Geoscience Reference
In-Depth Information
Fig. 6.1
Schematic of a linear time-invariant (LTI) system. h e input signal is transformed
into an output signal.
alters and smoothes the data with respect to its original form. For example
a i nite sized sediment sample is integrated over a certain period of time
and therefore smoothes the natural signal. Similarly, the measurement of
magnetic susceptibility in a sediment core with the help of a loop sensor
introduces signii cant smoothing since the loop response is integrated over
a section of the core.
h e characteristics of these natural i lters are ot en dii cult to determine,
whereas numerical i lters are designed with well-dei ned characteristics. In
addition, artii cial i lters are time invariant in most cases, whereas natural
i lters, such as mixing within the water body of a lake or bioturbation at the
water-sediment interface, may vary with time. An easy way to describe or
predict the ef ect of a i lter is to explore the i lter output from a simple input
signal as a sine wave, a square wave, a sawtooth function, a ramp function,
or a step function. Although there is an endless variety of such input signals,
most systems or i lters are described by their impulse response, i.e., the
output resulting from the input of a unit impulse.
h is chapter starts with a technical section (Section 6.2) on generating
periodic signals, trends, and noise, following on from Section 5.2 of the
previous chapter. Section 6.3 then considers linear time-invariant systems,
providing the mathematical background for i lters. h e succeeding sections
(6.4 to 6.9) deal with the design, the realization, and the application of linear
time-invariant i lters. Section 6.10 then considers the use of adaptive i lters
originally developed for use in the telecommunication industry. Adaptive
i lters automatically extract noise-free signals from duplicate measurements
on the same object. Such i lters can be used in a large number of applications,
for example to remove noise from duplicate paleoceanographic time series,
or to improve the signal-to-noise ratio of parallel color-intensity transects
across varved lake sediments (see Chapter 5, Fig. 5.1). Adaptive i lters
are also widely used in geophysics for noise canceling. We use the Signal
