...I suppose I should give a more useful answer than that, though.
Given any f(x): Is it possible to sample f(x) with a given sample-rate
> ignoring all frequencies (slopes) higher than SF/2?
Yes, but it isn't necessarily a good idea. The correct answer would be the
continuous convolution of f(x) with a sinc function scaled to match the
sampling rate of the signal. That is,
[image: Inline image 3]
TeX: f_{AA}(T) = \int_{-\infty}^{\infty}
f\bigg(\frac{T+t}{SR}\bigg)\frac{\sin(\pi t)}{\pi t} \delta t
Where T is the index of the sample and SR is the sampling rate.
But I think this is massive overkill. Even if you're happy to solve the
integral there and pay the CPU cost of crunching the numbers properly,
there's a strong possibility that you'll perceive artifacts resulting from
the time-domain "smear" in your signal. The theory here also breaks down
completely if your f(x) changes, such as shifting in fundamental frequency
--- although it holds if you interpolate between different waveforms.
Anyway, while I'm happy to illuminate this road I can't advise you to walk
it in good conscience.
– Evan Balster
creator of imitone <http://imitone.com>
On Thu, Sep 15, 2016 at 3:41 PM, Evan Balster <e...@imitone.com> wrote:
> General note: As things get clearer in the time domain, they tend to get
> fuzzier in the frequency domain. "Perfect" frequency-domain solutions
> (like sinc interpolation) tend to smear the signal in the time domain; that
> is the mirror image of the problem you're experiencing now. Don't forget
> that one of these is just a mathematical model for the other.
>
> It's my observation that when people ask for the best of both worlds on
> music-dsp, they tend to get overwhelmed by suggestions, opinions and
> academic papers devoted to overcoming a basic law of signal processing
> <https://en.wikipedia.org/wiki/Uncertainty_principle#Signal_processing>,
> when far simpler techniques will often suffice.
>
> So here's my suggestion: Don't be too clever. Just try smoothing out the
> waveforms a bit to combat the aliasing. It comes, roughly, from
> discontinuity in the waveform. Oversample and filter, or sample from
> "soft" square and sawtooth functions with a smoothing parameter built in.
>
>
> Given any f(x): Is it possible to sample f(x) with a given sample-rate
>> ignoring all frequencies (slopes) higher than SF/2?
>
>
> *This question is a pathway into madness.*
>
> – Evan Balster
> creator of imitone <http://imitone.com>
>
> On Thu, Sep 15, 2016 at 2:07 PM, André Michelle <andre.miche...@gmail.com>
> wrote:
>
>> I could phrase my question more general:
>>> Given any f(x): Is it possible to sample f(x) with a given sample-rate
>>> ignoring all frequencies (slopes) higher than SF/2?
>>>
>>>
>> Couldn't you just implement a brickwall FIR filter in your signal chain,
>> after whatever signal you're deciding to generate (prior to sampling, if
>> your in the analog domain)?
>>
>>
>> From what I understand it is impossible to get rid of aliased frequencies
>> after sampling.
>>
>> ~
>> André
>>
>>
>> _______________________________________________
>> dupswapdrop: music-dsp mailing list
>> music-dsp@music.columbia.edu
>> https://lists.columbia.edu/mailman/listinfo/music-dsp
>>
>
>
_______________________________________________
dupswapdrop: music-dsp mailing list
music-dsp@music.columbia.edu
https://lists.columbia.edu/mailman/listinfo/music-dsp
