Dear Alexander,

as you know, the window width, T, defines the frequency resolution (capability to distinguish a couple of harmonics as two separate spectral peaks). Frequency resolution of the rectangular window is =1/T, and the resolution is lower for the popular Hann window (about 1.6/T; see https://goo.gl/UeWcb1 ). Since the lower limit of the LF band is 0.04 Hz, you should select T > 1.6/0.02 =80 s for a frequency resolution of at least 0.02 Hz with the Hann window. The Kubios example proposes a longer window (T=150 s) to properly estimate also the VLF power.

You will probably use the Hann window (often erroneously dubbed "hanning") because it efficiently decreases power leakage from the main lobe to the side lobes. This, however, is obtained at the cost not only of a lower frequency resolution, as already mentioned, but also of a greater variability of the spectral estimate. In fact, the Hann window multiplies a block of N samples with a cosine wave, and the equivalent number of degree of freedoms of the statistical estimate is substantially lower than N. Therefore, the variance of the spectral estimate increases. With a 50% overlapping, most of the degrees of freedom that are lost in the upper half of the window “i” are recovered in the lower half of the successive window “i+1”. You could increase the overlapping (e.g., 90%) but in this case you also increase the computation time with only a little increase in the equivalent number of degree of freedoms. You could find further information on this report:

https://goo.gl/X1Kiqu

In conclusion, if you have to estimate LF and HF powers only from 5-minute long recordings, I would suggest to use Hann data windows of length T=120 s, with 75% overlapping. In this way, the resulting periodogram is the average of 7 FFT spectra, each with frequency resolution around 0.014 Hz.

Dear Paolo! Thank you for your detailed review on the problem of choosing a smoothing window. I would like to make only a few additions.

First, according to Nyquist-Shannon-Kotelnikov theorem, to select the frequency (period) of the smoothing window, use the rule f> 1/2fm, where 2fm is the maximum frequency of the detected frequency. Second, since we are not interested in the lower limit of low frequencies, but rather the main (central) frequency of the low frequency is 0.1 Hz, then formally we can use the width of the smoothing window 20 seconds. More preferably, of course, a wider window. Third, the choice of the window function usually depends on the preferences of the researcher, but the following should be kept in the mind: a) The central frequency of the low-frequency range is unstable over time and b) the use of the Gaussian window when using Fourier transformation looks like a logical solution.

And finally, about the overlapping of windows. Usually a value of 50% of the overlapping is generally adopted. In my work, I used the width of the window for 30 seconds, and the next window was dislocated forward for 1 second. Thus, the overlap was 97%. Increasing computation time unsubstantially in modern computers, but one can get time-varying analysis (Gabor transform).

Dear Alexander John,

the spectral density estimations of HR modulation are meaningful only at intervals were the homeostatic system under research could remain in unchanged (stationary) state. So to find the window width one need to find the maximim interval of stationarity.

Maybe you could find of some use my note in the discussion initiated by

Dr. Gheorghe Zaharia: https://www.researchgate.net/post/How_HRV_Heart_Rate_Variability_is_computed

Best regards,

M.G.