Digital Signal Processing Reference
In-Depth Information
Chapter 3
Deterministic Sampling for Quantification of Modeling
Uncertainty of Signals
Jan Peter Hessling
Additional information is available at the end of the chapter
http://dx.doi.org/10.5772/52193
1. Introduction
Statistical signal processing [1] traditionally focuses on extraction of information from noisy
measurements. Typically, parameters or states are estimated by various filtering operations.
Here, the quality of signal processing operations will be assessed by evaluating the statistical
uncertainty of the result [2]. The processing could for instance simulate, correct, modulate,
evaluate, or control the response of a physical system. Depending on the addressed task and
the system, this can often be formulated in terms of a differential or difference
signal processing
model equation
in time, with uncertain parameters and driven by an exciting input signal
corrupted by noise. The quantity of primary interest may not be the output signal but can be
extracted from it. If this uncertain dynamic model is linear-in-response it can be translated into
a linear digital filter for highly efficient and standardized evaluation [3]. A statistical model of
the parameters describing to which degree the dynamic model is known and accurate will be
assumed given, instead of being the target of investigation as in system identification [4].
Model
uncertainty
(of parameters) is then
propagated
to
model-ing uncertainty
(of the result)
.
The two
are to be clearly distinguished - the former relate to the input while the latter relate to the
output of the model.
Quantification of uncertainty of complex computations is an emerging topic, driven by the
general need for quality assessment and rapid development of modern computers. Applica‐
tions include e.g. various mechanical and electrical applications [5-7] using uncertain differ‐
ential equations, and statistical signal processing. The so-called brute force Monte Carlo
method [8-9] is the indisputable reference method to propagate model uncertainty. Its main
disadvantage is its slow convergence, or requirement of using many samples of the model
(large ensembles). Thus, it cannot be used for demanding complex models. The ensemble size
is a key aspect which motivates deterministic sampling. Small ensembles are found by
