Creating a new thread, to avoid completely hijacking Theo's thread.
Previous message here:
http://music.columbia.edu/pipermail/music-dsp/2015-June/073769.html
On 08-Jun-15 18:29, Sampo Syreeni wrote:
On 2015-06-08,vadim.zavalishin wrote:
The sampling theorem applies only to the class signals w
>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
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.
Previous message here:
http://music.columbia.edu/pipermail/music-dsp/2015-June/073769.html
On 08-Jun-15 18:29, Sampo Syreeni wrote:
On 2015-06-08,vadim.zavalishin
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 exponent
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
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 conce
>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 polynomia
On 10-Jun-15 21:26, Ethan Duni wrote:
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
On 2015-06-09, Ethan Duni wrote:
The Fourier transform does not exist for functions that blow up to +-
infinity like that. To do frequency domain analysis of those kinds of
signals, you need to use the Laplace and/or Z transforms.
Actually in the distributional setting polynomials do have Fou
On 11-Jun-15 11:00, Sampo Syreeni wrote:
I don't know how useful the resulting Fourier transforms would be to the
original poster, though: their structure is weird to say the least.
Under the Fourier transform polynomials map to linear combinations of
the derivatives of various orders of the delt
HI
While it's cute you all followed my lead to think about simple
continuous signals that are bandwidth limited, such that they can be
used as proper examples for a digitization/synthesis/reconstruction
discipline, I don't recommend any of the guys I've read from here to
presume they'll make
On 2015-06-11, Vadim Zavalishin wrote:
So they can be considered "kind of" bandlimited, although as I noted
in my other post, it seems to result in DC offsets in their restored
versions, if sinc is windowed.
Not really, if the windowing is done right. The DC offsets have more to
do with the
Sampo Syreeni писал 2015-06-11 15:55:
On 2015-06-11, Vadim Zavalishin wrote:
So they can be considered "kind of" bandlimited, although as I noted
in my other post, it seems to result in DC offsets in their restored
versions, if sinc is windowed.
Not really, if the windowing is done right. Th
On 2015-06-11, vadim.zavalishin wrote:
Not really, if the windowing is done right. The DC offsets have more
to do with the following integration step.
I'm not sure which integration step you are referring to.
The typical framework starts with BLITs, implemented as interpolated
wavetable loo
On 2015-06-09, robert bristow-johnson wrote:
so the way i resolve it in my head is say that (among other facts like
there really aren't any signals, remaining non-zero, that go off to t
= +/- infinity) i will model, in my imagination, the ideal impulse
function in time as having area of 1 (and
On 6/11/15 5:39 PM, Sampo Syreeni wrote:
On 2015-06-09, robert bristow-johnson wrote:
BTW, i am no longer much enamoured with BLIT and the descendents of
BLIT. eventually it gets to an integrated (or twice or 3 times
integrated) wavetable synthesis, and at that point, i'll just do
bandlimite
On 11-Jun-15 19:58, Sampo Syreeni wrote:
On 2015-06-11, vadim.zavalishin wrote:
Not really, if the windowing is done right. The DC offsets have more
to do with the following integration step.
I'm not sure which integration step you are referring to.
The typical framework starts with BLITs,
On 11 Jun 2015, at 19:58, Sampo Syreeni wrote:
> Now, I don't know whether there is a framework out there which can handle
> plain exponentials, a well as tempered distributions handle at most
> polynomial growth. I suspect not, because that would call for the test
> functions to be faster de
On 12-Jun-15 12:54, Andreas Tell wrote:
I think it’s not hard to prove that there is no consistent
generalisation of the Fourier transform or regularisation method that
would allow plain exponentials. Take a look at the representation of
the time derivative operator in both time domain, d/dt, and
On 12 Jun 2015, at 14:31, Vadim Zavalishin
wrote:
> On one hand cos(omega0*t) is delta(omega-omega0)+delta(omega+omega0) in the
> frequency domain (some constant coefficients possibly omitted). On the other
> hand, its Taylor series expansion in time domain corresponds to an infinite
> sum o
>The fact that the constant maps to a delta and the successive higher
>derivatives to monomials of equally higher order sort of correspond to
>the fact that in order to approximate something with such fiendishly
>local structure as a delta (corresponding in convolution to taking the
>value) and its
Ethan Duni писал 2015-06-12 23:43:
However, if
I'm following this correctly, it seems to me that the problem of
multiplication of distributions means that the whole basic set-up of
the
sampling theorem needs to be reworked to make sense in this context.
I.e.,
not much point worrying about whet
It seems mathematics haven't given or taken away anything: I didn't
invent them.
Of course it's possible to make models and approximations, and it would
be nice if some indication of error bounds could be given.
T.
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Upon a little bit more thinking I came to the conclusion that the
expressed in the earlier post (quoted below) idea should work.
Indeed, the windowed signal y(t) can be represented as a series of
windowed monomials, by simply windowing each of the terms of its Taylor
series separately. If the
On 2015-06-12, Ethan Duni wrote:
Thanks for expanding on that, this is quite interesting stuff.
However, if I'm following this correctly, it seems to me that the
problem of multiplication of distributions means that the whole basic
set-up of the sampling theorem needs to be reworked to make se
>Now that I read up on it... Actually no. Every tempered distribution has a
>Fourier transform, and if that's compactly supported, the original
distribution
>can be reconstructed via the usual Shannon-Whittaker sinc interpolation
>formula. That also goes for polynomials and sine modulated polynomia
On 2015-06-19, Ethan Duni wrote:
I guess what we lose is the model of sampling as multiplication by a
stream of delta functions, but that is more of a pedagogical
convenience than a basic requirement to begin with.
In fact even that survives fully. In the local integration framework
that the
>Of course some funky global, dual shit happens then: you
>actually need all of the samples from -inf to +inf in order to
>define any polynomial, and no finitely supported in time subset will
suffice.
Right, this is what I was getting at with the convergence line of thinking.
We theoretically need
On 6/19/15 5:03 PM, Sampo Syreeni wrote:
On 2015-06-19, Ethan Duni wrote:
I guess what we lose is the model of sampling as multiplication by a
stream of delta functions, but that is more of a pedagogical
convenience than a basic requirement to begin with.
pedagogical convenience, schmedagogi
On 2015-06-19, Ethan Duni wrote:
We theoretically need all samples from -inf to +inf in the regular
sampling theorem as well, [...]
Not exactly. If you take the typical sampling formula, with equidistant
samples, you need them all. But in theory pretty much any numerable
number of samples fr
On 2015-06-19, robert bristow-johnson wrote:
i thought that, because of my misuse of the Dirac delta (from a
mathematician's POV, but not from an EE's POV), i didn't think that
the "model of sampling as multiplication by a stream of delta
functions" was a living organism in the first place. i
On 6/19/15 7:22 PM, Sampo Syreeni wrote:
Nota bene, this is not EE stuff per se. This is heady math stuff, used
to formalize what you EEs wanted to do all along. It's the kind of
collaboration where us math freaks provide the rubber...and then you
EE folks can finally fuck your sister in peace
>Not exactly. If you take the typical sampling formula, with equidistant
samples, you need them all.
Yeah, that's what we're discussing isn't it?
>But in theory pretty much any numerable number of samples from any compact
interval will do.
Sure, but that's not going to help us with figuring out
On 2015-06-19, Ethan Duni wrote:
Not exactly. If you take the typical sampling formula, with
equidistant samples, you need them all.
Yeah, that's what we're discussing isn't it?
Are we? You can approximate any L_2 bandlimited function arbitrarily
closely with a finite number of samples. I d
After some googling I rediscovered (I think I already found out it one
year ago and forgot) the Paley-Wiener-Schwartz theorem for tempered
distributions, which is closely related to what I was aiming at. It
gives the sufficient and necessary condition of bandlimitedness in terms
of the exponent
On 2015-06-22, Vadim Zavalishin wrote:
After some googling I rediscovered (I think I already found out it one
year ago and forgot) the Paley-Wiener-Schwartz theorem for tempered
distributions, which is closely related to what I was aiming at.
It'll you land right back at the extended sampling
On 22-Jun-15 21:59, Sampo Syreeni wrote:
On 2015-06-22, Vadim Zavalishin wrote:
After some googling I rediscovered (I think I already found out it one
year ago and forgot) the Paley-Wiener-Schwartz theorem for tempered
distributions, which is closely related to what I was aiming at.
It'll you
On 2015-06-23, Vadim Zavalishin wrote:
It's just that you don't need any of that machinery in order to deal
with that mode of synthesis, and you can easily see from the
distributional theory that you can't do any better.
It seems I can do better. Because my question is not whether an
infinit
On 24-Jun-15 15:31, Sampo Syreeni wrote:
Certainly any signal with compact support isn't bandlimited. That's the
simplest form of the uncertainty principle. But even if you take a
strictly bandlimited "window" function with rapid falloff (a bandlimited
square/flattop convolved with itself a coupl
>I want to take a rectangular window. But then I can apply
>the BLEP method to the discontinuities in the function and
>the derivatives arising from this window. Now, do I get a
>bandlimited version in the result? If I apply the window to a
>polynomial (particularly a straight line, occuring in the
On 24-Jun-15 21:30, Ethan Duni wrote:
Could you expand a bit on exactly what it means to apply the BLEP method to
the discontinuities? I have a general grasp of the basic idea but I'm a bit
fuzzy on exactly what this means in practice. If you're getting a
truly band limited signal, then isn't the
Thanks for that Vadim, your pdf is quite helpful. I guess the kicker with
this approach is that we require knowledge of all of the signal's
derivatives on each side of every discontinuity? I also appreciate your
comment that min-phase BLEP disturbs the phase relationships and so gives
quite differe
On 2015-06-26, Ethan Duni wrote:
Thanks for that Vadim, your pdf is quite helpful.
Same here. Now I too think I know where we're getting at. Take the
simplest case of a rising sawtooth modulated by another one.
The usual sawtooth is generated by integrating a constant summed with
a careful
Sampo Syreeni писал 2015-06-28 18:39:
What makes all of that suspect is that at first it does seem to imply
that all of the interesting spectral information is in the
discontinuities.
That's until you begin considering analytic signals having infinitely
long Taylor series. Like a sine. Or an
On 2015-06-29, vadim.zavalishin wrote:
Roughly (or maybe even strictly?) speaking, the sums converge if and
only if the derivatives fall off sufficiently fast, which seems to be
equivalent to the relaxed (exponential growth on the complex plane,
instead of exponential growth on the imaginary a
| Because you want to distinguish between the waveforms which can be
antialiased by BLEP (at least theoretically) and those which can not.
But all waveforms can be "antialiased" by brick wall filtering, ie. sine
cardinal interpolation.
On Mon, Jun 29, 2015 at 6:49 PM, vadim.zavalishin <
vadim.zav
On 2015-06-29, Emanuel Landeholm wrote:
But all waveforms can be "antialiased" by brick wall filtering, ie.
sine cardinal interpolation.
The point is that you can't represent the continuous time waveforms in
the usual sampled form, and then apply a sinc filter. Which you need to
do in order
On 30-Jun-15 00:43, Sampo Syreeni wrote:
And even if what we've been talking about above does go as far as I
(following Vadim) suggested, exponential segments are still out of
the picture for now.
I would say the whole thread has been started mostly because of the
exponential segments. How are
On 6/29/15 6:43 PM, Sampo Syreeni wrote:
On 2015-06-29, Emanuel Landeholm wrote:
But all waveforms can be "antialiased" by brick wall filtering, ie.
sine cardinal interpolation.
The point is that you can't represent the continuous time waveforms in
the usual sampled form, and then apply a si
On 2015-06-30, Vadim Zavalishin wrote:
And even if what we've been talking about above does go as far as I
(following Vadim) suggested, exponential segments are still out of
the picture for now.
I would say the whole thread has been started mostly because of the
exponential segments. How are
On 2015-06-30, robert bristow-johnson wrote:
but wavetable synthesis *is* a framework that can do that for any
periodic (or quasiperiodic) signal.
How do you derive the hard bandlimited wavetable for an exponential,
rising segment? In closed form, so that your wavetable doesn't already
conta
So we're back where I started to make comments on a while ago. Hmm, I
knew that.
Let's go over the problem shortly again, and let me give one pointer for
you guys (and gals ?) who feel lost about the perfection many of us
probably would like.
It isn't that we cannot create frequency limited
On 7/5/15 10:09 PM, Sampo Syreeni wrote:
On 2015-06-30, robert bristow-johnson wrote:
but wavetable synthesis *is* a framework that can do that for any
periodic (or quasiperiodic) signal.
How do you derive the hard bandlimited wavetable for an exponential,
rising segment? In closed form, so
On 06-Jul-15 04:03, Sampo Syreeni wrote:
On 2015-06-30, Vadim Zavalishin wrote:
I would say the whole thread has been started mostly because of the
exponential segments. How are they out of the picture?
They are for *now* out, because I don't yet see how they could be
bandlimited systematicall
On Sun, Jul 5, 2015 at 9:09 PM, Sampo Syreeni wrote:
> On 2015-06-30, robert bristow-johnson wrote:
>
>> but wavetable synthesis *is* a framework that can do that for any periodic
>> (or quasiperiodic) signal.
>
>
> How do you derive the hard bandlimited wavetable for an exponential, rising
> segm
Charles Z Henry wrote:
...
y=conv(u, f_s*sinc(f_s*t) )
Think about it that that is a shifting integral with an sin(x)/x in it,
for which there isn't even an easy solution if f_s is really simple.
T.
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That's the point. You don't have to evaluate that integral, just
numerically integrate the ordinary differential equation that follows from
it to fill your wavetables.
Charles Z Henry wrote:
> ...
>
> y=conv(u, f_s*sinc(f_s*t) )
>
>
Think about it that that is a shifting integral with an sin(x)
On 10-Jul-15 19:50, Charles Z Henry wrote:
The more general conjecture for the math heads :
If u is the solution of a differential equation with forcing function g
and y = conv(u, v)
Then, y is the solution of the same differential equation with forcing function
h=conv(g,v)
I haven't got a solid
Vadim Zavalishin wrote:
...
How about the equation
u''=-w*u+g
where v is sinc and w is above the sampling frequency?
Aw man
You're now going to argue your every day signals are the exact outcome
of a differential equation, and ON TOP OF THAT are bandwidth limited ?
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dupswapdrop --
On Mon, Jul 13, 2015 at 3:28 AM, Vadim Zavalishin
wrote:
> On 10-Jul-15 19:50, Charles Z Henry wrote:
>>
>> The more general conjecture for the math heads :
>> If u is the solution of a differential equation with forcing function g
>> and y = conv(u, v)
>> Then, y is the solution of the same diffe
On Mon, Jul 13, 2015 at 8:39 AM, Charles Z Henry wrote:
> On Mon, Jul 13, 2015 at 3:28 AM, Vadim Zavalishin
> wrote:
>> On 10-Jul-15 19:50, Charles Z Henry wrote:
>>>
>>> The more general conjecture for the math heads :
>>> If u is the solution of a differential equation with forcing function g
>
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