Reconstruction and processing of bandlimited signals based on their discrete values

i Zusammenfassung Diese Arbeit befasst sich mit dem Wechselspiel zwischen analoger und diskreter Welt. Die Umwandlung von zeitdiskreten in zeitkontinuierliche Signale ist essenziell, da Informationen heutzutage fast ausschließlich digital verarbeitet werden, während reale Signale analog sind. Neben der bandbegrenzten Interpolation wird die Rekonstruktion von bandbegrenzen Signalen anhand ihrer Abtastwerte für verschiedene Signalräume untersucht. Es werden fundamentale Grenzen aufgezeigt und zahlreiche neue Resultate gewonnen, z.B. für nichtäquidistante Abtastung, Überabtastung und stochastische Prozesse. Für Anwendungen ist die Verarbeitung von Signalen durch lineare zeitinvariante Systeme von großer Bedeutung. Das klassische und distributionelle Konvergenzverhalten von verschiedenen Faltungs-Systemrepräsentationen wird analysiert. Im Fokus stehen Abtast-Systemrepräsentationen, die nur die Abtastwerte des Eingangssignals zur Berechnung des Systemausgangs verwenden. Abschließend wird der Einfluss von Quantisierung auf die Signalrekonstruktion und die Systemapproximation untersucht. Abstract This dissertation analyzes the interplay between the analog and the digital worlds. The conversion between discrete-time signals and continuous-time signals is important because today most information is processed digitally while real world signals are analog. Bandlimited interpolation is studied, as well as the reconstruction of bandlimited signals from their samples for different signal spaces. Fundamental limits are discovered and results are obtained in several directions, e.g., for non-equidistant sampling, oversampling, and stochastic processes. The processing of signals with linear time-invariant systems is important for applications. The classical and distributional convergence behavior of different convolution-type system representations is analyzed. Attention is paid to sampling-type representations that use only the samples of the input signal to compute the system output. Finally, the effects of quantization on the signal reconstruction and the system approximation are studied.This dissertation analyzes the interplay between the analog and the digital worlds. The conversion between discrete-time signals and continuous-time signals is important because today most information is processed digitally while real world signals are analog. Bandlimited interpolation is studied, as well as the reconstruction of bandlimited signals from their samples for different signal spaces. Fundamental limits are discovered and results are obtained in several directions, e.g., for non-equidistant sampling, oversampling, and stochastic processes. The processing of signals with linear time-invariant systems is important for applications. The classical and distributional convergence behavior of different convolution-type system representations is analyzed. Attention is paid to sampling-type representations that use only the samples of the input signal to compute the system output. Finally, the effects of quantization on the signal reconstruction and the system approximation are studied.