Fourier Spectral Analysis Object and Template – Multitaper Spectrum (Spectral Analysis Option)
The Multitaper Spectrum procedure uses a series of orthogonal data tapers to generate a Fourier Spectrum. This utilizes the information at the edges of the data set better and reduces the variance of the spectral estimate.
Spectrum Type
The frequency domain information can be output in a variety of formats. In the following table, Re is the real component of a given segment's real (single sided) FFT at a given frequency, Im is the imaginary component, 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(Re² + Im²) / n |
RMS Amplitude |
sqrt((Re² + Im²) / 2) / n |
Amplitude² |
(Re² + Im²) / n² |
dB |
20 * log10(sqrt(Re² + Im²) / n / Aref) Aref = Reference amplitude, which is assigned 0 dB |
Normalized dB |
20 * log10(sqrt(Re² + Im²) / n) - dBmax dBmax = dB value of the spectral line with maximum amplitude |
PSD - Power Spectral Density |
(Re² + Im²) / n² / δF / 2 |
TISA - Time Integral Amplitude² |
δX * (Re² + Im²) / n / 2 |
MSA - Mean Amplitude² |
(Re² + Im²) / n² / 2 |
SSA - Sum Amplitude² |
(Re² + Im²) / n / 2 |
Variance |
(Re² + Im²) / (n * σ²) / 2 |
Magnitude² |
Re²+Im² |
Magnitude |
sqrt(Re² + Im²) |
Third octaves (mean values) |
The mean of the amplitudes in a third octave band is formed. |
Third octaves (sums) |
The amplitudes in a third octave band are summed up. |
Third octaves (RMS) |
The square mean or RMS for each third octave band is calculated, i.e. the squared amplitudes are averaged and from this the square root is calculated. |
Third octaves (quadratic means) |
The square amplitudes in a third octave band are summed up. |
Octaves (mean values) |
The mean of the amplitudes in an octave band is formed. |
Octaves (sums) |
The amplitudes in an octave band are summed up. |
Octaves (RMS) |
The square mean or RMS for each octave band is calculated, i.e. the squared amplitudes are averaged and from this the square root is calculated. |
Octaves (quadratic means) |
The square amplitudes in an octave band are summed up. |
In an amplitude plot, you see the actual amplitude of sine components. In a normalized decibel plot, 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. The PSD TISA (time-integral squared amplitude power) is the actual integral under the curve defined by the square of the raw data.
For the third octaves and octaves spectral types an amplitude spectrum is computed first, which is then evaluated with the ThirdOctaveAnalysis or OctaveAnalysis FPScript functions respectively.
Windows
The Window adjustment determines the main lobe width of the evaluation window, and Number of DPSS windows determines the number of tapering windows to be used in the sequence. You can enter 0 to set this to the highest value possible. When used with a high dynamic range, however, you should not automatically use the largest number of windows possible. Instead, you should try to work with a lower number. For instance, with a window width of 4, the sixth and seventh windows are degraded instead of improved.
The list box Normalization offers two options to normalize after applying the window. Selecting Amplitude uses a normalization factor which compensates for the attenuation of amplitudes due to applying a window to the data. This selection is particularly suited for measuring peaks in the spectrum. Selecting Power has no affect in this case because the spectrum is always power normalized as long as amplitude normalization is not selected. The total power within the spectrum therefore always corresponds to the power of the data.
Parameters
The Best Exact N composite algorithm is used for the FFT.
The initial FFT length is equal to the data length. To zero pad, enter any value greater than the data size. You may also select from one of the FFT lengths in the drop down box or select Next power of two for the fastest FFT calculation. When any data tapering window is used, very little spectral leakage arises due to zero-padding.
An increase in the FFT length increases the number of frequency lines, but the characteristic multitaper peaks do not become sharper as a result. While zero padding can be helpful to isolate the mid frequencies of the peaks for a simple windowed FFT, it is of little use for the multitaper procedure.
To achieve this sharpening with a Multitaper spectrum, a longer data set at the same sampling rate would be required. For data that are rapidly changing, or when the time series is limited in size, a non-FFT procedure is usually required for good spectral resolution.
Options - Peaks (Analysis Wizard Only)
The peaks in the averaged spectrum are identified by a local maxima detection algorithm, which uses the F values and the spectrum. The frequency positions are determined using the F value maxima and the amplitudes are read out from the amplitude spectrum to these positions.
The maximum number of peaks can be specified directly or determined indirectly using a minimum F value. A found F value maximum must then exceed this minimum value in order to be accepted as a peak. The default is F = 2.
You can view the Y and/or X values of the peaks in the spectrum by pressing Toggle Labels.
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 window counts and window widths.
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.
Harmonic Table (Analysis Wizard Only)
The option Optional numeric results on page three of the Analysis Wizard produces a table of frequencies, amplitudes, PSD and F values for all peaks in the spectrum. Absolute and relative percents are also given for the component powers. These are often the quantities of interest when comparing strengths of signal components.