Sketching and Stacking With Random Fork Based Exact Signal Recovery Under Sample Corruption

In this paper, we propose a new technique for exact recovery of missing data due to impulsive noise in time-domain sampled acoustic waves, named as sketching and stacking with random fork (SSRF). Existing methods recover the original signal from the corrupted sequence based on the correlation between each element of the signal or between the bases constituting the signal such as in the field of interpolation or compressive sensing. In addition, a partially corrupted signal is retrieved through statistical approaches using a large amount of pre-measured data in the machine learning domain. As a new approach, we hypothesize that if there is a novel method of re-sampling and processing, which can extract the intrinsic information of a corrupted signal, then the original signal can be recovered without information loss. The mathematical backgrounds in our study are twofold; first, a signal made up of overlapped
$k$
damped sinusoidal waves can be represented as the superposition of
$2k$
number of geometric sequences according to Euler’s formula; second, this superposed signal can be decomposed into individual geometric sequences by well-transformed
$2k+1$
informative matrices. The proposed method is to extract the parameters of
$2k$
geometric sequences by transforming the non-corrupted samples into
$2k+1$
informative matrices, and retrieve the corrupted samples by extrapolating the obtained parameters. We reveal the condition where SSRF perfectly fulfills the exact reconstruction, and verify the quality of reconstruction by comparing the results with conventional schemes.