Transformations of Variables - Fourier Tab
Select the Fourier tab of the Transformations of Variables dialog box to access the options described here. The transformations on this tab are customarily performed on series (results) produced by a Single Spectrum (Fourier) Analysis or Cross-spectrum Analysis. Forward Fourier analysis and Inverse Fourier analysis are also used for constructing filters at particular frequencies (see below).
| Element Name | Description |
|---|---|
| OK (Transform selected series) | Click the OK (Transform selected series) button to transform the selected variable (series) and display a plot of the analysis. Transformed variables are appended to the active work area. If a transformation produces an invalid value (e.g., square root of a negative value, division by zero, etc.), STATISTICA assigns a missing data value; then, in a second pass, those missing data are replaced according to your selection on the Time Series Analysis Startup Panel - Missing Data tab. |
| Transformation | Use the options in the Transformation group box to select the desired type of transformation. |
| Tapering | The so-called process of split-cosine-bell Tapering is a recommended transformation of a series prior to spectrum analysis. It usually leads to a reduction of leakage in the periodogram. The rationale for this transformation is explained in detail in Bloomfield (1976, p. 80-94), see also Tapering. |
| Smoothing window | As described in greater detail in
Spectrum (Fourier) Analysis, the periodogram values computed in a spectrum analysis are subject to substantial random fluctuation. A clearer picture of underlying periodicities often only emerges when examining the spectral densities, that is, the frequency regions, consisting of many adjacent frequencies, that contribute most to the overall periodic behavior of the series. The spectral density estimates can be computed by smoothing the periodogram values with a weighted moving average. Suppose the moving average window is of width m as specified in the Span box (the value in the Span box determines the width of the smoothing window - the number entered must be odd and greater than or equal to 3); the following smoothers are those most commonly cited in the literature:
Daniell (or equal weight) window Tukey window, Hamming window Parzen window, Bartlett window If the User-defined window option button is selected, then click the arrow button |
| Real & imaginary part | Select the Real & imaginary part option button to perform a Fourier transformation on the input series, resulting in a real and imaginary part (i.e., two new series are created, representing complex numbers). The transformation is accomplished via the so-called
fast Fourier transform algorithm, or FFT for short (first popularized by J.W. Cooley and J.W. Tukey, 1965). The implementation of the FFT algorithm in the
Time Series module allows you to take full advantage of the savings afforded by this algorithm. On most standard computers, series with more than 100,000 cases can easily be analyzed. However, there are a few things to remember when analyzing series of that size.
The standard (and most efficient) FFT algorithm requires that the length of the input series is a power of 2. If this is not the case, then additional computations have to be performed. In order to still utilize the FFT algorithm, an implementation of the general approach described by Monro and Branch (1976) is used in the Time Series module. This method requires significantly more storage space; however, series of considerable length can still be analyzed very quickly, even if the number of observations is not equal to a power of 2. The Fourier transform and its inverse transformation (see below) are often used to construct frequency filters (see, for example, Bloomfield. 1976). For example, if a Single Spectrum (Fourier) Analysis of a series indicates no significant periodicities at high frequencies, then you may want to filter the series so as to completely eliminate all high frequency fluctuations. This can be accomplished by first performing the Fourier transformation, then modifying the Real & imaginary parts of the series at the respective frequencies, and finally recomputing the series via the Inverse Fourier transform. |
| Inverse Fourier transform | This is the inverse of the Fourier transform (using the same computational algorithms). Note that the currently highlighted variable is interpreted as the real part of the series, to select the imaginary part, click the Imaginary part (var) button (see below). |
| Imaginary part (var) | Click the Imaginary part (var) button to display the Currently available variables and transformations dialog box, in which you select the imaginary part of the series for the Inverse Fourier transformation. To select the variable, either double-click on the respective variable, or click once on the respective variable to highlight it and then click the OK button. |
to display the