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. 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	Sample block or example
Systolic	C	O(n)	O(mn2)	Implement Hardware-Efficient QR Decomposition Using CORDIC in a Systolic Array
Partial-Systolic	C	O(m)	O(n2)	Real Partial-Systolic QR Decomposition Real Partial-Systolic Matrix Solve Using QR Decomposition
Partial-Systolic with Forgetting Factor	C	O(n)	O(n2)	Fixed-Point HDL-Optimized Minimum-Variance Distortionless-Response (MVDR) Beamformer
Burst	O(n)	O(mn2)	O(n)	Real Burst QR Decomposition Real Burst Matrix Solve Using QR Decomposition

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.

Block Timing

The following table provides details on the timing for the QR decomposition blocks.

Block	validIn to ready (c cycles)	validIn to validOut (v cycles)
Real Partial-Systolic QR Decomposition	c = w + 8	v = c(m + n - 1)
Complex Partial-Systolic QR Decomposition	c = 2w + 15	v = c(m + n - 1)
Real Partial-Systolic Q-less QR Decomposition	c = w + 8	v = c(m + n - 1)
Complex Partial-Systolic Q-less QR Decomposition	c = 2w + 15	v = c(m + n - 1)
Real Partial-Systolic Q-less QR Decomposition with Forgetting Factor	c = w + 8	v = c(2n - 1)
Complex Partial-Systolic Q-less QR Decomposition with Forgetting Factor	c = 2w + 15	v = c(2n - 1)

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

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

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

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

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 a single, 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.

Fixed-Point Conversion
Design and simulate fixed-point systems using Fixed-Point Designer™.

Version History

Introduced in R2020b