Given a time series measured (or generated) by a known or an unknown dynami
cal system we address a series of problems which can be considered as advan
ced signal processing tasks, namely: (1) section-wise approximation of the
measured signal by pieces of trajectories from a chosen nonlinear dynamical
system (model); (2) signal restoration when the measured signal has been c
orrupted e.g. by quantization; (3) signal coding and compression. These tas
ks can be addressed using a new approach to the shadowing problem based on
nonlinear observability problem. Its goal is to reproduce initial condition
s for a dynamical system under consideration (approximating waveform genera
tor) giving rise to an orbit which is optimal in the sense of average dista
nce from the measured (or prescribed) transient output waveform. We present
the results obtained using the Chua's circuit as approximating waveform ge
nerator.