Complex Partial-Systolic QR Decomposition

QR decomposition for complex-valued matrices

• Library:

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

• 

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Description

The Complex Partial-Systolic 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 Partial-Systolic 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).

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 — Matrix R

matrix

Economy-size QR decomposition matrix R, returned as a matrix. R is an upper triangular matrix. The size of matrix R is n-by-n. R has the same data type as A.

Data Types: single | double | fixed point

C — Matrix C=Q'B

matrix

Economy-size QR decomposition matrix C=Q'B, returned as a matrix 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 and C 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

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

4 (default) | positive integer-valued scalar

The number of rows in input 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 input 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 input 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) | 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: positive integer-valued scalar

Default: 0

Model Examples

Implement Hardware-Efficient Complex Partial-Systolic QR Decomposition

How to use the Complex Partial-Systolic 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

Algorithms

Choosing the Implementation Method

Partial-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	Ready	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(mn2)	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 Partial-Systolic QR Decomposition blocks accept and process A and B matrices row by row. After accepting m rows, the block outputs the R and C matrices as vectors. The partial-systolic implementation uses a pipelined structure, so the block can accept new matrix inputs before outputting the result of the current matrix.

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, R1 is 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.

The Partial-Systolic Q-less QR Decomposition blocks accept and process the matrix A row by row. After accepting m rows, the block outputs the R matrices as single vectors. The partial-systolic implementation uses a pipelined structure, so the block can accept new matrix inputs before outputting the result of the current matrix.

For example, assume that the input A matrix is 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, R1 is 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.

The following table provides details of the timing for the Partial-Systolic QR Decomposition blocks.

Block	validIn to ready (cycles)	Last Row validIn to validOut (cycles)
Real Partial-Systolic QR Decomposition	wl + 7	(wl + 6)*n + 6
Complex Partial-Systolic QR Decomposition	wl + 9	(wl + 7.5)*2*n + 6
Real Partial-Systolic Q-less QR Decomposition	wl + 7	(wl + 6)*n + 3
Complex Partial-Systolic Q-less QR Decomposition	wl + 9	(wl + 7.5)*2*n + 3

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 A.

• If the data type of A is fixed point, then wl is the word length.

• If the data type of A is double, then wl is 53.

• If the data type of A is single, then wl is 24.

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	319908	425280	75.22
CLB Registers	250839	850560	29.49
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	3.299 ns
Slack	0.016 ns
Clock Frequency	301.45 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 Verilog and VHDL 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 R2020b

R2021a: Reduced HDL resource utilization

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