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I have time-series data with a sampling rate of 1 Hz, hence the Nyquist frequency is 0.5 Hz.

I am interested in the frequency band 0.01-0.1 Hz and I would like to apply a Periodogram to transform the time- into the frequency-domain to obtain the power spectrum of the data. Subsequently, I would cut the frequency band at both ends to 0.01 and 0.1 Hz, respectively

Question: I understand that a bandpass filter (or at least a low-pass filter) would be required to avoid aliasing when further processing the data in the time-domain. But what about computations in the frequncy-domain within the frequency band 0.01-0.1 Hz (that is way below the Nyquist frequency of 0.5 Hz)? Is it still recommend to bandpass (or low-pass) the time-series before applying a Periodogram, and if so why would it be recommended? Or would a bandpass filter be superfluous or possibly even detrimental to further computations in the frequency-domain, given the Nyquist frequency and my frequency band of interest?

More specifically, I am interested in computing the least-square linear regression (slope) of the log-log transformed frequency-domain (power spectrum). Based on my testing, the slope can slightly change depending on the chosen bandpass filter (such as Butterworth vs. Chebyshev filter) and its parameters, and the slope can further slightly change when comparing bandpassing vs. no bandpassing. This is not surprising, as I understand the differences between both filters. My question is just about the usage of a bandpass filter (that is maybe superfluous) in my specific case.

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  • $\begingroup$ ” I understand that a bandpass filter (or at least a low-pass filter) would be required to avoid aliasing when further processing the data in the time-domain.”…. Why? $\endgroup$
    – Jdip
    Jan 4 at 7:24
  • $\begingroup$ As I understand it, the raw time-series can contain higher frequencies above 0.5 Hz that can cause aliasing effects. $\endgroup$
    – Philipp
    Jan 4 at 7:52
  • $\begingroup$ No. What Nyquist tells you is that if you sample a signal at a sampling rate 1 Hz, only frequencies below 0.5 Hz will be distinguishable from each other. Any frequencies above will alias in the 0-0.5 Hz band. That is why sampling involves low passing the data first. When you have “time-series data with a sampling rate of 1 Hz”, the low-pass+sampling should have been done already and you shouldn’t have any aliasing. Do you have a reason to suspect your data comes from an improperly sampled signal? $\endgroup$
    – Jdip
    Jan 4 at 8:05
  • $\begingroup$ The data comes from fMRI BOLD. Problems are respiration/breathing at around 0.3 Hz that can alias into lower frequencies, as well as cardiac signals at around 1 Hz. I work with digital data and the data was not low-pass or bandpassed yet. I could re-phrase my question by asking: is cutting the frequency band to 0.01-0.1 Hz after the Fourier transform equivalent to first band-passing the time-series to 0.01-0.1 Hz and then applying the Fourier transform? Obviously not, because it depends on the filter and its parameters. Given my circumstances and aim, what is the better option of the two? $\endgroup$
    – Philipp
    Jan 4 at 8:13
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    $\begingroup$ I think you might need to review the basics of sampling theory. The low-pass stage of the ADC process is applied before the sampling operation. If it was not, your data is already aliased. No amount of filtering will reverse that. The reason I’m saying all this is that I don’t believe that you need to do any sort of filtering before processing your data with regards to aliasing issues. Unless you wrote your own ADC, I don’t see why your data would have any aliasing. $\endgroup$
    – Jdip
    Jan 4 at 8:25

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There was some confusion from the OP with regards to the need to low-pass the data because of potential aliasing post sampling. These have been cleared now.

For the OP’s purposes of checking if the log-log transformed frequency domain has a scale-free (power-law) shape, i.e if the frequency-domain has a linear shape after taking the logarithm on both axes, no filtering is needed.
Instead, the data can be first transformed to the log-log frequency domain representation, and the linearity can then be checked for in the frequency band of interest.

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