  This thesis consists of original contributions in the area of digital signal
processing. The reconstruction of signals sparse (highly concentrated) in
various transform domains is the primary problem analyzed in the thesis. The
considered domains include Fourier, discrete Hermite, one-dimensional and
two-dimensional discrete cosine transform, as well as various time-frequency
representations. Sparse signals are reconstructed using sparsity measures,
being, in fact, the measures of signal concentration in the considered domains.
The thesis analyzes the compressive sensing reconstruction algorithms and
introduces new approaches to the problem at hand. The missing samples influence
on analyzed transform domains is studied in detail, establishing the relations
with the general compressive sensing theory. This study provides new insights
on phenomena arising due to the reduced number of signal samples. The
theoretical contributions involve new exact mathematical expressions which
describe performance and outcomes of reconstruction algorithms, also including
the study of the influence of additive noise, sparsity level and the number of
available measurements on the reconstruction performance, exact expressions for
reconstruction errors and error probabilities. Parameter optimization of the
discrete Hermite transform is also studied, as well as the additive noise
influence on Hermite coefficients, resulting in new parameter optimization and
denoising algorithms. Additionally, an algorithm for the decomposition of
multivariate multicomponent signals is introduced, as well as an instantaneous
frequency estimation algorithm based on the Wigner distribution. Extensive
numerical examples and experiments with real and synthetic data validate the
presented theory and shed a new light on practical applications of the results.
