Implementing modified QR decomposition in hardware

1. A circuit, configured to implement a QR decomposition of a matrix A into two matrices Q and R using a Modified Gram Schmidt (MGS) process, wherein Q represents an orthonormal basis that spans a column space of A, and wherein R is a triangular matrix, the circuit comprising:

an input;

a specified set of hardware components, coupled to the input, wherein the specified set of hardware components are dedicated to computing matrix Q; and

an output;

wherein the circuit is configured to:

receive a first set of inputs at the input, wherein the first set of inputs includes the matrix A and a scaling factor σ;

compute matrix Q via the specified set of hardware components dedicated to computing matrix Q, based on the first set of inputs using the MGS process;

scale the identity matrix by the scaling factor σ, thereby generating scaled identity matrix σI;

receive a second set of inputs to the specified set of hardware components dedicated to computing matrix Q, wherein the second set of inputs is different from the first set of inputs, wherein the second set of inputs comprises the scaled identity matrix;

compute scaled matrix σR −1 via the specified set of hardware components dedicated to computing matrix Q, based on the second set of inputs using the MGS process;

unscale the scaled matrix σR −1 , thereby computing matrix R −1 , wherein matrix R −1 is the inverse of matrix R;

output the matrix Q and/or matrix R −1 via the output; and

use the matrix Q and/or matrix R −1 to perform signal estimation on a noisy channel.

2. The circuit of claim 1 , wherein the circuit is further configured to:

compute matrix A −1 based on the matrix R −1 and the matrix Q.

3. The circuit of claim 1 , wherein the circuit is further configured to solve a system of linear equations specified by the expression Ax=b based on the matrix R −1 , the matrix Q, and vector b, and wherein x is a vector.

4. The circuit of claim 3 , wherein in being configured to solve the system of linear equations specified by the expression Ax=b, the circuit is inherently configured to determine an estimated signal s est with a minimum mean square error (MMSE) with respect to an observed signal y on a noisy channel, wherein to determine the estimated signal s est the circuit is configured to:

receive a third set of inputs at the input, wherein the third set of inputs includes a channel matrix H and the scaling factor σ;

scale the identity matrix by the scaling factor σ, thereby generating a scaled identity matrix σI;

extend the channel matrix H with the scaled identity matrix σI, thereby generating extended channel matrix B;

compute matrix Q via the specified set of hardware components dedicated to computing matrix Q, based on the extended matrix B using the MGS process;

divide matrix Q into matrix Q 1 and matrix Q 2 ;

store the matrix Q 1 and matrix Q 2 ;

scale matrix Q 2 by σ, thereby computing Q 2 /σ; and

compute the estimated signal s est based on Q 1 , Q 2 /σ, and the observed signal y; and

output the estimated signal s est .

5. The circuit of claim 1 , wherein the circuit comprises an application specific integrated circuit (ASIC).

6. The circuit of claim 1 , wherein the circuit comprises a programmable hardware element.

7. The circuit of claim 1 , wherein the circuit implements a systolic array to perform the MGS process.

8. The circuit of claim 7 , wherein the systolic array is implemented to perform the MGS process based on one or more of:

a specified throughput requirement; or

a specified footprint requirement.

9. A method, comprising:

providing a circuit, wherein the circuit is configured to implement a QR decomposition of a matrix A into two matrices Q and R using a Modified Gram Schmidt (MGS) process, wherein Q represents an orthonormal basis that spans a column space of A, wherein R is a triangular matrix, and wherein the circuit comprises a specified set of hardware components dedicated to computing matrix Q;

providing a first set of inputs to the circuit, wherein the first set of inputs includes the matrix A and a scaling factor σ;

computing matrix Q via the specified set of hardware components dedicated to computing matrix Q, based on the first set of inputs using the MGS process;

storing the matrix Q;

scaling the identity matrix by the scaling factor σ, thereby generating scaled identity matrix σI;

providing a second set of inputs to the specified set of hardware components dedicated to computing matrix Q, wherein the second set of inputs is different from the first set of inputs, wherein the second set of inputs comprises the scaled identity matrix;

computing scaled matrix σR −1 via the specified set of hardware components dedicated to computing matrix Q, based on the second set of inputs using the MGS process;

unscaling the scaled matrix σR −1 , thereby computing matrix R −1 , wherein matrix R −1 is the inverse of matrix R;

storing the matrix R −1 ; and

using the matrix Q and/or matrix R −1 to perform signal estimation on a noisy channel.

10. The method of claim 9 , wherein the circuit is further configured to compute matrix A −1 based on the matrix R −1 and the matrix Q, the method further comprising:

computing, via the circuit, matrix A −1 based on the matrix R −1 and the matrix Q.

11. The method of claim 9 , wherein the circuit is further configured to solve a system of linear equations specified by the expression Ax=b, wherein x and b are respective vectors, the method further comprising:

solving, via the circuit, the system of linear equations specified by the expression Ax=b to determine x, based on matrix R −1 , the matrix Q, and vector b, and wherein x is a vector.

12. The method of claim 11 , wherein in being configured to solve the system of linear equations specified by the expression Ax=b, the circuit is inherently configured to determine an estimated signal s est with a minimum mean square error (MMSE) with respect to an observed signal y on a noisy channel, the method further comprising:

receiving a third set of inputs at the input, wherein the third set of inputs includes a channel matrix H and the scaling factor σ;

scaling the identity matrix by the scaling factor σ, thereby generating a scaled identity matrix σI;

extending the channel matrix H with the scaled identity matrix σI, thereby generating extended channel matrix B;

computing matrix Q via the specified set of hardware components dedicated to computing matrix Q, based on the extended matrix B using the MGS process;

dividing matrix Q into matrix Q 1 and matrix Q 2 ;

storing the matrix Q 1 and matrix Q 2 ;

scaling matrix Q 2 by σ, thereby computing Q 2 /σ; and

computing the estimated signal s est based on Q 1 , Q 2 /σ, and the observed signal y; and

outputting the estimated signal s est .

13. The method of claim 9 , wherein the circuit comprises an application specific integrated circuit (ASIC).

14. The method of claim 9 , wherein the circuit comprises a programmable hardware element.

15. The method of claim 9 , wherein the circuit implements a systolic array to perform the MGS process.

16. The method of claim 15 , wherein the systolic array is implemented to perform the MGS process based on one or more of:

a specified throughput requirement; or

a specified footprint requirement.

17. A circuit, configured to implement a QR decomposition of a matrix A into two matrices Q and R using a Modified Gram Schmidt (MGS) process, wherein Q represents an orthonormal basis that spans a column space of A, and wherein R is a triangular matrix, the circuit comprising:

an input;

a specified set of hardware components, coupled to the input, wherein the specified set of hardware components are dedicated to computing matrix Q; and

an output;

wherein the circuit is configured to:

receive a first set of inputs at the input, wherein the first set of inputs includes the matrix A;

compute matrix Q via the specified set of hardware components dedicated to computing matrix Q, based on the first set of inputs using the MGS process;

receive a second set of inputs to the specified set of hardware components dedicated to computing matrix Q, wherein the second set of inputs is different from the first set of inputs, wherein the second set of inputs comprises the identity matrix;

compute matrix R −1 via the specified set of hardware components dedicated to computing matrix Q, based on the second set of inputs using the MGS process, wherein matrix R −1 is the inverse of matrix R;

output the matrix Q and/or matrix R −1 via the output; and

use the matrix Q and/or matrix R −1 to perform signal estimation on a noisy channel.

18. The circuit of claim 17 , wherein the circuit is further configured to:

compute matrix A −1 based on the matrix R −1 and the matrix Q.

19. The circuit of claim 17 , wherein the circuit is further configured to solve a system of linear equations specified by the expression Ax=b based on the matrix R −1 , the matrix Q, and vector b, and wherein x is a vector.

20. The circuit of claim 19 , wherein in being configured to solve the system of linear equations specified by the expression Ax=b, the circuit is inherently configured to determine an estimated signal s est with a minimum mean square error (MMSE) with respect to an observed signal y on a noisy channel, wherein to determine the estimated signal s est the circuit is configured to:

receive a third set of inputs at the input, wherein the third set of inputs includes a channel matrix H and a scaling factor σ;

scale the identity matrix by the scaling factor σ, thereby generating a scaled identity matrix σI;

extend the channel matrix H with the scaled identity matrix σI, thereby generating extended channel matrix B;

compute matrix Q via the specified set of hardware components dedicated to computing matrix Q, based on the extended matrix B using the MGS process;

divide matrix Q into matrix Q 1 and matrix Q 2 ;

store the matrix Q 1 and matrix Q 2 ;

scale matrix Q 2 by σ, thereby computing Q 2 /σ; and

compute the estimated signal s est based on Q 1 , Q 2 /σ, and the observed signal y; and

output the estimated signal s est .

21. A method, comprising:

providing a circuit, wherein the circuit is configured to implement a QR decomposition of a matrix A into two matrices Q and R using a Modified Gram Schmidt (MGS) process, wherein Q represents an orthonormal basis that spans a column space of A, wherein R is a triangular matrix, and wherein the circuit comprises a specified set of hardware components dedicated to computing matrix Q;

providing a first set of inputs to the circuit, wherein the first set of inputs includes the matrix A;

computing matrix Q via the specified set of hardware components dedicated to computing matrix Q, based on the first set of inputs using the MGS process;

storing the matrix Q;

providing a second set of inputs to the specified set of hardware components dedicated to computing matrix Q, wherein the second set of inputs is different from the first set of inputs, wherein the second set of inputs comprises the identity matrix;

computing matrix R −1 via the specified set of hardware components dedicated to computing matrix Q, based on the second set of inputs using the MGS process, wherein matrix R −1 is the inverse of matrix R;

storing the matrix R −1 ; and

use the matrix Q and/or matrix R −1 to perform signal estimation on a noisy channel.

22. The method of claim 21 , wherein the circuit is further configured to compute matrix A −1 based on the matrix R −1 and the matrix Q, the method further comprising:

computing, via the circuit, matrix A −1 based on the matrix R −1 and the matrix Q.

23. The method of claim 21 , wherein the circuit is further configured to solve a system of linear equations specified by the expression Ax=b, wherein x and b are respective vectors, the method further comprising:

solving, via the circuit, the system of linear equations specified by the expression Ax=b to determine x, based on matrix R −1 , the matrix Q, and vector b, and wherein x is a vector.

24. The method of claim 23 , wherein in being configured to solve the system of linear equations specified by the expression Ax=b, the circuit is inherently configured to determine an estimated signal s est with a minimum mean square error (MMSE) with respect to an observed signal y on a noisy channel, the method further comprising:

receiving a third set of inputs at the input, wherein the third set of inputs includes a channel matrix H and a scaling factor σ;

scaling the identity matrix by the scaling factor σ, thereby generating a scaled identity matrix σI;

extending the channel matrix H with the scaled identity matrix σ, thereby generating extended channel matrix B;

computing matrix Q via the specified set of hardware components dedicated to computing matrix Q, based on the extended matrix B using the MGS process;

dividing matrix Q into matrix Q 1 and matrix Q 2 ;

storing the matrix Q 1 and matrix Q 2 ;

scaling matrix Q 2 by σ, thereby computing Q 2 /σ; and

computing the estimated signal s est based on Q 1 , Q 2 /σ, and the observed signal y; and

outputting the estimated signal s est .