9.10. Frequency Domain Analysis 563
Fig.9.9.1 also shows the result of moving averages for 3-point subsets of data.
It is evident that such a technique is not suitable for determining the baseline.
However there are other instances where this technique gives better results than the
simple all-data averaging. Actually moving average is most suitable for situations
where small scale fluctuations are superimposed on a large scale fluctuation. In
other words if the data contain high frequency components on top of low frequency
components then smoothing by simple average would smooth out the low frequency
components as well. On the other hand, moving average smoothing would retain
the low frequency components. In this respect we can say that moving average acts
like a low pass filter with a pass band that depends on the number of data points
selected in each set.
Simple and moving averaging are not the only smoothing techniques available.
In fact, with the availability of fast computers, the trend now is to use the more
involved techniques, such asexponential smoothing. The basic idea behind expo-
nential smoothing is to perform weighted averages instead of the simple averages.
The weights are assigned in an exponentially deceasing fashion, hence the name
exponential smoothing.
9.10FrequencyDomainAnalysis.......................
Sometimes it is desired that the time series data are analyzed in the frequency
domain rather than the time domain. This desire is generally driven by the need to
obtain frequency and phase content of the signal. For example, one may want to see
if the time series data from a detector has a 50 or 60Hzcomponents, which would
indicate coupling of the system with a power line. Similarly one might be interested
in determining the time periodicities in the signal, which stand out as frequency
peaks in the frequency domain. Another application of frequency domain analysis is
the filtration of digitized pulse, which allows the frequencies to be selectively filtered
out.
The frequency domain analysis is also sometimes referred to asspectral analysis.
However spectral analysis is a much broader term that involves more involved types
of analysis as well, such as fractal dimensional analysis. Here by frequency domain
analysis we mean transformation of the signal into frequency domain through Fourier
transformation and then analyzing the resulting power spectrum. A power spectrum
is the variation of the square of the Fourier transform amplitude with respect to
frequency.
The basic idea behind Fourier transformation is that any signal can bedecomposed
into a sum of simple sinusoidal functions with coefficients that represent amplitudes.
In other words, no matter how complex a signal is, it can be represented by a sum
of sinusoids. The resulting function is known as a Fourier series. We will not go into
the details of this series but will concentrate on how a time varying function can
be transformed into frequency space. Any functiong(t) can be Fourier transformed
according to
F(f)=∫∞
−∞g(t)e−i^2 πftdt, (9.10.1)wherefrepresents frequency inHz. This equation is good for continuous function
that can be evaluated analytically and therefore if we want to transform experimental