Re: [music-dsp] Sampling theorem extension
On 09-Jun-15 19:23, Ethan Duni wrote: Could you give a little bit more of a clarification here? So the finite-order polynomials are not bandlimited, except the DC? Any hints to what their spectra look like? How a bandlimited polynomial would look like? Any hints how the spectrum of an exponential function looks like? How does a bandlimited exponential look like? I hope we are talking about one and the same real exponential exp(at) on (-infty,+infty) and not about exp(-at) on [0,+infty) or exp(|a|t). The Fourier transform does not exist for functions that blow up to +- infinity like that. I understood from Sampo Syreeni's answer, that Fourier transform does exist for those functions. And that's exactly the reason for me asking the above question. To do frequency domain analysis of those kinds of signals, you need to use the Laplace and/or Z transforms. Equivalently, you can think of doing a regular Fourier transform after applying a suitable exponential damping to the signal of interest. This will handle signals that blow up in one direction (like the exponential), but signals that blow up in both directions (like polynomials) remain problematic. Not good enough. If we're talking about unilateral Laplace transform, then it introduces a discontinuity at t=0, which immediately introduces further non-bandlimited partials into the spectrum. I'm not sure how you suppose to answer the question of the original signal being bandlimited in this case. With bilateral Laplace transform it's also complicated, because the damping doesn't work there, except possibly at one specific damping setting (for an exponent, where for polynomials it doesn't work at all), yielding a DC. I'm not fully sure, how to analytically extend this result to the entire complex plane and whether this will make sense in regards to the bandlimiting question. That said, I'm not sure why this is relevant? Seems like you aren't so much interested in complete exponential/polynomial functions over their entire domain, but rather windowed versions that are restricted to some small time region? I am specifically interested in the functions on the entire real axis. Further in my original email there is an explanation of the reasons. Regards, Vadim -- Vadim Zavalishin Reaktor Application Architect | R&D Native Instruments GmbH +49-30-611035-0 www.native-instruments.com -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
Re: [music-dsp] Sampling theorem extension
On 09-Jun-15 22:08, robert bristow-johnson wrote:
a Nth order polynomial, f(x), driven by an x(t) that is bandlimited to B
will be bandlimited to N*B. if you oversample by a ratio of at least
(N+1)/2, none of the folded images (which we call "aliases") will reach
the original passband and can be filtered out with an LPF (at the high
sampling rate) before downsampling to the original rate. with 4x
oversampling, you can do a 7th-order polynomial and avoid non-harmonic
aliased components.
We are not talking about signals being fed through the polynomials. We
are talking about the polynomials as the signals.
I'm failing to see how Euler equation can relate exponentials of a real
argument to sinusoids of a real argument? Any hints here?
let f(x) be defined to be f(x) = e^(j*x)/(cos(x) + j*sin(x))
> .
I'm failing to see how Euler equation can relate exponentials of a *real
argument* to sinusoids of a *real argument*
I hope we are talking about
one and the same real exponential exp(at) on (-infty,+infty) and not
about exp(-at) on [0,+infty) or exp(|a|t).
oh, (assuming you meant e^(-|a*t|)), them's are in the textbooks.
Did I correctly understand you? The Fourier transform of exp(at) where a
and t are real and t is from -infty to +infty is in the textbooks? Any
hints how it looks like?
Yes, this is what I was referring to. Currently I'm interested in the
class of functions which are representable as a sum of a real function,
which, if analytically extended to the complex plane, is entire and
isolated derivative discontinuity functions (non-bandlimited versions of
BLEPs BLAMPs etc).
i think, if you allow for dirac impulses (or an immeasureably
indistinguishable approximation of width 10^(-44) second), any finite
and virtually bandlimited function will do. if you insist on being
strict with your mathematics, i can't help you anymore (it's been more
than 3 decades since i cracked open any Real Analysis or Complex
Variables or Functional Analysis textbook)
The problem currently is not the impulses, but the entire (complex
analytical) part of the signal.
BTW, i am no longer much enamoured with BLIT and the descendents of
BLIT.
I'm not sure how BLEP is a descendant of BLIT
Because, if the continuous part is bandlimited, then we have "just" to
replace the discontinuities by their bandlimited versions (the essence
of the BLEP approach) and the remaining question is only: if there are
infinitely many discontinuities at a given point, whether the sum of
their bandlimited versions will converge.
they don't. imagine a perfect brickwall filter with sinc(t) as its
impulse response. now drive the sonuvabitch with
{ (-1)^n n>=0
x[n] = {
{ -(-1)^n n<0
which is x[n] = -1, +1, -1, +1, +1, -1, +1, -1,
^
|
|
x[0]
and see what you get. it ain't BIBO.
Interesting observation. I might need to think a little bit more about
this :)
However I'm not sure how this is related to the convergence of the BLEPs
in the context which we are talking about.
Regards,
Vadim
--
Vadim Zavalishin
Reaktor Application Architect | R&D
Native Instruments GmbH
+49-30-611035-0
www.native-instruments.com
--
dupswapdrop -- the music-dsp mailing list and website:
subscription info, FAQ, source code archive, list archive, book reviews, dsp
links
http://music.columbia.edu/cmc/music-dsp
http://music.columbia.edu/mailman/listinfo/music-dsp
Re: [music-dsp] Sampling theorem extension
robert bristow-johnson wrote: On 6/9/15 4:32 AM, Vadim Zavalishin wrote: Creating a new thread, to avoid completely hijacking Theo's thread. it's a good idea. I agree that there was the possibility of an unstable offense resolution, but I wasn't aware people were being afraid of that concept. Look, it's a matter of decent engineer interactions and more so: decent scientific ausbildung, scientific method, and let's say "respect after failure". Moreover, for a lot of people who either are glad not to have to delve much in the mathematics, or who didn't get the chance to access proper higher education in the various fields, it might be the intellectual robbery that could be imminent isn't clear, and certainly disgusting. I was glad to have been informed, as undergrad student, of the underpinnings I've put forward here, and maintain there are a few main things about sampling that I think some people ought to know, and as it so happens do know a bit about now. It's decent for academic engineers to follow a path where first they score enough points in the undergrad realm, then get taught decent mutual respect and communication modes about engineering subjects, then hopefully learn how to master a subject in science, and then they're off to be a decent, usually on the cool side, person with the ability to get into engineer problems at the appropriate level and deal with scientific sides to their work. I've been around a European (at that time) top university long enough to know why that is so, and what's wrong with all kinds of funny and slightly interesting nerdy students trying to work themselves to a position of power, and I won't condone it in general, if I can in all decency help it. So when people work on a subject, get corrected at undergrad level (the same where many have passed through and are satisfied and usually successful with, and where the subjects taught are centuries old and tried) it's not proper to just happily go on and act as if a personal and professional sense of honor can be seconded to some end that will justify all inter-engineer injustice, and in the end social interactions with all people matter not enough to be a solid and recognizable person. Anyhow, as a summary once more: the only proper way to sample a signal (with the obvious conditions luckily reiterated regularly) and process it or play it back properly is based on a theory that cannot be internally reversed or made into a local signal processing idea, while maintaining general applicability. And I know there are some signal precautions and some modes of processing possible that IMO have been thought about at least in the 60s, and maybe before I was born. Unfortunately a lot of software and DSP code is just as limited as it is and that's not going to change if enthusiastic and clearly extremely immature mathematicians are going to try out new "theories" or engage in opportunistic word games. It just is no different, even if I'd want it to be. So once more: it doesn't matter what you do in sample space much, if I don't see sinc function reconstruction preferably with a quantification of the errors involved, I'm not going to ratify the ideas as scientifically proper enough to make a theoretic strong point with, let alone history. Maybe I am actually sorry as a person that there are so many errors in the often promoted as "perfect" digital signal processing domain, but that doesn't change anything! So about that idea (not really of mine) to think about the effect of the DACs interpolation and smoothing filters: that's real, but still you need the *properly reconstructed signal* first, and THEN on top of that make sure the signal wurst-ing that goes on in the DAC comes out the way you want. Terribly complicated as that seems, to me it's rather basic. T. -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
Re: [music-dsp] Sampling theorem extension
>If we're talking about unilateral Laplace transform,
No, the full-blown ("bilateral") Laplace and Z transforms.
>With bilateral Laplace transform it's also complicated, because the
>damping doesn't work there, except possibly at one specific damping
>setting (for an exponent, where for polynomials it doesn't work at
>all), yielding a DC
Why isn't that sufficient? Do you need a bigger region of convergence for
something? Note that the region of convergence for a DC signal is also
limited to the real line/unit circle (for Laplace/Z respectively). I'm
unclear on exactly what you're trying to do with these quantities.
>I'm not fully sure, how to analytically extend this result to the entire
>complex plane and whether this will make sense in regards to the
>bandlimiting question.
I'm not sure why you want to do that extension? But, again, note that you
have the same issue extending the transform of a regular DC signal to the
entire complex plane - maybe it would be enlightening to walk through what
you do in that case?
>I am specifically interested in the functions on the entire real axis.
>Further in my original email there is an explanation of the reasons.
I guess I'm not following what the goal is here. I get that you're
interested in extending functions to the entire complex plane, but I'm not
sure why that is, and I'm also unclear on why you need to handle naked
exponential signals (rather than windowed versions thereof, wherein the
Fourier transform is well defined to begin with)? Note that the "window"
could still have infinite support - it only needs to ensure that the result
is in L2. Why are we starting with Fourier transforms and then extending
those analytically to the whole complex plane - isn't that simply the
Laplace/Z transform, in the first place?
Also I have to say that I do not see much prospect for doing frequency
analysis of entire polynomials like this, since they do not admit Laplace/Z
transforms. They don't have valid spectra anywhere on the complex plane, so
I don't see what there is to extend?
E
On Wed, Jun 10, 2015 at 1:25 AM, Vadim Zavalishin <
vadim.zavalis...@native-instruments.de> wrote:
> On 09-Jun-15 19:23, Ethan Duni wrote:
>
>> Could you give a little bit more of a clarification here? So the
>>> finite-order polynomials are not bandlimited, except the DC? Any hints
>>> to what their spectra look like? How a bandlimited polynomial would look
>>> like?
>>>
>>
>> Any hints how the spectrum of an exponential function looks like? How
>>> does a bandlimited exponential look like? I hope we are talking about
>>> one and the same real exponential exp(at) on (-infty,+infty) and not
>>> about exp(-at) on [0,+infty) or exp(|a|t).
>>>
>>
>> The Fourier transform does not exist for functions that blow up to +-
>> infinity like that.
>>
>
> I understood from Sampo Syreeni's answer, that Fourier transform does
> exist for those functions. And that's exactly the reason for me asking the
> above question.
>
>
> To do frequency domain analysis of those kinds of
>> signals, you need to use the Laplace and/or Z transforms. Equivalently,
>> you
>> can think of doing a regular Fourier transform after applying a suitable
>> exponential damping to the signal of interest. This will handle signals
>> that blow up in one direction (like the exponential), but signals that
>> blow
>> up in both directions (like polynomials) remain problematic.
>>
>
> Not good enough. If we're talking about unilateral Laplace transform, then
> it introduces a discontinuity at t=0, which immediately introduces further
> non-bandlimited partials into the spectrum. I'm not sure how you suppose to
> answer the question of the original signal being bandlimited in this case.
> With bilateral Laplace transform it's also complicated, because the damping
> doesn't work there, except possibly at one specific damping setting (for an
> exponent, where for polynomials it doesn't work at all), yielding a DC. I'm
> not fully sure, how to analytically extend this result to the entire
> complex plane and whether this will make sense in regards to the
> bandlimiting question.
>
>
>> That said, I'm not sure why this is relevant? Seems like you aren't so
>> much
>> interested in complete exponential/polynomial functions over their entire
>> domain, but rather windowed versions that are restricted to some small
>> time
>> region?
>>
>
> I am specifically interested in the functions on the entire real axis.
> Further in my original email there is an explanation of the reasons.
>
> Regards,
> Vadim
>
>
>
> --
> Vadim Zavalishin
> Reaktor Application Architect | R&D
> Native Instruments GmbH
> +49-30-611035-0
>
> www.native-instruments.com
> --
> dupswapdrop -- the music-dsp mailing list and website:
> subscription info, FAQ, source code archive, list archive, book reviews,
> dsp links
> http://music.columbia.edu/cmc/music-dsp
> http://music.columbia.edu/mailman/listinfo/music-dsp
>
--
dupswapdrop -- the music-dsp mailing list and w
