” 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?

As I understand it, the raw time-series can contain higher frequencies above 0.5 Hz that can cause aliasing effects.

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?

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?

My point is that if the data has been sampled properly, you will not have aliasing. How was your data sampled? In other words, where are you getting your data from? A database? A software package that acquires fMRI BOLD data?

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.

Thanks, I understand (but not fully, so one more question to clarify things): Would you then say it is meaningless to apply a bandpass filter in my case? Instead, I could just apply the FFT on the time-series, then "cut off" the lower frequencies below 0.01 Hz and the upper frequencies above 0.1 Hz (frequencies I am not interested in) and proceed with any computation in the frequency-domain? Conversley, applying a bandpass filter can change the power spectrum, such as at the edges of the chosen frequency band, due to filter roll-off etc. I could avoid that because the filter is not required.

Hi, thanks for taking your time

I think the question you should ask yourself is: will the frequencies below 0.01 hz and above 0.1 hz bother you with your processing? If not, there’s no need to do any filtering/cutting. If yes, then the next question is what kind of processing do you intend to do in the frequency domain?

My aim is as follows: I am interested to find out if the log-log transformed frequency domain has a scale-free (power-law) shape. That is, if the frequency-domain has a linear shape (approximately) after the taking the logarithm on both axes. And I am only interested in a specific frequency range, such as 0.01-0.1 Hz in this specific example now

One option to achieve this task is simply by cutting off frequencies below 0.01 and above 0.1 Hz in the script (after the transformation into the frequency-domain). In this case, no bandpass filter would be applied

And I was wondering if that would be sufficient, or if it would me methodologically bad to achieve my aim this way

Based on how I understood your answers, a bandpass filter would not be required for my aim

Got it! Don’t cut anything. Do your log log transformation and THEN look at the linearity of the spectrum in your band of interest.

I’ll write a quick answer, feel free to upvote/accept if it helped ;)

Thank you. I should have better explained my problem right from the beginning. But I thought I keep it simple to not confuse people. In the end, I did not provide enough information, I guess

No worries!

Just a note in case you are intereted: the problem with applying a bandpass filter on the time-series first and then doing the FFT is that even with a cheby filter with a sharp transition at the frequency edges and a relatively high filter order always "bends" the log-log power spectrum at its edges. And those "bends" due to the filter roll-off destroy the scale-free distribution in the log-log spectrum. hence, NOT applying a bandpass filter yields "better" results

Makes sense Philipp. And that’s been my point all along, why would you apply a filter if all you’re interested in is looking at the frequency domain with no interest in converting back to the time domain

Thanks!