Butterworth

As I understand it right now it is used to forecast the data.

Yes, that's a common case for AR(p) models; but in order to forecast, its parameters should be estimated and it is done over the observations you provide to it. Therefore you can have so-called "fitted values" and use them as the "denoised" version of the signal at hand. This is because AR(p) is this:

I recommend that you filter with filtfilt() because filfilt() has exactly zero delay at all frequencies. It achieves this because it filters the signal forward and then backward. The time delays on the forward pass are cancelled out by the equal but opposite delays when it goes through backward. Note that because you are filtering twice, the final attenuation at each frequency is twice as much (if measured in dB) as what you get from a single pass. Therefore you get 6 dB attenuation at the original cutoff frequency, instead of the usual 3 dB. For most applications, including yours I believe, this is not a problem. If it is a problem, then you can adjust the cutoff of the initial filter definition, so that you get 3 dB attenuation at the desired cutoff after both passes.

See attached code and figure for a comparison of delay with filt() and filtfilt(). The figure shows that the raw and filtfilt() signals both transition through y=0 at the exact same time, which reflects the absence of any delay with filtfilt(). The figure also shows that the high frequency noise in the raw signal has been removed by the filter.

If for some reason you do not want to use filtfilt(), then a very good approximation to the delay of a lowpass Butterworth filter of order n, in the passband, is

Delay = Kn/Fc (Delay in s, Fc=cutoff freq in Hz)

where Kn is a constant that depends on the filter order, n:

n=2,3,4,6,8

Kn=0.225,0.318,0.416,0.615,0.816

For more orders and the derivation, see my paper "A general solution for the time delay...", J Biomechanics (2007), https://pubmed.ncbi.nlm.nih.gov/16545388/.

When applying a Butterworth (or any IIR) filter, each output sample is computed based on previous output samples,

I think this project can help you: https://github.com/soygabimoreno/RT

Specifically, you can dig deeper into this class: https://github.com/soygabimoreno/RT/blob/master/app/src/main/java/com/appacoustic/rt/domain/calculator/processing/FilterIIR.kt

For audio filtering in Android, there is a standard equalizer AudioEffect library that looks usable.

FWIW, Java is a painful language to do signal processing in. If you need something beyond Android's AudioEffects, consider using Android NDK and writing in C++.

Yes, filtering a signal with Chebyshev or Butterworth filter shifts the signal. This is known as the "group delay" of the filter: let φ(ω) be the filter's phase response, then the group delay is the derivative

τ_g(ω) = −dφ(ω)/dω.

The group delay is a function of frequency ω, meaning that in general each frequency might experience a different shift, i.e. dispersion. For a linear phase filter like the 3-tap moving average filter, this does not happen; group delay is constant over frequency. But for causal IIR filters, some variation in group delay is unavoidable. For popular IIR designs like Chebyshev or Butterworth, the group delay varies slowly across frequency except around the filter's cutoff.

Like you said, you can advance the signal by some number of samples in post-processing to counteract the shift. Since group delay generally varies with frequency, the number of samples to advance by is a choice about where you want it to match best. Something like the average group delay over the passband is a reasonable place to start.

Alternatively, another way to address shifting is "forward-backward filtering". Apply the filter running forward, then again going backward. That way the shift introduced by the forward pass is counteracted in the backward pass. You can use Matlab's filtfilt or scipy.signal's filtfilt to perform forward-backward filtering.

It doesn't make any sense at all. If you look at your levels now:

The frequency response of the Butterworth filter is not real-valued. When plotting the complex-valued response using plt.plot(), only the real component is shown. You should see a warning:

The initial transient that you see in your graphs is the filter's step response as a sudden input is applied to the filter in its resting state. If you had just connected a physical instrument including such a bandpass filter, the instrument's sensor might have picked up input data samples jumping from 0 (while the probe is disconnected) to the first sample's value of ~0.126V. The response of the instrument's filter would have then shown a similar transient.

However, you are probably more interested in the steady-state response of the instrument after it is no longer disturbed by these external factors (such as the probe being connected), and has had time to settle to the properties of the signal of interest.

One way to achieve this is to use a data sample that is sufficiently long and discard the initial transient. Another approach is to force the filter's initial internal state to something close to what might be expected had a signal of similar amplitude been applied for some time before your first input sample. This can be done for example by setting the initial condition with scipy.signal.lfilter_zi.

Now, I assume you've used butter_bandpass_filter from SciPy Cookbook, which doesn't take care of the filter's initial conditions. Fortunately, it can be easily modified to this end: