Geoscience Reference
In-Depth Information
h e unit impulse is the most popular synthetic signal used to study the
performance of a i lter. h e output of the i lter, i.e., the impulse response,
describes the characteristics of a i lter very well. Moreover, the output of a
linear time-invariant i lter can be described by the superposition of impulse
responses that have been scaled by multiplying the output of the i lter by the
amplitude of the input signal.
6.3 Linear Time-Invariant Systems
Filters can be described as systems with an input
x
(
t
) and output
y
(
t
). We will
therefore i rst describe the characteristics of systems in general before then
considering i lters. Important characteristics of a system are
•
Continuity
- A system with continuous inputs
x
(
t
) and outputs
y
(
t
) is a
continuous system. Most natural systems are continuous. However, at er
sampling natural signals we obtain discrete data series and model these
natural systems as discrete systems, with discrete inputs and outputs.
•
Linearity
- For linear systems, the output
y
(
t
) of the linear combination of
several input signals
x
i
(
t
), where
is the same as the linear combination of the outputs
y
i
(
t
):
Important properties of linearity are scaling and additivity (
superposition
),
which allow the input and output to be multiplied by a constant
k
i
, either
before or at er transformation. Superposition allows additive components
of the input to be extracted and transformed separately. Fortunately,
many natural systems follow a linear pattern of behavior. Complex linear
signals such as additive harmonic components can be separated out and
transformed independently. Milankovitch cycles provide an example of
linear superposition in paleoclimate records, although there is an ongoing
debate about the validity of this theory. Numerous nonlinear systems
also exist in nature, which do not possess the properties of scaling and
additivity. An example of a linear system is
x = (1:100)';
y = 2*x;
