Fast Fourier Transform (FFT) in IDL
Introduction:
When processing a time series of data, one of the main analysis consists of
finding whether the series presents some periodic behaviour. One way to analyze
the periodicity is to do a spectral analysis. In this work, I try to understand
some of the features presented by the IDL FFT routine and give some tentative
notes on its use.
IDL provides a buildin function called FFT to be used as
result = FFT(series)
Data:
A time series result of a sum of various sinusoidal functions of different
frequencies is generated.
"generate_sin_data.pro" is a simple IDL routine which generates a long time series
with sampling every half an hour (30 min). The series is made of 4 sinusoidal
frequencies at 0.5, 1, 10, and 36 day period (2, 1, 0.1, and 0.0278 cycle per day).
Series of 120 days and of 360 days are studied in this work.
Analysis:
The 120day series as well as its corresponding power spectrum are illustrated in
Figure 1. In this case, both spectrum with and without Hanning windowing are shown.

Figure 1. 
Clear narrow peaks corresponding to 2 frequencies (2 and 1 cycles per day) are observed.
Another frequency 0.1 cycle per day is also present but with a certain degree of
spread. Those three frequencies correspond to 12, 24, and 240 hour (0.5, 1, and 10 day)
period. Finally, there is the very spread peak at 0.025 frequency (or 40 day period).
As we can see, the first 2 frequencies are well defined and set to the right
original frequencies, the third is also defined at the right original frequency;
however, it starts showing some leakage phenomenon. The last one is defined with
more leakage than the previous spike, its period is actually shifted by 4 days.
From this data, we can see that a series of 120 days allows to clearly extract
periods of less than few days. At 10 day period, the spectrum starts to show
leakage. The period which corresponds to 1/3rd of the length of the series are
there but not well defined anymore.
When processing a series of 360 days of data, the corresponding power spectrum
graph is given in Figure 2. One can see that some noises show up at high frequencies.
In addition, the low frequency peak now shows up at 0.0278 cycle per day (~ 35.98
days closer to the real frequency). The way this frequency is spread is the same
as in Fig. 1 for frequency at 10 days.

Figure 2. 
My Conclusion:
The following conclusion is drawn by looking at the result. A furher and more
indepth investigation might be necessary to confirm it.
We are tempted to conclude that for a series of data, periods above 1/12 of
the length of the data will start to show up with some sort of leakage
(illustrated as spreading in the power spectrum graph).
The noises at high frequencies can be an artifact of the IDL FFT routine. Even by
lowpass filtering the data, we still have almost the same power spectrum graph.
Notes:
The following IDL routines were written and used to carry out the above study:
generate_sin_data.pro
compute_fft.pro
Useful links:
http://local.wasp.uwa.edu.au/~pbourke/miscellaneous/dft/
http://www.davidpace.com/physics/graduateschool/idlbutterworth.htm