Implement Hardware-Efficient Complex Partial-Systolic Matrix Solve Using QR Decomposition

This example shows how to implement a hardware-efficient least-squares solution to the complex-valued matrix equation AX=B using the Complex Partial-Systolic Matrix Solve Using QR Decomposition block.

Define Matrix Dimensions

Specify the number of rows in matrices A and B, the number of columns in matrix A, and the number of columns in matrix B.

```m = 300; % Number of rows in matrices A and B n = 10; % Number of columns in matrix A p = 1; % Number of columns in matrix B ```

Generate Random Least-Squares Matrices

For this example, use the helper function `complexRandomLeastSquaresMatrices` to generate random matrices A and B for the least-squares problem AX=B. The matrices are generated such that the real and imaginary parts of the elements of A and B are between -1 and +1, and A is full rank.

```rng('default') [A,B] = fixed.example.complexRandomLeastSquaresMatrices(m,n,p); ```

Select Fixed-Point Data Types

Use the helper function `complexQRMatrixSolveFixedpointTypes` to select fixed-point data types for input matrices A and B, and output X such that there is a low probability of overflow during the computation.

The real and imaginary parts of the elements of A and B are between -1 and 1, so the maximum possible absolute value of any element is sqrt(2).

```max_abs_A = sqrt(2); % Upper bound on max(abs(A(:)) max_abs_B = sqrt(2); % Upper bound on max(abs(B(:)) precisionBits = 24; % Number of bits of precision T = fixed.complexQRMatrixSolveFixedpointTypes(m,n,max_abs_A,max_abs_B,precisionBits); A = cast(A,'like',T.A); B = cast(B,'like',T.B); OutputType = fixed.extractNumericType(T.X); ```

Open the Model

```model = 'ComplexPartialSystolicQRMatrixSolveModel'; open_system(model); ```

The Data Handler subsystem in this model takes complex matrices A and B as inputs. It sends rows of A and B to QR block using the AMBA AXI handshake protocol. The `validIn` 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 `validIn` and `ready` signals are high. You can set a delay between the feeding in rows of A and B in the Data Handler to emulate the processing time of the upstream block. `validIn` remains high when `rowDelay` is set to 0 because this indicates the Data Handler always has data available.

Set Variables in the Model Workspace

Use the helper function `setModelWorkspace` to add the variables defined above to the model workspace. These variables correspond to the block parameters for the Complex Partial-Systolic Matrix Solve Using QR Decomposition block.

```numSamples = 1; % Number of sample matrices rowDelay = 1; % Delay of clock cycles between feeding in rows of A and B fixed.example.setModelWorkspace(model,'A',A,'B',B,'m',m,'n',n,'p',p,... 'regularizationParameter',0,... 'numSamples',numSamples,'rowDelay',rowDelay,'OutputType',OutputType); ```

Simulate the Model

```out = sim(model); ```

Construct the Solution from the Output Data

The Complex Partial-Systolic Matrix Solve Using QR Decomposition block outputs data one row at a time. When a result row is output, the block sets `validOut` to true. The rows of X are output in the order they are computed, last row first, so you must reconstruct the data to interpret the results. To reconstruct the matrix X from the output data, use the helper function `matrixSolveModelOutputToArray`.

```X = fixed.example.matrixSolveModelOutputToArray(out.X,n,p,numSamples); ```

Verify the Accuracy of the Output

To evaluate the accuracy of the Complex Partial-Systolic Matrix Solve Using QR Decomposition block, compute the relative error.

```relative_error = norm(double(A*X - B))/norm(double(B)) %#ok<NOPTS> ```
```relative_error = 4.2851e-05 ```