Page 126 - MATLAB Recipes for Earth Sciences

120 6 Signal Processing

a certain period of time and therefore smoothes the natural signal. Similarly,
the measurement of magnetic susceptibility with the help of a loop sensor

introduces significant smoothing since the loop integrates over a certain sec-
tion of the sediment core.

In most cases, the characteristics of these natural filters are difficult to de-

termine. Numerical filters, however, are designed with well-defi ned charac-

teristics. In addition, artifi cial filters are time invariant in most cases, while
natural filters, such as ocean mixing or bioturbation, may change with time.

An easy way to describe or predict the effect of a filter is to explore the

fi lter output of a simple input signal, such as a sine wave, a square wave, a
sawtooth, ramp or step function. Although there is an endless variety of such
signals, most systems or filters are described by their impulse response, i.e.,

the output of a unit impulse.
The chapter starts with a more technical section on generating periodic
signals, trends and noise, similar to Chapter 5.1. Chapter 6.3 is on linear
time-invariant systems, which provide the mathematical background for fi l-
ters. The following Chapters 6.4 to 6.9 are on the design, the realization and

the application of linear time-invariant filters. Chapter 6.10 then suggests

the application of daptive filters originally developed in telecommunication

automatically. Adaptive filters extract noisefree signals from duplicate mea-

surements on the same object. Such filters can be used in a large number of
applications, such as noise removal from duplicate paleoceanographic time
series or to improve the signal-to-noise ratio of parallel color-intensity tran-
sects across varved lake sediments (see Chapter 5, Fig. 5.1). Moreover, such

filters are also widley-used in geophysics for noise canceling.

6.2 Generating Signals

MATLAB provides numerous tools to generate basic signals that can be
used to illustrate the effects of filters. In the previous chapter we have gener-

ated a signal by adding together three sine waves with different amplitudes
and periods. In the following example, the time vector is transposed for the
purpose of generating column vectors.

t = (1:100)';
x = 2*sin(2*pi*t/50) + sin(2*pi*t/15) + 0.5*sin(2*pi*t/5);
plot(t,x), axis([0 100 -4 4])

Frequency-selective filters are very common in earth sciences. They are
used for removing certain frequency bands from the data. As an example,