The fast Fourier transform

Fourier techniques are widely used in science. The basic idea of these techniques is that just about any variation can be expressed as the sum of regular sine waves with different periods. This is true even if the variation is not periodic. However, for a periodic signal, only relatively few frequencies will be present (frequency is the inverse of period here). Using Fourier transforms does have complications, however, because we really need continuous data. The basic "compromise" technique is the fast Fourier transform (FFT), which settles for equally spaced data points. The output of an FFT is normally called a periodogram or power spectrum, because it gives the "power" at any particular frequency (period).

The program powerts calculates the power spectrum of a single time series taken from a Ginga data file. The results are written to a file so they can be displayed or analysed further. An example of powerts is given below:

qcl>powerts
 
** POWERTS Version 1.1 Mon Oct  6 20:58:37 1997 **
 
POW_DATIN Data file name =mkn335_91a.time
 Number of time samples    393
 Number of PHA samples     1
 Number of IDENTs          2
 Sample duration  :        128
POW_IPHA PHA subscript = 1 =
POW_IDENT Ident subscript = 1 =
 Number of missing samples 305
 Mean data value           -0.0005417928438295
 Dimension chosen for DFT  1024
 Noise power estimated from variances 172.33609260212 (cts/sec)**2/Hz
 Average power in spectrum 153.3974091716 (cts/sec)**2/Hz
qcl>

The program powerts writes the results to a file with the same root name as the input time series, but with an extension ".pspe" (i.e. mkn335_91a.pspe in the above example). The QCL procedure pl_power.qcl can be used display the power spectrum. The following is an example of its use:

qcl>.pl_power
QCL_CL Power spectrum file name = mkn335_91a.pspe =
QCL_CL graphics device = xw =
QCL_CL Title for plot = mkn335_91a.pspe =
Starting /soft1/xanadu/OSF1_4.0_alpha/bin/pgxwin_server.
    Graphics device /xw open
    HDS_NODE mkn335_91a.pspe
    GRP_XMODE            0
    GRP_XNAM1 FREQUENCY
    GRP_XNAM2 DUMMY1
    GRP_YMODE            0
    GRP_YNAM1 POWERSPECTRUM
    GRP_YNAM2 DUMMY2
    LABX  X                                                           
    LABY  Y                                                           
    Number of data points          512
    Range of data points
    X  0.762939E-05 to  0.390625E-02
    Y  0.221402     to   1188.87    
QCL_CL Bottom of Y axis = 86.168046301060 =
QCL_CL Top of Y axis = 4755.4800000000 =
QCL_CL Low X = 7.62939453125000E-06 =
QCL_CL High X = 3.90625000000000E-03 =
    Mapping   7.6293945E-06  3.9062500E-03   86.16805       4755.480
    Position  0.762940E-05   172.336    
    Position  0.390625E-02   172.336    

  Device still open ...
  Type CLOSE to finish
qcl>

Note that periods of ~96 minutes (frequency of ~1.7 x 10^-4 Hz) are spurious; that is about the orbital period of the Ginga satellite.