On a Sparse Approximation of Compressible Signals

Abstract

Many physical phenomena can be modeled by compressible signals, i.e., the signals with rapidly declining sample amplitudes. Although all the samples are usually nonzero, due to practical reasons such signals are attempted to be approximated as sparse ones. Because sparsity of compressible signals cannot be unambiguously determined, a decision about a particular sparse representation is often a result of comparison between a residual error energy of a reconstruction algorithm and some quality measure. The paper explores a relation between mean square error (MSE) of the recovered signal and the residual error. A novel, practical solution that controls the sparse approximation quality using a target MSE value is the result of these considerations. The solution was tested in numerical experiments using orthogonal matching pursuit (OMP) algorithm as the signal reconstruction procedure. The obtained results show that the proposed quality metric provides fine control over the approximation process of the compressible signals in the mean sense even though it has not been directly designed for use in compressed sensing methods such as OMP.

Equalizer Parameters’ Adjustment Based on an Oversampled Channel Model for OFDM Modulation Systems

“…When these conditions are met, iterative compressive sensing algorithms such as OMP (Orthogonal Matching Pursuit) [14,15] and CoSaMP (Compressive Sampling Matching Pursuit) [16] perform well. However, if the synchronization is not perfect, the observed channel response is the result of convolving the actual channel response (which has a limited number of coefficients) with the baseband filter response (which is theoretically infinite) of the transmission system [17,18].…”

“…A reliable stopping criterion occurs when the residue power is less than the noise power; however, the receiver needs to estimate the noise level [27]. An alternative approach involves comparing the power of the residue in consecutive iterations and terminating when the value changes minimally [17,18].…”

Estimation of Compressible Channel Impulse Response for OFDM Modulated Transmissions

“…The choice of the length U requires a little bit more investigation. The research in [16] reveals that the best estimate of compressible signal in terms of MSE is its sparse approximation. This means that only large-amplitude components of the signal should be recovered, and their number is inversely related to noise power.…”

“…However, the channel compressibility nature has not been exploited there. The authors' method is mathematically explained using a relation between estimation and approximation errors-the idea with a detailed description presented in [16]. A simulation comparative analysis with other popular DFT-based methods shows quality improvement of the uncoded transmission in terms of Bit Error Rate (BER) and Mean Square Error (MSE) measures in low and middle ranges of signal-to-noise ratio (SNR).…”