Geoscience Reference
In-Depth Information
x = (1:100)';
y = zeros(size(x));
plot(x,y)
h e output
y
is zero for all inputs
x
, and the output therefore does not
contain any information about the input.
•
Causality
- h e system response only depends on present and past inputs
x
(0),
x
(-1), …, whereas future inputs
x
(+1),
x
(+2), … have no ef ect on
the output
y
(0). All real-time systems, such as telecommunication systems,
must be causal since they cannot have future inputs available to them. All
systems and i lters in MATLAB are indexed as causal. In earth sciences,
however, numerous non-causal i lters are used. h e i ltering of images
and signals extracted from sediment cores are examples where the future
inputs are available at the time of i ltering. Output signals have to be
delayed at er i ltering in order to compensate for the dif erences between
causal and non-causal indexing.
•
Stability
- A system is stable if the output
y
(
t
) of a i nite input
x
(
t
) is
also i nite. Stability is critical in i lter design, where i lters ot en have
the disadvantage of provoking divergent outputs. In such cases, the i lter
design has to be revised and improved.
Linear time-invariant (LTI) systems are very popular as a special type of
i lter. Such systems have all the advantages that have been described above,
as well as being easy to design and use in many applications. h e following
Sections 6.4 to 6.9 describe the design, realization and application of LTI-
type i lters to extract specii c frequency components from signals. h ese
i lters are mainly used to reduce the noise level in signals. Unfortunately,
however, many natural systems do not behave as LTI systems in that the
signal-to-noise ratio ot en varies with time. Section 6.10 describes the
application of adaptive i lters that automatically adjust their characteristics
in a time-variable environment.
6.4 Convolution, Deconvolution and Filtering
Convolution is a mathematical description of a system transformation.
Filtering is an application of the convolution process. A running mean of
length i ve provides an example of such a simple i lter. h e output of an
arbitrary input signal is
