Fourier Spectral Analysis Object and Template – Fourier Spectrum (Spectral Analysis Option)
The Fourier Spectrum spectral procedure provides the Fourier Spectrum of the windowed data. The data set has to be equidistantly sampled (constant sampling rate).
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, In the is the imaginary component, δF is the frequency spacing in 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) |
Phase |
arctan(Im / Re) |
Phase unwrapped |
arctan(Im / Re), unwrapped to avoid discontinuities |
Complex |
complex(Re, Im) |
Real component |
Re |
Imaginary component |
Im |
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 (square sums) |
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 (square sums) |
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
FlexPro offers a variety of tapering windowsto 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 will be disabled for fixed windows.
The list box Normalization offers two options to normalize after applying the window. Selecting Amplitude normalizes to the gain of the used window function, i.e. the sum of all values is divided by their number. 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 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.
Increasing the zero padding will increase the number of frequency channels, which for small size data sets can aid in more accurately determining the center frequencies of spectral peaks. This will not change the basic shape of the spectrum, however. If a given peak is defined by only three frequency bins when an exact 64 point FFT is made, a 1064 point FFT will basically fill in this same shape. It is a type of interpolation, since zero padding cannot sharpen the peaks. To achieve this sharpening with an FFT, a longer 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.
When a data tapering window is used, then there is very little spectral Leakage.
Options - Peaks (Analysis Wizard Only)
The spectral peaks are identified by a local maxima detection algorithm. Both the amplitude and the frequency locations of the detected peaks are based upon a cubic spline bin interpolation procedure.
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 current 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.
Harmonic Table (Analysis Wizard Only)
The Optional numeric results option on page three of the Analysis Wizard produces a table of frequencies, (cosine-based) phases and PSD 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.