# Minimum discernible frequency in power spectral analysis

Say I have a signal of length 20s that contains signal from various (unknown) biological sources, e.g. heartbeat (~0.2Hz), respiration (~1Hz), and possibly som very-low frequency oscillations (~0.05Hz). My sampling frequency is fs=10Hz. I conduct a periodogram PSD estimate (hanning window, mean-subtracted, N_FFT=500), yielding a spectrum with frequencies 0:0.02:5 Hz.

Say I have several 20s recordings in two different settings and I want to conduct a statistical analysis of the spectral power in specific frequency bins or bands. Is there a lower limit to which frequencies I can use to statistically infer on biology?

For example, should I assume there to be a specific number of periods of frequency present in the window? E.g., if I say that at least 2 periods should be present in the window, the lowest detectable frequency is 0.1Hz. Or can I just as easily infer on the spectral power of frequencies as low as my resolution allows?

## 1 Answer

If the goal is to map out a range of low frequencies, then CWT is preferred over STFT, as it zooms logarithmically and will provide far more detail (examples). If the goal is a few specific frequencies, then a targeted STFT will work (i.e. design windows for said frequencies rather than doing a linear sweep).

What remains is boundary effects: large windows demand data beyond the span of original input. The answer is padding, but no single perfect scheme: any method (including zero padding) is akin to imposing a statistical prior, i.e. assumption to best-fit the source process.

• I've done much work on this subject and can recommend reflect as the best general scheme, if nothing else is known. Related post, discussion
• However, if the goal is instantaneous frequency/amplitude localization, zero might be favorable (see examples).
• For an advanced approach, see boundary wavelets.

Best results will integrate domain knowledge into the application - i.e., use the "best guess" for how the source behaves outside the measurement. If the source isn't sufficiently oscillatory to begin with, methods other than CWT/STFT may be favorable.