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5.2 Generating Signals 87

samples equally spaced along the sediment core are therefore not equally
spaced on the time axis. In this case, the quantity

where T is the full length of the time series and N is the number of data points,
represents only an average sampling interval. In general, a time series y(t )
k
of a process can be represented as a linear sum of a long-term component or
trend y (t ), a periodic component y (t ) and a random noise y (t ).
tr k p k n k

The long-term component is a linear or higher-degree trend that can be ex-

tracted by fitting a polynomial of a certain degree and subtracting the values
of this polynomial from the data (see Chapter 4). Noise removal will be
described in Chapter 6. The periodic – or cyclic in a mathematically less
rigorous sense – component can be approximated by a linear combination
of cosine (or sine) waves that have different amplitudes A , frequencies f and
i i
phase angles ψ .
i

The phase angle ψ helps to detect temporal shifts between signals of the
same frequency. Two signals y and y of the same period are out of phase if
1 2
the difference between ψ and ψ is not zero (Fig. 5.2).
1 2
The frequency f of a periodic signal is the inverse of the period τ. The
Nyquist frequency f is half the sampling frequency f and provides a maxi-
Nyq s
mum frequency the data can produce. As an example, audio compact disks
(CDs) are sampled at frequencies of 44,100 Hz (Hertz, which is 1/second).
The corresponding Nyquist frequency is 22,050 Hz, which is the highest
frequency a CD player can theoretically produce. The limited performance

of anti-alias filters used by CD players again reduce the frequency band and
cause a cutoff frequency at around 20,050 Hz, which is the true upper fre-
quency limit of a CD player.
We generate synthetic signals to illustrate the use of time-series analysis
tools. While using synthetic data we know in advance which features the
time series contains, such as periodic or stochastic components, and we can
introduce artifacts such as a linear trend or gaps. This knowledge is particu-
larly important if you are new to time series analysis. The user encounters
plenty of possible effects of parameter settings, potential artifacts and errors