Bandwidth of a finite impulse response filter

Question

In a machine learning application, a model learns to apply a filter h(n) via convolution to a 1-dimensional input signal s(n) (e.g. sampled at Fs = 16 kHz). The filter size (filter length?) is limited to a fixed number, e.g. k = 100, so the model cannot learn a filter that exceeds this duration but it is free to insert zeros and make the filter "shorter".

Does this mean that the filter size limits the duration of the "slowest" signal component that falls into the passing bandwidth and hence Fs/k = 160 Hz is the minimal bandwidth that h(n) can represent (assuming it's a band pass filter)?

Answer 1

Yes, the length of a filter limits the "frequency-selectiveness". If you remember modern physics, that's mathematically very related to the Heisenberg uncertainty principle: something that is temporally short can't be very sharp in the frequency domain, and vice versa.

You can find one example of such a constraint in this answer, though the formula only applies to specific filter constraint.

Generally, filter design is one of the oldest application fields of evolutionary / ML algorithms. You'll be delighted to find that even some of the very "classical", non-ML algorithms for bandpass filter design make extensive use of a loss / reward function while optimizing a filter.

If you're doing modern machine learning for filer design, there's a lot of modern literature, but I'd really start by reading up on the Remez exchange (or just Remez) algorithm, to give you a starting direction. Understanding that relatively simple algorithm and its convergence will give you a solid base!

Especially when later writing a report on it, that would be the most basic benchmark for your ML approach that I could think of. Compare yourself to that!