Upper frequency limit and Nyquist Coefficient

The keyword ufreq allows to determine the upper limit of the frequency interval to be considered.

An alternative method is the automatic determination of this limit by means of the Nyquist Coefficient (keyword nycoef). For equidistantly sampled time series with sampling interval width $\delta t$, there is a uniquely defined Nyquist Frequency

$$f_\nu := \frac{1}{2\,\delta t}\, .$$ (5)

In case of non-equidistant sampling, each sampling interval between two consecutive time values may be considered to produce its individual Nyquist Frequency, whence this limit is ambiguous. In this case, the Nyquist Coefficient for an arbitrarily given frequency is introduced as the fraction of sampling intervals in the time domain the individual Nyquist Frequency of which is higher than the frequency under consideration. This provides to select an upper frequency limit by specifying a minimum Nyquist Coefficient. E.g., specifying a Nyquist Coefficient of 0.5 (which is the default value) guarantees that at least half of the information contained by the spectrum in the considered frequency range is below the Nyquist limit.

Additional information is available by setting the keyword nyscan in the .ini file. If this keyword is specified, SIGSPEC creates a file nyscan.dat in the project directory containing the Nyquist Coefficients over the specified frequency range.

Example. The sample project limits illustrates the use of the keyword ufreq. The line

ufreq 5

in the file limits.ini restricts all computations performed by SIGSPEC to frequencies below 5 cycles per day. The spectrum limits/s000000.dat (sig and amplitude) is displayed in Fig.7. A comparison with the screen output in Example SigSpecNative, p., where no restrictions to the frequency range apply, shows that the screen output in this example contains one line less:

   1 freq 3.13205  sig 9.54539  rms 0.00449592  csig 9.54539
   2 freq 3.98471  sig 7.43085  rms 0.00422861  csig 7.42753
   3 freq 2.664  sig 4.60182  rms 0.0040257  csig 4.60117

The signal component at 5.4 cycles per day is not detected, because it is outside the specified frequency range.

Example. The sample project nyos illustrates the use of the keywords nycoef and nyscan for the V photometry of IC4996#89. The line

nycoef 0.99

in the file nyos.ini provides an upper frequency limit of 110.77 cycles per day. The keyword nyscan is given, and the file nyos/nyscan.dat contains the Nyquist Coefficients for frequencies from 0 to 110.77 cycles per day, as displayed in Fig.8.

Figure 8: The file nyos/nyscan.dat contains the Nyquist coefficients depending on frequency for the sample project nyos.