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

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

Since R2020b

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

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).

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

Ports

Input

A(i,:) — Rows of matrix A

vector

Rows of matrix A, specified as a vector. A is an infinitely tall matrix of streaming data. 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. The size of matrix R is n-by-n. 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

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

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 Partial-Systolic QR Decomposition with Forgetting Factor blocks accept and process the matrix A row by row. After accepting the first m rows, the block starts to output the R matrix as a single vector. From this point, for each row input, the block calculates a R matrix. 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 matrix A is 3-by-3. Additionally assume that validIn asserts before ready, meaning that the upstream data source is faster than the Q-less 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.

• validIn to validOut — From a successful row input to the block starting to output the corresponding solution.

The following table provides details of the timing for the Partial-Systolic Q-less QR Decomposition with Forgetting Factor blocks.

Block	validIn to ready (cycles)	validIn to validOut (cycles)
Real Partial-Systolic Q-less QR Decomposition with Forgetting Factor	wl + 7	(wl + 6)*n + 3
Complex Partial-Systolic Q-less QR Decomposition with Forgetting Factor	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).

In R2023a: The table below shows a summary of the resource utilization results.

This example data was generated by synthesizing the block on a Xilinx^® Zynq^®-7 ZC706 evaluation board (-2 speed grade).

The following parameters were used for synthesis.

• Block parameters:

  • m = 10

  • n = 10

  • p = 1

  • Matrix A dimension: 10-by-10

  • Matrix B dimension: 10-by-1

• Input data type: sfix18_En12

Resource	Usage
LUT	91496
LUTRAM	2000
Flip Flop	54609
BRAM	31

In R2022b: The following tables show the post place-and-route resource utilization results and timing summary, respectively.

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:

  • n = 16

  • p = 1

  • Matrix A dimension: inf-by-16

  • Matrix B dimension: 16-by-1

• Input data type: sfix16_En14

• Target frequency: 300 MHz

Resource	Usage	Available	Utilization (%)
CLB LUTs	327009	425280	76.89
CLB Registers	236476	850560	27.80
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 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 R2020b

R2023a: Smart unrolling for improved resource utilization

When you update the diagram, the loop which composes the partial-systolic pipeline is unrolled. This updated internal architecture removes dead operations in simulation and generated code, resulting in a significant decrease in the number of hardware resources required. This block simulates with clock and bit-true fidelity with respect to library versions of these blocks in previous releases.

Resource	R2022b	R2023a
LUT	162888	91496
LUTRAM	3620	2000
Flip Flop	100309	54609
BRAM	45	31

This example data was generated by synthesizing the block on a Xilinx Zynq-7 ZC706 evaluation board (-2 speed grade).

The following parameters were used for synthesis.

• Block parameters:

  • m = 10

  • n = 10

  • p = 1

  • Matrix A dimension: 10-by-10

  • Matrix B dimension: 10-by-1

• Input data type: sfix18_En12

R2022a: Support for Tikhonov regularization parameter

The Complex Partial-Systolic Q-less QR Decomposition with Forgetting Factor block 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.