FOURlER ANALYSIS! IN A NUTSHELL
FASTER and BETTER
by Anthony Warren Ph.D.
For those individuals who are currently using Fourier Spectrum analysis, this article describes some
techniques that make the analysis easier to perform and the spectral plots easier to interpret. (The reader
may also want to review the January articles on Fourier Analysis 1 , or the May article on its use in
selecting moving average parameters 2 .) Part I. of this article attempts to shed some light on the question,
"How many data points should I use for doing Spectrum Analysis?". In a nutshell, the answer is that the
data length should cover at least one cycle of the longest cycle in the data (i.e., for most stocks and
commodities, one trading year). Part II. describes a simple preprocessing calculation that reduces the time
needed to obtain a Spectrum plot. This makes it practical to generate Spectrum plots for a long time
series, say one to two years of data, on a regular basis.
Part I. Using Long Time Series for High Resolution Spectrum Plots
One of the most important considerations in generating Spectrum plots is how to resolve spectral peaks,
i.e. how to find the frequency (rate of oscillation in cycles/year), or alternatively, how to find the cycle
length of the highest power peaks in the spectrum (which ultimately should give you an idea of when to
trade). The ability to resolve spectral peaks is directly proportional to the length of time over which the
data are collected. For example, when using three months of data (one quarter of a year), the lowest
frequency components in the Fourier Spectrum are at 0, 4, 8,...cycles/yr. Consequently, we can resolve
spectral peaks with an accuracy of only about four cycles/year, and probably less due to noise averaging
and other effects. By contrast, with a full year's data we can resolve spectral peaks with an accuracy close
to one cycle/year. This higher resolution is valuable for optimizing the filter constants (cycle length, Trix
alpha, etc.) used in many technical analyses.