How to apply an anti-aliasing filter before downsampling

Question

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).

Answer 1

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.

Answer 2

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.)