ompdecomp

Decompose signal using orthogonal matching pursuit

Syntax

[coeff,dictatom,atomidx,errnorm] = ompdecomp(X,dict)

[coeff,dictatom,atomidx,errnorm] = ompdecomp(X,dict,'MaxSparsity',nm)

[coeff,dictatom,atomidx,errnorm] = ompdecomp(X,dict,'NormWeight',wts)

Description

[coeff,dictatom,atomidx,errnorm] = ompdecomp(X,dict) computes the decomposition matrices coeff and dictatom of the signal X. The product of the decomposition matrices, dictatom x coeff, approximates X. The atoms in dictatom are selected from dict. atomidx are the indices in dict corresponding to dictatom. errnorm is the decomposition error. The decomposition is based on an orthogonal matching pursuit (OMP) algorithm that minimizes the Frobenius norm ||X – dictatom x coeff||.

[coeff,dictatom,atomidx,errnorm] = ompdecomp(X,dict,'MaxSparsity',nm) also specifies the maximum sparsity nm.

[coeff,dictatom,atomidx,errnorm] = ompdecomp(X,dict,'NormWeight',wts) minimizes the weighted Frobenius norm ||wts^1/2(X – dictatom x coeff)|| using the weights wts.

Examples

Decompose ULA Beamsteering Weights

Given a set of optimal, full-digital, beamforming weights for an 8-element uniform linear array, decompose the weights into a product of analog and digital beamforming weights. Assume two RF chains. Show that the combined weights achieve a performance similar to the optimal weights.

Specify the optimal, full-digital, beamforming weights.

wopt = steervec((0:7)*0.5,[20 -40]);

Create a dictionary of steering vectors.

stvdict = steervec((0:7)*0.5,-90:90);

Perform the decomposition using OMP. Set the maximum sparsity to two.

[wbb,wrf,wdictidx,normerr] = ompdecomp(wopt,stvdict,'MaxSparsity',2);

Compare the beam patterns derived from the optimal weights and the hybrid weights. The plot shows that the decomposition of wopt into wrf and wbb is almost exact.

plot(-90:90,abs(sum(wopt'*stvdict)),'-', ...
    -90:90,abs(sum((wrf*wbb)'*stvdict)),'--','LineWidth',2)
xlabel('Angles (degrees)')
ylabel('Amplitude')
legend('Optimal','Hybrid')

Input Arguments

X — Input data

complex-valued N-by-N_c matrix

Input data to be decomposed, specified as a complex-valued N-by-N_c matrix.

Data Types: double | single
Complex Number Support: Yes

dict — Dictionary of atoms

complex-valued matrix

Dictionary of atoms, specified as a complex-valued matrix. The function uses a subset of atoms from the dictionary to construct the data.

Data Types: double
Complex Number Support: Yes

nm — Maximum sparsity

1 (default) | positive integer

Maximum sparsity of the decomposition, specified as a positive integer. The decomposition stops when the sparsity of nm is achieved.

Example: 5

Dependencies

Use this argument with the syntax specifying 'MaxSparsity'.

Data Types: double | single

wts — Norm weights

N-by-N identity matrix (default) | complex-valued N-by-N matrix

Norm weights used by OMP to minimize the weighted Frobenius norm of ||wts^1/2 x (X – dictatom x coeff)||, specified as a complex-valued N-by-N matrix.

Example: 5

Dependencies

Use this argument with the syntax specifying 'NormWeight'.

Data Types: double | single
Complex Number Support: Yes

Output Arguments

coeff — Coefficients of basis atoms

N_s-by-N_c matrix

Coefficients of basis atoms, returned as an N_s-by-N_c matrix. The rows represent the coefficients for the corresponding atoms in dictatom. N_s represents the number of atoms selected from the dictionary and is a measure of signal sparsity.

Data Types: double | single
Complex Number Support: Yes

dictatom — Signal basis atoms

N-by-N_s matrix

Signal basis atoms, returned as an N-by-N_s matrix. The columns are the atoms forming the basis of the signal. These atoms are a subset of the dictionary specified in dict. N_s represents the number of selected atoms and is a measure of signal sparsity.

Data Types: double | single
Complex Number Support: Yes

atomidx — Indices of selected atoms

integer-valued length-N_s row vector

Indices of the atoms selected from the dictionary dict, returned as a length-N_s row vector where dict(:,atomidx) = dictatom.

Data Types: double | single

errnorm — Norm of decomposition error

0 (default) | nonnegative scalar

Norm of the decomposition error, returned as a nonnegative scalar.

Data Types: double | single

More About

Hybrid Beamforming Weights

In the context of hybrid beamforming, the coeff argument represents digital weights. dictatom represents analog weights and dict is a collection of steering vectors that can be used as analog weights.

References

[1] Ayach, Omar El et al. "Spatially Sparse Precoding in Millimeter Wave MIMO Systems" IEEE Trans on Wireless Communications. Vol. 13, No. 3, March 2014.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced in R2019b