# Narrowest passband dsp filter one can apply

I was wondering how one would know the narrowest pass-band digital filter one can apply, given the original signal's sampling frequency. For example on a signal coming out from a 14 bit ADC acquiring let's say at 100Msps (maybe a first Hilbert filter was applied there...)

The kind of application I'm thinking of is frequency finding: imagine I have a filter bank, where each filter is 'as thin as possible' so by looking at the 'response' I get in at the output of each filter, I can see in which frequency(ies?) the signal gives the most energy (or gives energy at all).

Supposing unlimited computing power, what's the 'thinnest' individual filter one can build here? What's the limit? I'm guessing it has to with shannon/nyquist but I don't see how to make the link.

Sorry my vocabulary is trash, and I'm a noob on dsp topics.

What's the limit?

I think it's primarily "how long are you willing to wait".

As far as I know there is no theoretical limit on how narrow you can make a bandpass, but you'll run into the limit of the time/frequency "uncertainty" principle: the more something is defined in frequency, the less it is defined in time.

Roughly speaking: a filter that has 1Hz bandwidth will take about 1 second to "get into gear", i.e. approaching a steady state response. An FIR filter of 1Hz bandwidth sampled at 100 MHz will need 100 million taps (give or take an order of magnitude). An IIR filter will have poles extremely close to the unit circle, but with enough numerical precision could still be stable.

The biggest issue IMO would be that it's just not very useful. The narrower the filter, the lower the information content of the output would be. You'll just get a sine wave with a slow amplitude modulation. The narrower the filter, the slower the amplitude variation will get and in the extreme case you'll just end up with a steady state sine wave, which is kind of boring.

• //I think it's primarily "how long are you willing to wait".// and how much buck you wanna pay for the bang.  good answer though. Commented Aug 22, 2021 at 0:20
• If only one frequency is of interest, wouldn't the principles of a locked-in amplifier be more fit? Commented Aug 22, 2021 at 8:23
• Oh OK so it's 'just' a fucntion of how many samples I want to / can integrate. Makes sense. By not very useful, do you mean that a bank of very narrow filters wouldn't give me more information on the actual frequency? The narrower filters wouldn't give more information? Up to a point? Commented Aug 22, 2021 at 9:23
• @a concerned citizen, wouldn't I need a reference signal for that? Or maybe I could 'generate' one (or several?) perfect narrow-band signals (1 for each narrow passband filter?) to 'demodulate'? Real noob here. Commented Aug 22, 2021 at 9:54
• @TouisteurEmporteUneVache: The question here is: what's your input and what exactly do you want to do with the output. The output of an extremely narrow filter will just be a sine wave. You already know that upfront, so what's the point in filtering ? Commented Aug 22, 2021 at 13:55