Well, this is sort of philosophical in a sense, but when those samples are used to represent band-limited signals, they are strictly speaking not impulses, but rather data-points ("samples") on the unique band-limited signal that passes through those data-points... and if we replace all those data-points with scaled cardinal sines that we add together, we obtain said band-limited signal, though we could also obtain the same signal by fitting a polynomial (of infinite degree) through all the data-points (with infinite zero-padding, since "band-limited" implies compact support of the spectrum which means we can't have compact support in time)."Samples aren't [or don't represent] impulses"—OK, then why do we test an LTI system's impulse response by feeding it a single non-zero sample? You feed an impulse to obtain an impulse response, no?
But it turns out it does not matter in LTI cases, because LTI systems can be represented by convolution and convolution is commutative, so the result is the same whether we convolve by the cardinal sine before (samples are data-points of a band-limited signal) or after (samples are impulses and we band-limit afterwards when we reconstruct into continuous time) an LTI system.
Once we do something non-linear (or time-varying.. but that's basically the same thing) this is no longer true and treating the samples as impulses predicts the aliasing we observe because of the periodicity of the spectrum of a sampled signal treated as impulses... and this is where things get slightly philosophical, because now we need to choose what the "correct" behaviour should look like: are we willing to accept aliasing as the natural result of "samples as impulses" view, or are we going to try to go into extra lengths to try and obtain results similar to what we would have obtained had we reconstructed the band-limited signal, applied the non-linear process in continuous-time, then band-limited and sampled the results?
I'd generally argue that for musical purposes you (usually) would prefer the latter and go into some extra pains to try to make it happen... but for LTI purposes (eg. linear filter design) it's all irrelevant, because convolution is commutative.
Statistics: Posted by mystran — Mon Mar 25, 2024 2:40 am