V.1:2 (26-28): A MiniGuide to Fourier Spectrum Analysis by ANTHONY WARREN, PH.D./STAFF WRITER

A MiniGuide to Fourier Spectrum Analysis by ANTHONY WARREN, PH.D./STAFF WRITER

The basic idea behind Fourier Analysis of time series is to decompose the data into a sum of sinusoids of varying cycle length, each cycle being a fraction of a common or fundamental cycle length. (For example, figure 1 shows a time series consisting of a linear trend and two major cycles, and figure 2 shows the decomposition into component sinusoids.) Originally, cyclic analysis was applied to physical phenomena to describe the behavior of complex vibrations, as for example, the multiple vibrations created by plucking a stringed instrument. The analysis of such systems is elegantly described by the behavior of the longest or fundamental cycle, and the response to the first, second and higher order harmonics of the fundamental. Later, Fourier and others extended this analysis and showed that any finite data segment or curve could be approximated arbitrarily well by a series of sinusoids, each of which is periodic over the data interval. This method is the basis of Fourier Analysis of sampled data (time series) and of conventional spectrum analysis. (Note: the Fourier approximation of a curve or time series is periodic, even if the data is not!)

In order to understand Fourier Spectrum Analysis we briefly review the properties of an individual sinusoid. A sinusoid may be uniquely characterized at any point in time by its amplitude or maximum value, by its frequency or rate of vibration, and by its phase. (See figure 3.) The period or cycle length of the sinusoid is the number of trading days per year (assume one year is 260 trading days) divided by the frequency, i.e. a sinusoid with a frequency of 10 cycles per year has a period of 260 / 10 = 26 days. Fourier analysis decomposes the data into a sum of sinusoids of appropriate amplitude, frequency, and phase. Fourier Spectrum analysis is a condensation of this data transform, whereby the amplitude squared or power in each sinusoid is plotted versus each sinusoid frequency. (Phase information is thus lost in the spectrum representation of the data.) For example, the amplitude spectrum of the cyclic data in figure 1 consists of two data spikes at appropriate frequencies, as shown in figure 4.