  In this paper, we present an assortment of both standard and advanced Fourier
techniques that are useful in the analysis of astrophysical time series of very
long duration -- where the observation time is much greater than the time
resolution of the individual data points. We begin by reviewing the operational
characteristics of Fourier transforms (FTs) of time series data, including
power spectral statistics, discussing some of the differences between analyses
of binned data, sampled data, and event data, and briefly discuss algorithms
for calculating discrete Fourier transforms (DFTs) of very long time series. We
then discuss the response of DFTs to periodic signals, and present techniques
to recover Fourier amplitude "lost" during simple traditional analyses if the
periodicities change frequency during the observation. These techniques include
Fourier interpolation which allows us to correct the response for signals that
occur between Fourier frequency bins. We then present techniques for estimating
additional signal properties such as the signal's centroid and duration in
time, the first and second derivatives of the frequency, the pulsed fraction,
and an overall estimate of the significance of a detection. Finally, we present
a recipe for a basic but thorough Fourier analysis of a time series for
well-behaved pulsations.
