Fourier Cross Spectral Analysis Object and Template – Transfer Function (Spectral Analysis Option)
The Transfer Function procedure estimates the frequency domain Transfer Function between an input signal and an output signal. This procedure uses multiple segments that are typically used with some measure of overlap. The algorithm uses Fast Fourier transforms. The data streams must be uniformly sampled (constant sample increment) and of the same length.
An averaged transfer function is computed 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 transfer function can be output in an variety of formats. In the following table, Re is the real component of the transfer function at a given frequency, Im is the imaginary component, and n is the data stream size:
Spectrum Type |
Formula/Description |
|---|---|
Amplitude |
sqrt(Re² + Im²) |
dB |
10 * log10(Re² + Im²) |
Normalized dB |
10 * log10(Re² + Im²) - dBmax dBmax = dB value of the spectral line with maximum power |
Phase |
arctan(Im / Re) |
Phase unwrapped |
arctan(Im / Re), unwrapped to avoid discontinuities |
Complex |
complex(Re, Im) |
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.
Parameters
The Best Exact N composite algorithm is used for the FFT.
The length of individual data segments, Segment Length and 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. Values for the overlap that produce the minimum variance are reported to be in the range of 50 to 70%.
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 transfer function 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.