Complex Partial-Systolic Q-less QR Decomposition with Forgetting Factor

Q-less QR decomposition for complex-valued matrices with infinite number of rows

• Library:

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

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Description

The Complex Partial-Systolic Q-less QR Decomposition with Forgetting Factor block uses QR decomposition to compute the economy size upper-triangular R factor of the QR decomposition A = QR, without computing Q. A is an infinitely tall complex-valued matrix representing streaming data.

When the regularization parameter is nonzero, the Complex Partial-Systolic Q-less QR Decomposition with Forgetting Factor block initializes the first upper-triangular factor R to λI_n before factoring in the rows of A, where λ is the regularization parameter and I_n = eye(n).

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 m ≥ n. 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

validIn — Whether inputs are valid

Boolean scalar

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

scalar | vector

Economy size QR decomposition matrix R, returned as a scalar or vector. R is an upper triangular matrix. R has the same data type as A.

Data Types: single | double | fixed point

validOut — Whether output data is valid

Boolean scalar

Whether the output data is valid, specified as a Boolean scalar. This control signal indicates when the data at output port R is valid. When this value is 1 (true), the block has successfully computed the matrix R. 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 columns in matrix A — Number of columns in input matrix A

4 (default) | positive integer-valued scalar

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

Forgetting factor — Forgetting factor applied after each row of matrix is factored

0.99 (default) | real positive scalar

Forgetting factor applied after each row of the matrix is factored, specified as a real positive scalar. The output is updated as each row of A is input indefinitely.

Programmatic Use

Block Parameter: forgettingFactor

Type: character vector

Values: positive integer-valued scalar

Default: 0.99

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

Model Examples

Implement Hardware-Efficient Complex Partial-Systolic Q-less QR with Forgetting Factor

How to use the Complex Partial-Systolic Q-less QR Decomposition with Forgetting Factor block.

Open Script

Determine Fixed-Point Types for Q-less QR Decomposition

Use fixed.qlessqrFixedpointTypes to determine fixed-point types for computation of Q-less QR decomposition.

Open Live Script

Algorithms

Q-less QR Decomposition with Forgetting Factor

The Complex Partial-Systolic Q-less QR Decomposition with Forgetting Factor block implements the following recursion to compute the upper-triangular factor R of continuously streaming n-by-1 row vectors A(k,:) using forgetting factor α. It's as if matrix A is infinitely tall. The forgetting factor in the range 0 < α < 1 prevents it from integrating without bound.

$$\begin{array}{c}{R}_0=\mathrm{zeros}(n,n)\\ \left[\sim ,R_1\right]=\mathrm{qr}\left(\left[\begin{array}{c}{R}_0\\ A\left(1,:\right)\end{array}\right],0\right)\\ R_1=\alpha R_1\\ \left[\sim ,R_2\right]=\mathrm{qr}\left(\left[\begin{array}{c}{R}_1\\ A\left(2,:\right)\end{array}\right],0\right)\\ R_2=\alpha R_2\\ ⋮\\ \left[\sim ,R_k\right]=\mathrm{qr}\left(\left[\left[\begin{array}{c}{R}_{k-1}\\ A\left(k,:\right)\end{array}\right]\right],0\right)\\ R_k=\alpha R_k\\ ⋮\end{array}$$

Q-less QR Decomposition with Forgetting Factor and Tikhonov Regularization

The upper-triangular factor R_k after processing the k^th input A(k,:) is computed using the following iteration.

$$\begin{array}{c}{R}_0=\lambda I_n\\ \left[~,R_1\right]=\mathrm{qr}\left(\left[\begin{array}{c}{R}_0\\ A\left(1,:\right)\end{array}\right],0\right)\\ R_1=\alpha R_1\\ \left[~,R_2\right]=\mathrm{qr}\left(\left[\begin{array}{c}{R}_1\\ A\left(2,:\right)\end{array}\right],0\right)\\ R_2=\alpha R_2\\ ⋮\\ \left[~,R_k\right]=\mathrm{qr}\left(\left[\begin{array}{c}{R}_{k-1}\\ A\left(k,:\right)\end{array}\right],0\right)\\ R_k=\alpha R_k\\ ⋮\end{array}$$

This is mathematically equivalent to computing the upper-triangular factor R_k of matrix A_k, defined as follows, though the block never actually creates A_k.

$$A_k=\left[\begin{array}{c}{\alpha }^k\lambda I_n\\ \left[\begin{array}{cccc}{\alpha }^k& & & \\ & {\alpha }^{k-1}& & \\ & & \ddots & \\ & & & \alpha \end{array}\right]A\left(1:k,:\right)\end{array}\right]$$

Forward and Backward Substitution

When an upper triangular factor is ready, then forward and backward substitution are computed with the current input B to produce output X.

$$X=R_k\backslash \left(R_k^{\text{'}}\backslash B\right)$$

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