Averaged Properties of the Residual Error in Sparse Signal Reconstruction

Greedy algorithms, like orthogonal matching pursuit (OMP), compressive sampling matching pursuit, and their modifications, are the popular methods of compressive sensing in sparse signal reconstruction. They are iterative in manner and often use a scaled value of the system noise energy as a reference threshold to decide when to stop the searching procedure. This letter presents an analytical analysis of the residual error and also of the difference of the successive residual errors in a general case of the greedy searching procedure, provided that the measurement error is an independent and identically distributed Gaussian stationary process. The obtained results show that the mean energy of the mentioned errors, just at the moment of perfect recovery of the signal support, depends only on the sparsity order and/or on the number of measurement samples. The results contributed to designing a practical stopping rule, which is a combination of those two errors. Numerical experiments with the new rule implemented in the OMP algorithm show that the number of executed iterations can be less spread around the true value of the signal sparsity in the mean sense. A quality of the sparse signal estimation is also preserved.