Spectral Analysis of Geophysical Data - Advances in Data, Methods, Models and Their Applications in Geoscience - page 39

W(ω) of the rectangular pulse, w(t) as 2

. Thus ( ) = ()∗

. This shows
that truncating a signal brings about a spectrum modification expressed by the convolution
operation between the two spectra, F(ω) and W(ω). The truncation of a signal, therefore,
introduces a smoothing effect whose severity depends on the window length. The shorter
the window length, the greater the degree of smoothing and vice-versa. The truncated
Fourier transform F tr (ω) is often called the average or weighted spectrum (Blackman &
Tukey, 1959). Since all observational data or signals have finite length, truncation effect can
never be avoided.
In order to minimize spectral distortion from the signal truncation, other types of time
windows may be applied. In general, a window which tapers off gradually towards both
ends of the signal introduces less distortion than a window which has near-vertical sides
(like the box-car function). A least distortive time window should have the following
properties:
a. The time interval must be as long as possible. This implies that its corresponding
Fourier transform or the spectral window has its energy concentrated to its main lobe.
b. The shape must be as smooth as possible and free of sharp corners. The more smooth
the time window is, the smaller the side lobes of the corresponding spectral window
become (the box-car function is a dirty window!).
At this juncture, we shall mention some popular time windows. These include the box-car
(rectangular), Bartlett (triangular), Blackman, Daniell, Hamming, Hanning (raised cosine),
Parzan, Welch and tapered (rectangular windows). Excellent treatise on spectral windows
can be consulted.
In general, the Fourier transforms of time domain windows have central main lobe and side
lobes in each transform and the magnitudes of the side lobes emphasize the differences
between them. Ideally, the main lobe width should be narrow, and the side lobe amplitude
should be small.
Windows are also extensively used in designing filters and the window parameters (side
lobe amplitude, transition width and stopband attenuation) must be used for the design.

5.6 Fourier spectrum of observational data
To compute the spectrum of a function f(t) which obeys Dirichlet conditions, Fourier
transform is applied to it directly, particularly the use of Fourier integral equations. We can
use the basic theorems already presented to evaluate the transforms.
The common types of functions which are usually subjected to Fourier analysis are those
obtained by some kind of physical measurements. Functions which represent observational
data are normally converted into digital form (if presented as a continuous plot in profile or
map forms) so that their spectra can be computed by numerical Fourier transformation.
Observational functions are usually not continuous and not infinite as the theory of Fourier
transformation demands. For this reason, observational spectra suffer from two types of
distortions: (1) truncation effect (by a window function) and (2) digitization effect. When a
signal is digitized, its spectrum becomes periodic and so the original spectrum (scaled by
the inverse of sampling period) becomes repetitive with the same frequency as that used in
sampling the signal. Coarse digitization results in distorted spectrum. The extent of
digitization effect depends on the sampling frequency as well as on the cut-off frequency of
the signal (see Section 3).