Digital estimation of continuous-time signals using factor graphs

A digital signal may represent noisy, quantized, discrete-time observations of a continuous-time signal. If the continuous-time signal is bandlimited to half the sampling rate, and the samples are taken without noise and with infinite precision, the continuous-time signal can be fully reconstructed from its digital samples. In this thesis we introduce a model of a continuous-time signal that is not bandlimited in the traditional sense, but its spectrum is shaped by the transfer function of a known, finite-order linear system/ filter. Among other applications the continuous-time system/ filter may represent the analog parts of a receiver, or it may represent a sensor that delivers a filtered version of the actual force applied. We use Forney factor graphs to represent the continuous-time signal and its noisy, quantized, discrete-time observations. The factor graph approach, which is well developed for discrete-time systems, is extended in this thesis for continuous-time systems. We use message passing algorithms in the factor graph to estimate both, the input and output signal of the continuous-time system/ filter at arbitrary time instances. The estimation of the output signal is essentially a Kalman smoother. The estimate of the input signal (which appears to be new) can be viewed as a generalized bandlimited reconstruction of a continuous-time signal. Two applications are described in this thesis. We show how our model can be extended to increase the quality of a digital signal that was sampled with clock jitter. We also introduce a general view on the widely used sigma-delta converter: an A/D converter may consist of any unstable linear system with feedback. The feedback ensures that the internal variables of the system stay in a predefined range. We use our proposed model to represent the components of such an “Unstable Linear Filter ADC” and we can then estimate its input signal given quantized, discrete-time observations of some internal variables of the system.