We have been discussing how does the parameter estimation depends on the length per FFT segment. In other words, after we collected a series of data, would it be better for us to divide it into many segments so that we have many averages, or should we use long FFT segments so that we have more frequency bins?
My conclusions are that:
1). We need to make sure that the segment length is long enough with T_seg > min[ Q_i / f_i ], where f_i is the resonant frequency of the i'th resonant peak and the Q_i its quality factor.
2). Once 1) is satisfied, the result depends weakly on the FFT length. There might be a weak hint preferring a longer segment length (i.e., want more freq bins than more averages) though.
To reach the conclusion, I performed the following numerical experiment.
I considered a simple pendulum with resonant frequency f_1 = 0.993 Hz and Q_1 = 6.23. The value of f_1 is chosen such that it is not too special to fall into a single freq bin. Additionally, I set an overall gain of k=20. I generated T_tot = 512 s of data in the time domain and then did the standard frequency domain TF estimation. I.e., I computed the CSD between excitation and response (with noise) over the PSD of the excitation. The spectra of excitation and noise in the readout channel are shown in the first plot.
In the second plot, I showed the 1-sigma errors from the Fisher matrix calculation of the three parameters in this problem, as well as the determinant of the error matrix \Sigma = inv(Fisher matrix). All quantities are plotted as functions of the duration per FFT segment T_seg. The red dotted line is [Q_1/f_1], i.e., the time required to resolve the resonant peak. As one would expect, if T_seg <~ (Q_1/f_1), we cannot resolve the dynamics of the system and therefore we get nonsense PE results. However, once T_seg > (Q_1/f_1), the PE results seem to be just fluctuating (as f_1 does not fall exactly into a single bin). Maybe there is a small hint that longer T_seg is better. Potentially, this might be due to that we lose less information due to windowing? To be investigated further...
I also showed the Fisher estimation vs. MCMC results in the last two plots. Here each dot is an MCMC posterior. The red crosses are the true values, and the purple contours are the results of the Fisher calculations (3-sigma contours). The MCMC results showed similar trends as the Fisher predictions and the results for T_seg = (32, 64, 128) s all have similar amounts of scattering << the scattering of the T_seg=8 s results. Though somehow it showed a biased result. In the third plot, I manually corrected the mean so that we could just compare the scattering. The fourth plot showed the original posterior distribution.