposed upper triangular matrix of the lower triangular

matrix (L);

calculating a permuted, inverted input matrix

by multiplying the conjugated transposed upper triangular matrix and the lower triangular matrix in the form

L^HL; and

providing the inverse input matrix (G^-1) by permuting the permuted, inverted input matrix (G^-1).

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.