# How to apply an anti-aliasing filter before downsampling

I have an eeg signal with two channels (f3m2 and f4m1) which is divided into epochs. I want to augment the data by taking every 5th sample. I can re-use the discarded samples to create 5 versions of the data (take every 5th sample starting at sample 0, then take every 5th sample starting at sample 1, etc.). I have written my own code to downsample the data starting at sample 0, 1 etc. but I know that in order to avoid aliasing I need to first apply a low pass filter.

How do I decide what filter to use (FIR, IRR, etc.)? How do I decide which cutoff frequency to use? How would I implement this?

My code is in python and I have been considering using the scipy.signal library (but am open to any python libraries).

• "I can re-use the discarded samples to create 5 versions of the data" Why? After filtering and downsampling, won't those all contain exactly the same information? Feb 12 '21 at 1:39

Anti-aliasing filtering is applied just as any other LTI filtering: If your input data is $$x[n]$$, and the impulse response is $$h[n]$$, then your output will be
$$y[n] = x[n] \star h[n]$$
where $$\star$$ is the convolution operation, a.k.a. the anti-aliasing filtering in this context.
Your impulse response $$h[n]$$, ideally, corresponds to a lowpass brickwall filter, with a cutoff frequency of $$\omega_c = \frac{\pi}{M}$$ radians per sample, where $$M=5$$ is the downsampling ratio in your particular application. You may design $$h[n]$$ by any suitable method, but for maintaining high accuracy, you better use tight specs on the desiged filter, this can be achieved by a sufficiently long FIR linear-phase filter, or similar IIR filter. The latter will be more efficient to implement but less accurate due to its nonlinear phase.
Then you can decimate $$y[n]$$ by $$M$$ as $$v[n]= y[Mn]$$ to get the downsampled sequence.
If you're using scipy.signal and processing signals offline, then you can just use decimate which handles the filtering for you. It also does zero-phase filtering by default, which you probably want for an EEG signal to avoid shifting the shape of the waveforms? (I know that's desirable for EKG, not sure about EEG.)