Description  The data set X is analyzed for its power spectral density (PSD), i.e. the mean square amplitude. The result is stored in Spc. xsiz must be at least 2*specsiz, and specsiz has to be an integer power of 2. Internally, X is divided into (xsiz/specsiz)−1 segments and the average over the spectra of the individual segments is calculated. Each segment of length 2*specsiz yields the PSD for specsiz+1 frequencies (see VF_FFT). In order to keep specsiz an integer power of 2, there are only specsiz points stored in Spc and the last one, the PSD at the Nyquist frequency f_{Nyquist} = 0.5 / sampling_interval, is given as the return value of the function. It may either be neglected (by calling the function like a void function) or stored as the last element in Spc by calling the function as
Spc[specsiz] = VF_spectrum( Spc, specsiz, X, xsiz, Win );
in this case, Spc must have a length of specsiz+1.
Win is a window that is applied to the data segments. The size of the Win vector must be 2*specsiz. Within the VectorLib library, three functions are available that give suitable Windows: VF_Welch, VF_Parzen, and VF_Hann. A square window (i.e. no windowing at all) is achieved by setting all elements of Win to 1.0 using VF_equ1. Use of the square window is not recommended here, though.
You may wish to test the quality of the calculated spectrum by applying Parseval's theorem (provided you called VF_spectrum as in the above example and stored the PSD for the Nyquist frequency):
1.0/xsize * VF_ssq( X, xsize ) must be about equal to VF_sum( Spc, specsiz+1 ).
If the deviation between both results is large, the sampling interval in X probably is too large.
Internally, VF_spectrum allocates and frees additional workspace memory. For repeated calls, this would be inefficient. In such a case, it is recommended to use VFb_spectrum instead. The size of Buf must be ≥ 2*xsiz + 2*specsiz. Additionally, Buf must be 128bit (P8) or 256bit (P9) aligned. This usually means you can only take a vector allocated by the VF_vector family as Buf.
For an example for VF_spectrum, see the demo program VDEMO.
