The Uneven Data Fourier Spectral Analysis generates a Lomb-Scargle Periodogram for data with unevenly spaced X values and for data containing void values.
Spectrum Type
The frequency domain information can be output in a variety of formats. In the following table, mag² is the magnitude-squared of the Lomb spectrum at a given frequency, δF is the frequency spacing of the spectrum, n is the data set size, δX is the sampling interval, and σ² is the data set variance.
Spectrum Type |
Formula/Description |
|---|---|
Amplitude |
sqrt(mag²) / n |
RMS |
sqrt(mag² / 2) / n |
Amplitude² |
mag² / n² |
dB |
20 * log10(sqrt(mag²) / n / Aref) Aref = Reference amplitude, which is assigned 0 dB. |
Normalized dB |
20 * log10(sqrt(mag²) / n / Amax dBmax = dB value of the spectral line with greatest amplitude. |
PSD - Power Spectral Density |
mag² / n² / δF / 2 |
TISA - Time Integral Amplitude² |
δX * mag² / n / 2 |
MSA - Mean Amplitude² |
mag² / n² / 2 |
SSA - Sum Amplitude² |
mag² / n / 2 |
Variance |
mag² / (n * σ²) / 2 |
Magnitude² |
mag² |
Magnitude |
sqrt(mag²) |
In an amplitude plot, you see the actual amplitude of sine components. With the normalized dB display, the highest peak is at 0 dB. A peak at -3 dB would have half the power, and a peak at -6 dB would have half the amplitude.
Windows
FlexPro offers a variety of Data Tapering Window to reduce the Spectral Leakage. The Window adjustment field is used to set the spectral width, and thus the dynamic range, of adjustable windows. This field is disabled for fixed width windows.
Only continuous window functions are available in this procedure. For this reason, the Chebyshev and Slepian DPSS windows are not available. The Chebyshev (approximated) window is a continuous approximation to the Chebyshev window that was designed expressly for this procedure.
The list box Normalization offers two options to normalize after applying the window. Selecting Amplitude normalizes for the gain of the used window function in the meaning of the time integral of the window function is divided by its X range. This compensates the damping of the amplitudes caused by applying the window. This is especially useful to measure peaks within the spectrum. If you select Power, the loss of power is compensated. The ratio of the sum of the squared data before and after applying the window is used as a normalization factor. The total power within the spectrum therefore always corresponds to the power of the data before applying the window.
Parameters
The Number of Frequencies for the spectrum sets the length of the positive frequencies in the computed spectrum. Unlike the FFT, this does not include an initial zero frequency. You can choose any number up to 65536. You may also select from one of the fast lengths in the drop down box.
Increasing this number of evaluated frequencies in the spectrum produces results similar to zero padding an FFT. This can aid in more accurately determining the center frequencies of spectral peaks. As with the FFT, however, this will not change the basic shape of the spectrum.
The Nyquist Multiple field sets the frequency upper limit of the spectrum and refers to the Nyquist frequency resulting from the average sampling interval. You can exceed the Nyquist frequency by up to a factor of four in order to display the spectral components of signal components with an above-average sampling rate.
FlexPro creates a uniformly spaced frequency spectrum similar to the FFT.
Options - Peaks (Analysis Wizard Only)
The spectral peaks are identified by a local maxima detection algorithm. The peaks can be set with a maximum peak count or a dB threshold below the largest peak. Peaks are ranked by interpolated amplitude. Note that a target signal component count may not be realized as fewer peaks than this target may be detected.
You can view the Y and/or X values of the peaks in the spectrum by pressing Toggle Labels.
Options - Set/Clear Reference (Analysis Wizard Only)
This function lets you compare various spectral procedures and settings. You can view a copy of the currently displayed spectrum in the lower pane by pressing Set Reference. Next, you can adjust additional settings that affect the display in the upper pane. With Clear Reference you can remove the copy and the time signal will appear again.
Options - White Noise Critical Limit (Analysis Wizard Only)
FlexPro offers peak-type Critical Limits to determine the statistical significance of the highest peak present in the spectrum. These limits are computed for all windows, including those with adjustable parameters.
Note that the exponential distribution limits traditionally a part of the Lomb-Scargle periodogram are not used in this procedure. FlexPro implements peak-type critical limits as opposed to the more commonly used confidence limits.
The critical limits for this procedure were generated from Monte Carlo trials that used uniformly spaced abscissae. Although researchers have reported little difference in significance levels between uniformly spaced data and randomly spaced data, you should consider these critical limits approximate when data are not evenly sampled.
Harmonic Table (Analysis Wizard Only)
The Optional numeric results on page three of the Analysis Wizard produces a table with the frequency, amplitude, PSD, power, normalized to a sum of 100 and power, normalized to a maximum of 100 for all peaks in the spectrum.