Sparse Representations of Random Signals

Sparse (fast) representations of deterministic signals have been well studied. Among other types there exists one called adaptive Fourier decomposition (AFD) for functions in analytic Hardy spaces. Through the Hardy space decomposition of the $L^2$ space the AFD algorithm also gives rise to sparse representations of signals of finite energy. To deal with multivariate signals the general Hilbert space context comes into play. The multivariate counterpart of AFD in general Hilbert spaces with a dictionary has been named pre-orthogonal AFD (POAFD). In the present study we generalize AFD and POAFD to random analytic signals through formulating stochastic analytic Hardy spaces and stochastic Hilbert spaces. To analyze random analytic signals we work on two models, both being called stochastic AFD, or SAFD in brief. The two models are respectively made for (i) those expressible as the sum of a deterministic signal and an error term (SAFDI); and for (ii) those from different sources obeying certain distributive law (SAFDII). In the later part of the paper we drop off the analyticity assumption and generalize the SAFDI and SAFDII to what we call stochastic Hilbert spaces with a dictionary. The generalized methods are named as stochastic pre-orthogonal adaptive Fourier decompositions, SPOAFDI and SPOAFDII. Like AFDs and POAFDs for deterministic signals, the developed stochastic POAFD algorithms offer powerful tools to approximate and thus to analyze random signals.

Amplitude spaces of mono-components from Blaschke products and an intrinsic multiresolution analysis

A mono-component is a real-variable and complex-valued analytic signal with nonnegative frequency components. The amplitude of an analytic signal is determined by its phase in a canonical amplitude-phase modulation. This paper investigates the amplitude spaces of analytic signals in terms of the Blaschke products with zeros in [Formula: see text]. It is proved that these amplitude spaces are invariant under the Hilbert transform and form a multiresolution analysis in the Hilbert space of signals with finite energy.

Design and implementation of the subsystem subject to emission of multicomponent high-frequency signals to ensure its reliability

Telecommunication systems, especially digital ones, are mostly known to be immune to noise given their extensive range of applications. This study aimed to investigate the methods and tools used for the analysis of multicomponent signals input to high-frequency digital subsystems, including the analysis of changes in its electrical behavior. This research mainly focuses on analyzing a high-frequency telecommunication subsystem, recording the results, investigating the system behavior against signals with different amplitudes and phases, detecting the received signals, and measuring the phase differences. The study extended the mono-component signals to multi-component signals and accurately extracted the statistical signal specifications using analytic signals in the time-frequency domain. To this end, a method was proposed based on the switch matrix to relate the different components and parameters, and also a mathematical model based on the state-space equations was employed to evaluate the nonlinear system modes. Given that the decoupling of measurement parameters is a problem to be tackled from multiple aspects, the costs and test durations were also taken into calculations in addition to considering all the detection methods for interference signals, reliability and time under test.