Complex Burst QR Decomposition

QR decomposition for complex-valued matrices

Libraries:
Fixed-Point Designer HDL Support / Matrices and Linear Algebra / Matrix Factorizations

Description

The Complex Burst QR Decomposition block uses QR decomposition to compute R and C = Q'B, where QR = A, and A and B are complex-valued matrices. The least-squares solution to Ax = B is x = R\C. R is an upper triangular matrix and Q is an orthogonal matrix. To compute C = Q', set B to be the identity matrix.

When Regularization parameter is nonzero, the Complex Burst QR Decomposition block transforms $[\begin{matrix} λ I_{n} \\ A \end{matrix}]$ in-place to $R = Q' [\begin{matrix} λ I_{n} \\ A \end{matrix}]$ and $[\begin{matrix} 0_{n, p} \\ B \end{matrix}]$ in-place to $C = Q' [\begin{matrix} 0_{n, p} \\ B \end{matrix}]$ where λ is the regularization parameter, QR is the economy size QR decomposition of $[\begin{matrix} λ I_{n} \\ A \end{matrix}]$ , A is an m-by-n matrix, p is the number of columns in B, I_n = eye(n), and 0_n,p = zeros(n,p).

Examples

Implement Hardware-Efficient Complex Burst QR Decomposition

How to use the Complex Burst QR Decomposition block.

Open Script

Determine Fixed-Point Types for QR Decomposition

Use fixed.qrFixedpointTypes to determine fixed-point types for computation of QR decomposition.

Open Live Script

Ports

Input

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A(i,:) — Rows of matrix A
vector

Rows of matrix A, specified as a vector. A is an m-by-n matrix where m ≥ 2 and n ≥ 2. If B is single or double, A must be the same data type as B. If A is a fixed-point data type, A must be signed, use binary-point scaling, and have the same word length as B. Slope-bias representation is not supported for fixed-point data types.

Data Types: single | double | fixed point
Complex Number Support: Yes

B(i,:) — Rows of matrix B
vector

Rows of matrix B, specified as a vector. B is an m-by-p matrix where m ≥ 2. If A is single or double, B must be the same data type as A. If B is a fixed-point data type, B must be signed, use binary-point scaling, and have the same word length as A. Slope-bias representation is not supported for fixed-point data types.

Data Types: single | double | fixed point
Complex Number Support: Yes

validIn — Whether inputs are valid
`Boolean` scalar

Whether inputs are valid, specified as a Boolean scalar. This control signal indicates when the data from the A(i,:) and B(i,:) input ports are valid. When this value is 1 (true) and the value at ready is 1 (true), the block captures the values on the A(i,:) and B(i,:) input ports. When this value is 0 (false), the block ignores the input samples.

After sending a true validIn signal, there may be some delay before ready is set to false. To ensure all data is processed, you must wait until ready is set to false before sending another true validIn signal.

Data Types: Boolean

restart — Whether to clear internal states
`Boolean` scalar

Whether to clear internal states, specified as a Boolean scalar. When this value is 1 (true), the block stops the current calculation and clears all internal states. When this value is 0 (false), and the validIn value is 1 (true), the block begins a new subframe.

Data Types: Boolean

Output

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R(i,:) — Rows of matrix R
scalar | vector

Rows of the economy size QR decomposition matrix R, returned as a scalar or vector. R is an upper triangular matrix. The size of the matrix R is min(m,n)-by-n. R has the same data type as A.

Data Types: single | double | fixed point

C(i,:) — Rows of matrix C=Q'B
scalar | vector

Rows of the economy size QR decomposition matrix C=Q'B, returned as a scalar or vector. C has the same number of rows as R. C has the same data type as B.

Data Types: single | double | fixed point

validOut — Whether output data is valid
`Boolean` scalar

Whether the output data is valid, returned as a Boolean scalar. This control signal indicates when the data at output ports R(i,:) and C(i,:) is valid. When this value is 1 (true), the block has successfully computed the R and C matrices. When this value is 0 (false), the output data is not valid.

Use fixed.getQRDecompositionModel(A,B) to generate a template model containing a Complex Burst QR Decomposition block for complex-valued input matrices A and B.

Algorithms

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Choosing the Implementation Method

Systolic implementations prioritize speed of computations over space constraints, while burst implementations prioritize space constraints at the expense of speed of the operations. The following table illustrates the tradeoffs between the implementations available for matrix decompositions and solving systems of linear equations.

Implementation	Throughput	Latency	Area
Systolic	C	O(n)	O(mn²)
Partial-Systolic	C	O(m)	O(n²)
Partial-Systolic with Forgetting Factor	C	O(n)	O(n²)
Burst	O(n)	O(mn)	O(n)

Where C is a constant proportional to the word length of the data, m is the number of rows in matrix A, and n is the number of columns in matrix A.

For additional considerations in selecting a block for your application, see Choose a Block for HDL-Optimized Fixed-Point Matrix Operations.

AMBA AXI Handshake Process

This block uses the AMBA AXI handshake protocol [1]. The valid/ready handshake process is used to transfer data and control information. This two-way control mechanism allows both the manager and subordinate to control the rate at which information moves between manager and subordinate. A valid signal indicates when data is available. The ready signal indicates that the block can accept the data. Transfer of data occurs only when both the valid and ready signals are high.

Block Timing

The Burst QR Decomposition blocks accept and process A and B matrices row by row synchronously. After accepting m rows, the block outputs the R and C matrices row by row continuously. The matrices are output from the last row to the first row.

For example, assume that the input A and B matrices are 3-by-3. Additionally assume that validIn asserts before ready, meaning that the upstream data source is faster than the QR decomposition.

In the figure,

A1r1 is the first row of the first A matrix, R1r3 is the third row of the first R matrix, and so on.
validIn to ready — From a successful row input to the block being ready to accept the next row.
Last row validIn to validOut — From the last row input to the block starting to output the solution.
validOut to ready — From the block starting to output the solution to the block ready to accept the next matrix input.

The following table provides details of the timing for the Burst QR Decomposition blocks.

Block	`validIn` to `ready` (cycles)	Last Row `validIn` to `validOut` (cycles)	`validOut` to `ready` (cycles)
Real Burst QR Decomposition	(wl + 5)*min(m,n) + 2	(wl + 5)*min(m,n) + 2	min(m,n) + 1
Complex Burst QR Decomposition	(wl2 + 11)min(m,n) + 2	(wl2 + 11)min(m,n) + 2	min(m,n) + 1

In the table, m represents the number of rows in matrix A, and n is the number of columns in matrix A. wl represents the word length of the input data.

If the data type of A is double, then wl is 53.
If the data type of A is single, then wl is 24.
If the data types of A and B are fixed point, then wl is given by
```
max(A.WordLength + ~issigned(A), B.WordLength + ~issigned(B))
```

Hardware Resource Utilization

This block supports HDL code generation using the Simulink^® HDL Workflow Advisor. For an example, see HDL Code Generation and FPGA Synthesis from Simulink Model (HDL Coder) and Implement Digital Downconverter for FPGA (DSP HDL Toolbox).

This example data was generated by synthesizing the block on a Xilinx^® Zynq^® UltraScale™ + RFSoC ZCU111 evaluation board. The synthesis tool was Vivado^® v.2020.2 (win64).

The following parameters were used for synthesis.

Block parameters:
- m = 16
- n = 16
- p = 1
- Matrix A dimension: 16-by-16
- Matrix B dimension: 16-by-1
Input data type: sfix16_En14
Target frequency: 300 MHz

The following tables show the post place-and-route resource utilization results and timing summary, respectively.

Resource	Usage	Available	Utilization (%)
CLB LUTs	22713	425280	5.34
CLB Registers	22469	850560	2.64
DSPs	0	4272	0.00
Block RAM Tile	0	1080	0.00
URAM	0	80	0.00

	Value
Requirement	3.3333 ns
Data Path Delay	3.149 ns
Slack	0.166 ns
Clock Frequency	315.72 MHz

References

[1] "AMBA AXI and ACE Protocol Specification Version E." https://developer.arm.com/documentation/ihi0022/e/AMBA-AXI3-and-AXI4-Protocol-Specification/Single-Interface-Requirements/Basic-read-and-write-transactions/Handshake-process

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Slope-bias representation is not supported for fixed-point data types.

HDL Code Generation
Generate VHDL, Verilog and SystemVerilog code for FPGA and ASIC designs using HDL Coder™.

HDL Coder™ provides additional configuration options that affect HDL implementation and synthesized logic.

HDL Architecture

This block has one default HDL architecture.

HDL Block Properties

General
ConstrainedOutputPipeline	Number of registers to place at the outputs by moving existing delays within your design. Distributed pipelining does not redistribute these registers. The default is `0`. For more details, see ConstrainedOutputPipeline (HDL Coder).
InputPipeline	Number of input pipeline stages to insert in the generated code. Distributed pipelining and constrained output pipelining can move these registers. The default is `0`. For more details, see InputPipeline (HDL Coder).
OutputPipeline	Number of output pipeline stages to insert in the generated code. Distributed pipelining and constrained output pipelining can move these registers. The default is `0`. For more details, see OutputPipeline (HDL Coder).

Restrictions

Supports fixed-point data types only.

Version History

Introduced in R2019b

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R2022a: Support for Tikhonov regularization parameter

The Complex Burst QR Decomposition block now supports the Tikhonov Regularization parameter.

R2021a: Reduced HDL resource utilization

This block now has an improved algorithm to reduce resource utilization on hardware-constrained target platforms.

Complex Burst QR Decomposition

Description

Examples

Implement Hardware-Efficient Complex Burst QR Decomposition

Determine Fixed-Point Types for QR Decomposition

Ports

Input

A(i,:) — Rows of matrix A vector

B(i,:) — Rows of matrix B vector

validIn — Whether inputs are valid Boolean scalar

restart — Whether to clear internal states Boolean scalar

Output

R(i,:) — Rows of matrix R scalar | vector

C(i,:) — Rows of matrix C=Q'B scalar | vector

validOut — Whether output data is valid Boolean scalar

ready — Whether block is ready Boolean scalar

Parameters

Number of rows in matrices A and B — Number of rows in matrices A and B 4 (default) | positive integer-valued scalar

Programmatic Use

Number of columns in matrix A — Number of columns in matrix A 4 (default) | positive integer-valued scalar

Programmatic Use

Number of columns in matrix B — Number of columns in matrix B 1 (default) | positive integer-valued scalar

Programmatic Use

Regularization parameter — Regularization parameter 0 (default) | real nonnegative scalar

Programmatic Use

Tips

Algorithms

Choosing the Implementation Method

AMBA AXI Handshake Process

Block Timing

Hardware Resource Utilization

References

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using Simulink® Coder™.

HDL Code Generation Generate VHDL, Verilog and SystemVerilog code for FPGA and ASIC designs using HDL Coder™.

Version History

R2022a: Support for Tikhonov regularization parameter

R2021a: Reduced HDL resource utilization

See Also

Blocks

Functions

Topics

A(i,:) — Rows of matrix A
vector

B(i,:) — Rows of matrix B
vector

validIn — Whether inputs are valid
`Boolean` scalar

restart — Whether to clear internal states
`Boolean` scalar

R(i,:) — Rows of matrix R
scalar | vector

C(i,:) — Rows of matrix C=Q'B
scalar | vector

validOut — Whether output data is valid
`Boolean` scalar

ready — Whether block is ready
`Boolean` scalar

Number of rows in matrices A and B — Number of rows in matrices A and B
`4` (default) | positive integer-valued scalar

Number of columns in matrix A — Number of columns in matrix A
`4` (default) | positive integer-valued scalar

Number of columns in matrix B — Number of columns in matrix B
`1` (default) | positive integer-valued scalar

Regularization parameter — Regularization parameter
0 (default) | real nonnegative scalar

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

HDL Code Generation
Generate VHDL, Verilog and SystemVerilog code for FPGA and ASIC designs using HDL Coder™.