Complex Burst QR Decomposition

R2026b

QR decomposition for complex-valued matrices

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Complex Burst QR Decomposition block

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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 $$\left[\begin{array}{c}\lambda I_n\\ A\end{array}\right]$$ in-place to $$R=Q\text{'}\left[\begin{array}{c}\lambda I_n\\ A\end{array}\right]$$ and $$\left[\begin{array}{c}{0}_{n,p}\\ B\end{array}\right]$$ in-place to $$C=Q\text{'}\left[\begin{array}{c}{0}_{n,p}\\ B\end{array}\right]$$ where λ is the regularization parameter, QR is the economy size QR decomposition of $$\left[\begin{array}{c}\lambda I_n\\ A\end{array}\right]$$, 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

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

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.

Data Types: Boolean

ready — Whether block is ready

Boolean scalar

Whether the block is ready, returned as a Boolean scalar. This control signal indicates when the block is ready for new input data. When this value is 1 (true), and the validIn value is 1 (true), the block accepts input data in the next time step. When this value is 0 (false), the block ignores input data in the next time step.

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

Parameters

Main

Number of rows in matrices A and B — Number of rows in matrices A and B

4 (default) | positive integer-valued scalar

The number of rows in matrices A and B, specified as a positive integer-valued scalar.

Programmatic Use

Block Parameter: m

Type: character vector

Values: positive integer-valued scalar

Default: 4

Number of columns in matrix A — Number of columns in matrix A

4 (default) | positive integer-valued scalar

The number of columns in input matrix A, specified as a positive integer-valued scalar.

Programmatic Use

Block Parameter: n

Type: character vector

Values: positive integer-valued scalar

Default: 4

Number of columns in matrix B — Number of columns in matrix B

1 (default) | positive integer-valued scalar

The number of columns in input matrix B, specified as a positive integer-valued scalar.

Programmatic Use

Block Parameter: p

Type: character vector

Values: positive integer-valued scalar

Default: 1

Regularization parameter — Regularization parameter

0 (default) | real nonnegative scalar

Regularization parameter, specified as a nonnegative scalar. Small, positive values of the regularization parameter can improve the conditioning of the problem and reduce the variance of the estimates. While biased, the reduced variance of the estimate often results in a smaller mean squared error when compared to least-squares estimates.

Programmatic Use

Block Parameter: regularizationParameter

Type: character vector

Values: real nonnegative scalar

Default: 0

Implementation

Constant multiplication — Constant multiplication implementation

CSD (default) | Multiplier

Constant multiplication implementation, specified as one of these values:

• CSD — Canonical Signed Digit (CSD) technique, which uses only shift-add operations.

• Multiplier — Multiplication operation, *.

Tips

• Use this parameter to help balance use of different resources on hardware.

• For more information on the CSD technique, see Constant Multiplier Optimization to Reduce Area (HDL Coder).

Programmatic Use

To set the block parameter value programmatically, use the set_param function.

To get the block parameter value programmatically, use the get_param function.

Parameter:	ConstMultiplier
Values:	CSD (default) | Multiplier
Data Types:	string | char

CORDIC inverse gain data type — Data type of CORDIC inverse gain

Inherit: Same word length as input (default) | fixdt(0,16) | <data type expression>

Data type of inverse CORDIC gain, specified as Inherit: Same word length as input, fixdt(0,16), or as a user-specified data type expression.

Tips

• Use this parameter to fine-tune the quantization of the internal gain value and trade off between hardware resource utilization and numeric precision.

Programmatic Use

To set the block parameter value programmatically, use the set_param function.

To get the block parameter value programmatically, use the get_param function.

Parameter:	MultiplierDataTypeStr
Values:	Inherit: Same word length as input (default) | fixdt(0,16) | <data type expression>
Data Types:	string | char

Tips

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

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	High	O(nlog2(m))	O(mn2)
Partial-Systolic	Medium	O(mn)	O(n2)
Burst	Low	O(mn)	O(n)

Where m is the number of rows in matrix A and n is the number of columns in matrix A. Regardless of architecture, a larger word length results in lower throughput, larger latency, and larger area.

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	(wl*2 + 11)*min(m,n) + 2	(wl*2 + 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.2025.1 (glnx64).

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

• CORDIC inverse gain data type: Inherit: Same word length as input

Hardware resource utilization results are provided for both CSD and Multiplier implementations for comparison.

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

Constant multiplication: CSD

Resource	Usage	Available	Utilization (%)
CLB LUTs	15034	425280	3.54
CLB Registers	9969	850560	1.17
DSPs	0	4272	0.00
Block RAM Tile	0	1080	0.00
URAM	0	80	0.00

	Value
Requirement	3.3333 ns (300 MHz)
Data Path Delay	1.624 ns
Slack	1.692 ns
Clock Frequency	609.26 MHz

Constant multiplication: Multiplier

Resource	Usage	Available	Utilization (%)
CLB LUTs	11051	425280	2.60
CLB Registers	6709	850560	0.79
DSPs	68	4272	1.59
Block RAM Tile	0	1080	0.00
URAM	0	80	0.00

	Value
Requirement	3.3333 ns (300 MHz)
Data Path Delay	1.712 ns
Slack	1.692 ns
Clock Frequency	582.30 MHz

References

[1] "AMBA AXI and ACE Protocol Specification Version E." https://developer.arm.com/documentation/ihi0022/e/

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 in the design. Distributed pipelining does not redistribute these registers. The default value 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 value 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 value is 0. For more details, see OutputPipeline (HDL Coder).

Restrictions

Supports fixed-point data types only.

Version History

Introduced in R2019b

R2026b: QR Decomposition and Matrix Solve Blocks: Improved resource utilization and configurable multiplication method

Several improvements have been made to the Real Burst QR Decomposition, Complex Burst QR Decomposition, Real Burst Matrix Solve Using QR Decomposition, and Complex Burst Matrix Solve Using QR Decomposition blocks:

• HDL resource utilization has been further optimized to require fewer hardware resources. The reduction in resource utilization includes changes to both the algorithm and the implementation. Numeric outputs are not bit-exact with previous releases, but have the same noise floor.

• These blocks now allow you to choose the implementation of the constant multiplication between a multiplication operation or the previously existing canonical signed digit (CSD) technique, which uses only shift-add operations.

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.