The two questions in science ``why" and ``how" are hereby answered in the context of statistical signal processing applied to vibration analysis and ultrasonic testing for fault detection and characterization in critical materials such as rolling bearings and thin layered media. Both materials are of interest in industrial processes. Therefore, assuring the best operating conditions on rolling bearings and product quality in thin layered materials is important.

The methods defended in this thesis are for retrieval of the impulsive signals arising from such equipments and materials, representing either faults or responses to an excitation. As the measurements collected via sensors usually consist of signals masked by some unknown systems and noise, retrieving the information-rich portion is often challenging. By exploiting the statistical characteristics due to their natural structure, a linear system is designed to recover the signals of interest in different scenarios. Suppressing the undesired components while enhancing the impulsive events by iteratively adapting a filter is the primary approach here. Signal recovery is accomplished by optimizing objectives (skewness and $\ell_1$-norm) quantifying the presumed characteristics, rising the question of objective surface topology and probability of ill convergence. To attack these, mathematical proofs, experimental evidences and comprehensive discussions are presented in the contributions each aiming to answer a specific question.

The aim in the theoretical study is to fill a gap in signal processing by providing analytical and numerical results especially on \emph{skewness} surface characteristics on a signal model (periodic impulses) build on harmonically related sinusoids. With understanding the inner workings and the conditions to suffice, the same approach is applied to different class of signals in ultrasonic testing, such as aperiodic finite energy signals (material impulse response) and a very short duration impulse as an excitation. A similar optimization approach aiming to enhance another attribute, \emph{sparseness}, is experimented numerically on the aforementioned signals as a case study. To summarize, two different objectives each quantifying a certain characteristic are optimized to recover signals carrying valuable information buried in noisy vibration and ultrasonic measurements.

Considering the fact that a research is qualified as successful if it creates more questions than it answers and lets ideas flourish creating scientific value, the presented work aims to achieve this in statistical signal processing. Analytical derivations assisted with experiments form the basis for observations, discussions and further questions to be studied and directed on similar phenomena arising from different sources in nature.