Fourier Cross Spectral Analysis Object and Template – Cross Periodogram (Spectral Analysis Option)
The Cross Periodogram spectral procedure generates spectra that reflect the common power across two distinct signals. This procedure uses multiple segments that are typically used with some measure of overlap. If the data length is small, and the cross periodogram resolution is insufficient, the Cross Spectrum procedure uses the full data length.
Cross Periodograms are computed using Fourier transforms. The data streams must be uniformly sampled (constant sample increment) and of the same length.
A periodogram computes an averaged frequency cross spectrum by taking paired FFTs of multiple (and usually overlapping) segments of the two data streams. The segmenting results in a smaller size data record, and consequently in a reduced spectral resolution. However, the averaging reduces the variance that would arise from using only one pair of FFTs. Data are assumed stationary. To check stationarity, use the Short-Time Fourier Transform spectral analysis for both of the signals.
Spectrum Type
The frequency domain information can be output in a variety of formats. In the following table, Re is the real component of the non-normalized cross spectrum at a given frequency, Im is the imaginary component, δF is the frequency spacing of the spectrum, n is the data segment size, δX is the sampling interval, and σ² is the geometric average of the variance of the two data sets. The mathematical basics can be found in Cross Spectral Measurements.
Spectrum Type |
Formula/Description |
|---|---|
Amplitude |
sqrt(absolute(Re)) / n |
RMS |
sqrt(absolute(Re) / 2) / n |
Amplitude² |
>absolute(Re) / n² |
dB |
20 * log10(sqrt(absolute(Re)) / n / Aref) Aref = Reference amplitude, which is assigned 0 dB |
dB normalized |
20 * log10(sqrt(absolute(Re)) / n) - dBmax dBmax = dB value of the spectral line with maximum power |
PSD - Power Spectral Density |
absolute(Re) / n² / δF / 2 |
TISA - Time Integral Amplitude² |
δX * absolute(Re) / n / 2 |
MSA - Mean Amplitude² |
absolute(Re) / n² / 2 |
SSA - Sum Amplitude² |
absolute(Re) / n / 2 |
Variance |
absolute(Re) / (n * σ²) / 2 |
Magnitude² |
>absolute(Re) |
Magnitude |
sqrt(absolute(Re)) |
For the dB normalized type, the averaged spectrum will have a maximum at 0 dB. However, the associated peak will probably be slightly positive as a consequence of the bin interpolation.
Note also that averaging dB values is not an arithmetic average of the individual spectra, but more of a logarithmically weighted average.
With a simple average of power, intermittent harmonics or noise bursts in one or more segments can overwhelm an otherwise low power trend. When dB values are averaged, the influence of intermittent elements is significantly lessened. A dB average is thus a robust measure of power, although it may miss a harmonic that appears only briefly in time. An arithmetic average in power, or even amplitude, is more likely to catch an intermittent harmonic component. This is one of the reasons it is important to first use the Short-Time Fourier Transform (STFT) or Continuous Wavelet Transform (CWT) analysis object when stationarity is not assured.
Windows
FlexPro offers a variety of tapering windows 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 will be disabled for fixed windows.
The list box Normalization offers two options to normalize after applying the tapering 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 tapering 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 tapering 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 tapering window.
Parameters
The Best Exact N composite algorithm is used for the FFT.
The length of individual data segments, Segment Lengthand the amount of overlap, Overlap %, can be specified. You should set the segment size based on the resolution needed. You can enter 0 for the segment length to set it to the data length / 4. Values for the overlap that produce the minimum variance are reported to be in the range of 50 to 70%. With the setting Gap in samples the data continue to be ignored. This setting should only be selected for very long time series with slowly changing spectral content.
To accommodate zero padding, the FFT Length can be specified separately. Zero padding occurs when you set the FFT length to a value greater than the segment length. You can enter 0 for the FFT length to set it to the segment length. When a data tapering window is used, then zero-padding causes very little spectral leakage. Zero-padding is especially useful for interpolating peak frequencies with this algorithm, given the loss in resolution incurred by the reduced size of the segments.
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 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 signals will appear again.