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I have a question about the use of anti-aliasing filters prior to sampling a finite time signal and if they actually help anything.

The point of the anti-aliasing filter is to remove high-frequency components to reduce aliasing. It's know a signal cannot be both band-limited and time-limited thus time limited signals are first low-passed to "reducing aliasing",

This sounds good but it seems to create its own problems in so far as it will create infinite(or near infinite) duration output as a response to a finite input.

We can't sample forever, no matter what the signals samples will be finite in quantity. So aren't we creating as big of a problem as we are solving?

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    $\begingroup$ Would you be satisfied by a proof that if a finite-valued finite-time signal/function is ideally low-pass filtered using a sinc impulse response, then the resulting band-limited signal/function can be reconstructed to arbitrary accuracy from a finite number of its samples? I think such a proof could be constructed. $\endgroup$
    – Olli Niemitalo
    Apr 19, 2018 at 6:04

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Math and Physics are not the same thing. They can be very close but infinity is hard to realize. Going from analog to digital, a more accurate description is an average of the waveform over a small time window and the average is in turn quantized. Filters don’t have infinite stop bands but 80 dB is usually more than good enough. There is always a small error, but one can bound it.

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  • $\begingroup$ Sure but the issue is more fundamental: you are reducing sampling rate at the expense of increasing sampling duration. In the extreme case it's possible the increase in sampling duration will make practical truncations impossible. $\endgroup$
    – FourierFlux
    Apr 9, 2018 at 7:19
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    $\begingroup$ The Nyquist rate is a bound, a minimum. You can’t achieve a brick wall anti aliasing filter so one has to sample faster than the Nyquist rate. One always aliases. Under the ideal conditions you alias zero. In reality you choose a filter and rate that minimizes the aliasing below some level. Nature also does its own filtering as well because most natural signals fall off at higher frequencies. If you have very short duration signals, you will probably need to go around 5x a nominal Nyquist rate. A modern audio A/D converter actually samples at hundreds Nyquist, at one bit. $\endgroup$
    – user28715
    Apr 9, 2018 at 9:10
  • $\begingroup$ Your also not limited to Sinc interpolation. You can reconstruct with splines as well. One is not constrained to periodic sampling either. You need to have criteria to control the error. There is no perfect anything. $\endgroup$
    – user28715
    Apr 9, 2018 at 9:13
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By allowing a little error, you can get away with a limited length filter, and it will introduce less error in the frequency domain than sampling the signal directly without filtering first.

Here is an illustrative only example using finite impulse response (FIR) windowed sinc filtering, which is compared to sinc filtering and direct sampling of a rectangular pulse. The length of the rectangular pulse is 10, the cutoff frequency of the sinc is $\pi$. The window function is Blackman–Nuttall and is of length 12 sampling periods at the critical sampling frequency of the sinc.

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Figure 1. Sinc (red) and windowed sinc (blue) filter impulse response.

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Figure 2. Rectangular pulse (grey), filtered with the sinc (red) and with the windowed sinc (blue) filter. Both ways of filtering introduce time-domain error in the pulse.

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Figure 3. Power spectra of the rectangular pulse (grey) filtered with the sinc (red, goes to $-\infty\text{ dB}$ at $\omega = \pi$) and with the windowed sinc (blue) filter. The vertical axis in decibels.

Sinc filter gives no frequency domain error at all for frequencies below $\pi$. The windowed sinc filter gives very little error at frequencies below about 2 (pass band) and has a very high attenuation at frequencies above about 5 (stop band).

Let's consider that the signal of interest is below frequency 2 and we sample the windowed sinc filtered signal at a sampling frequency of 7. The frequencies between 3.5 and 5 will alias to between frequencies 3.5 and 2. Only the greatly attenuated stop band aliases to the band of interest, introducing very little error. So in this case we could get away with a limited length filter and a limited sampling frequency. If we were to sample the rectangular pulse directly, there would be plenty of aliasing to the band of interest.

To continue with a different aspect, consider that we want to measure the start time of the pulse. If we don't low-pass filter, all start times between the time of a sample and the time of the next sample will result in identical sampled signals. If we low-pass filter, the start time can be resolved at arbitrary accuracy by increasing the filter length as the error in the measured start time will approach zero when the FIR filter length approaches infinity. So it pays of to low-pass filter even if the filter is not perfect.

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  • $\begingroup$ Hi, this is a bit late as I got pre-occupied, your answer is great but it doesn't really address my issue: A low-pass filtered signal will have a length of infinite duration, which means it would be impossible to sample it in a finite amount of time, which means you're going to lose information, since even if you have an 100% accurate quantizer, you're not sampling the signal forever. So I'm still not seeing how a low-pass filtered signal can be digitized more accurately than an aliased signal(in general). $\endgroup$
    – FourierFlux
    Apr 13, 2018 at 5:32

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