Spectral AnalysisSnapshot of the front panel of the simulator:
Description of the simulated systemThere are several functions or algorithms for calculation of spectral information. One widely used function is the power spectral density (PSD), which is the square of the mean absolute value of the discrete-time Fourier transform (DFT) of the signal:
where f is the frequency in Hz. f is actually a discrete frequency given by
where k is a frequency index: k = 0,..,N-1, where N is the number of samples of the time series (signal) x. The frequency resolution or frequency increment df is
where f_{s} is the sampling frequency [Hz] given by
where T_{s} is the sampling interval. PSD (and DFT) gives spectral information only for the frequency range from 0 up to the Nyquist frequency f_{N} defined as half the sampling frequency:
PSD also gives spectral information in the frequency range larger than f_{N}, i.e. from f_{N} to f_{s}, but this information is equal to the spectral information for the frequency range from 0 to f_{N} - but mirrored. Since DFT, and hence the PSD, is calculated from a number of samples of the signal x random noise in the signal is averaged, and hence the effect of such noise on the calculation of PSD is averaged (out). This implies that the PSD is able to show the spectral contents of very noisy signals. In the simulator the signal x for which the PSD is calculated, is a sum of the following four signals: Two sinusoids, x_{1} and x_{2}, a uniformly distributed random signal r, and a bias (constant signal) of value B:
The random component may represent measurement noise. What is the PSD of a sinusoid and of a constant signal (both are components of the signal x used in the simulator)? It can be shown that a sinusoid of amplitude A and frequency f_{1} have PSD of value A^{2}/4 at the frequency f_{1} (and the same value at the mirrored frequency f_{s} - f_{1}), and zero PSD at other frequencies. This can be used to calculate the amplitude and the frequency of an assumed sinusoid. Thus, a sinusoid appears as a spike at frequency f_{1} in a PSD plot). It can also be shown that a constant signal of value B has PSD value B^{2} at frequency 0 and PSD value 0 at other frequencies. In the simulator PSD is calculated using the LabVIEW function Power Spectrum PtByPt (Point-by-point). MotivationSpectral analysis is used to show the frequency - or spectral - contents of a signal. One important appliaction is detection of vibrations of machine bearings. TasksRun the simulator. Unless otherwise stated the random signal r can be suppressed (by setting the maximum amplitude R equal to 0). Also the bias can be suppressed (with B = 0).
Updated 12. January 2008. Developed by Finn Haugen. E-mail: finn@techteach.no. |