**Series in
Signal and Information Processing, Vol. 22
edited by Hans-Andrea Loeliger**

Lukas Bolliger

**Digital Estimation of Continuous-Time Signals**

**Using Factor Graphs**

1.
Auflage/1^{st} edition 2012, XIV, 92 Seiten/pages, € 64,00.

ISBN 3-86628-432-6, 978-3-86628-432-6

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.

**Keywords**: continuous-time signal, Forney factor graph,
sigma-delta converter, digital signal, clock jitter, Kalman
smoother.

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order directly from: Hartung.Gorre@t-online.de

Reihe "Series

in Signal and Information Processing