This example shows how to implement a hardware-efficient least-squares solution to the complex-valued matrix equation AX=B using the Complex Burst Matrix Solve Using QR Decomposition block.
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 = 50; % Number of rows in matrices A and B n = 10; % Number of columns in matrix A p = 1; % Number of columns in matrix B
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 has rank r.
rng('default') r = 3; % Rank of A [A,B] = fixed.example.complexRandomLeastSquaresMatrices(m,n,p,r);
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);
model = 'ComplexBurstQRMatrixSolveModel'; open_system(model);
The Data Handler subsystem in this model takes complex matrices A and B as inputs. The
ready port triggers the Data Handler. After sending a true
validIn signal, there may be some delay before
ready is set to false. When the Data Handler detects the leading edge of the
ready signal, the block sets
validIn to true and sends the next row of A and B. This protocol allows data to be sent whenever a leading edge of the
ready signal is detected, ensuring that all data is processed.
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 Burst Matrix Solve Using QR Decomposition block.
numSamples = 1; % Number of sample matrices fixed.example.setModelWorkspace(model,'A',A,'B',B,'m',m,'n',n,'p',p,... 'numSamples',numSamples,'OutputType',OutputType);
out = sim(model);
The Complex Burst 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
X = fixed.example.matrixSolveModelOutputToArray(out.X,n,p,numSamples);
To evaluate the accuracy of the Complex Burst Matrix Solve Using QR Decomposition block, compute the relative error.
relative_error = norm(double(A*X - B))/norm(double(B)) %#ok<NOPTS>
relative_error = 5.5331e-06