COMMISSION 27 OF THE I. A. U.
           INFORMATION BULLETIN ON VARIABLE STARS
                       Number 2285

                                                       Konkoly Observatory
                                                       Budapest
                                                       1983 March 1

                                                       HU ISSN 0374-0676
 

         THE HIGH FREQUENCY LIMIT TO FOURIER ANALYSIS.

               A REMINDER OF THE NYQUIST FREQUENCY



  This note is a reminder to astronomers using Fourier Analysis (to determine 
the frequencies present in variable star data) that one cannot extract
from a continuously sampled data set any frequency greater than 1/2 Delta where
Delta is the sampling interval. The reason for this is that aliases arise due
to beating between the sampling frequency, 1/Delta, and any real frequency, f,
present in the data. Thus, in the power spectrum of data varying with the
frequency f, peaks will occur at f, 1/Delta + f, 1/Delta - f, 2/Delta + f, 2Delta - f, etc.
The high frequency limit to any meaningful frequency search exists where the
real frequency f and the lowest frequency alias 1/Delta - f overlap, that is,
when f = 1/Delta - f which yields the Nyquist frequency of f = 1/2Delta.
   I have illustrated this problem in Figures 1 and 2 which are amplitude
spectra of artificial data. These data have sampling interval of Delta = 0.05
hour (or a sampling frequency of 20 hour^-1) and a time span of 5 hours. In
Figure 1 there is a real frequency of 1 hour^-1 present in the data with an
amplitude of 10 m mag. In Figure 2 there is a real frequency at 7 hour^-1.
Note that the amplitude spectra shown in Figures 1 and 2 would look identical 
if the real frequency in the original data were at the frequency of any
of the aliases. Also note that the aliases under discussion here occur
for continuously sampled data sets and are not the same as the aliases which
arise in the Fourier analysis of astronomical data with time gaps. Aliases
caused by time gaps in the data are simply cycle count ambiguities across
the gaps.


                             [FIGURE 1]

 Figure 1  An amplitude spectrum of 5 hours of artificial data sampled at a
 frequency of 1/Delta = 20 hour^-1 which gives a Nyquist frequency of
 1/2Delta = 10 hour^-1. The data were generated with a frequency of
 1 hour^-1 with an amplitude of 10 m mag. The noise is due to
 three decimal truncation of the generated magnitudes.


  The impetus for the note comes from two recent issues of IBVS in which
astronomers have performed Fourier analyses beyond the Nyquist frequency
and obtained results which are very probably wrong. Burki et al. (1982)
Fourier analysed data sampled at "1 measurement per night" for HR3562 and
derived "three significant periods" all less than 2 days which is the Nyquist


                             [FIGURE 2]

 Figure 2  An amplitude spectrum of the same data set as in figure 1 except
 with a generating frequency of 7 hour^-1. One can easily see from
 these two figures that the highest frequency which can be unambiguously
 extracted from such a diagram is the Nyquist frequency
 where the real frequency and the lowest frequency alias overlap.


limit to their analysis. Musielok and Kozar (1982) suggest periods of 5.4
and 5.9 minutes for data they have on 21 Com. Yet, if their 450 measurements 
obtained during 30 hours of observation are uniformly spaced, then
their data spacing is about 4 minutes/measurement which limits their search
to periods greater than 8 minutes.



                             D. W. KURTZ
                             Department of Astronomy,
                             University of Cape Town,
                             Rondebosch. 7700.
                             Cape. SOUTH AFRICA.


References:

Burki, G., Burnet, M., Magalhaes, A.S., North, P., Rufener, F. and
      Waelkens, C., 1982, I.B.V.S., no. 2211.
Musielok, B. and Kozar, T., 1982, I.B.V.S., no. 2237.