Complex Burst Q-less QR Decomposition
Libraries:
Fixed-Point Designer HDL Support /
Matrices and Linear Algebra /
Matrix Factorizations
Description
The Complex Burst Q-less QR Decomposition block uses QR decomposition to compute the economy size upper-triangular R factor of the QR decomposition A = QR, where A is a complex-valued matrix, without computing Q. The solution to A'Ax = B is x = R\R'\b.
When Regularization parameter is nonzero, the Complex Burst Q-less QR Decomposition block computes the upper-triangular factor R of the economy size QR decomposition of where λ is the regularization parameter.
Examples
Implement Hardware-Efficient Complex Burst Q-less QR Decomposition
How to use the Complex Burst Q-less QR Decomposition block.
Determine Fixed-Point Types for Q-less QR Decomposition
Use fixed.qlessqrFixedpointTypes
to determine fixed-point types for
computation of Q-less QR decomposition.
Ports
Input
A(i,:) — Rows of complex matrix A
vector
Rows of complex matrix A
, specified as a vector.
A is a m-by-n matrix where m ≥ 2 and n ≥ 2. If A is a fixed-point data type,
A must be signed and use binary-point scaling. 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(i,:) — Rows of upper-triangular 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. The output at R(i,:)
has
the same data type as the input at A(i,:)
.
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(i,:)
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 rows in matrix A — Number of rows in matrix A
4
(default) | positive integer-valued scalar
Number of rows in input matrix A, 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
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 |
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 |
Tips
Use fixed.getQlessQRDecompositionModel(A)
to generate a template model
containing a Complex Burst Q-less QR Decomposition block for complex-valued
input matrix A
.
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 | 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 Burst Q-less QR Decomposition blocks accept and process the matrix A row by row. After accepting m rows, the block outputs the matrix R row by row continuously. The matrix is output from the last row to the first row.
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,R1r3
is the third row of the first R matrix, and so on.validIn
toready
— From a successful row input to the block being ready to accept the next row.Last row
validIn
tovalidOut
— From the last row input to the block starting to output the solution.validOut
toready
— 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 Q-less QR Decomposition blocks.
Block | validIn to ready (cycles) | Last Row validIn to validOut
(cycles) | validOut to ready (cycles) |
---|---|---|---|
Real Burst Q-less QR Decomposition | (wl + 5)*min(m,n) + 2 | (wl + 5)*min(m,n) + 2 | min(m,n) + 1 |
Complex Burst Q-less 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 type of A is fixed point, then wl is the word length.
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
Matrix A dimension: 16-by-16
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 | 21137 | 425280 | 4.97 |
CLB Registers | 21157 | 850560 | 2.49 |
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 | 3.18 ns |
Slack | 0.134 ns |
Clock Frequency | 312.57 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.
This block has one default HDL architecture.
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
|
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
|
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
|
Supports fixed-point data types only.
Version History
Introduced in R2020aR2022a: Support for Tikhonov regularization parameter
The Complex Burst Q-less 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.
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