Complex Partial-Systolic QR Decomposition
QR decomposition for complex-valued matrices
- Library:
Fixed-Point Designer / Matrices and Linear Algebra / Matrix Factorizations
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.
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.
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.
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) | non-negative scalar
Regularization parameter, specified as a non-negative 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:
k |
| Type: character vector |
| Values: positive integer-valued scalar |
Default:
0 |
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) | |
| 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) |
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™.
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.
This block has a single, 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.
Fixed-Point Conversion
Design and simulate fixed-point systems using Fixed-Point Designer™.
A and B must be signed, use binary-point scaling, and have the same word length. Slope-bias representation is not supported for fixed-point data types.
See Also
Blocks
- Complex Burst QR Decomposition | Complex Partial Systolic Q-less QR Decomposition | Real Partial Systolic QR Decomposition