On Mon, Jul 13, 2015 at 8:39 AM, Charles Z Henry czhe...@gmail.com wrote:
On Mon, Jul 13, 2015 at 3:28 AM, Vadim Zavalishin
vadim.zavalis...@native-instruments.de 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
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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On Mon, Jul 13, 2015 at 3:28 AM, Vadim Zavalishin
vadim.zavalis...@native-instruments.de 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
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
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)/x
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
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 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
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
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
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, 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
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
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
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
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
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 polynomials
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 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
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,
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
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
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
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
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 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,
On 11 Jun 2015, at 19:58, Sampo Syreeni de...@iki.fi 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
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
On 12 Jun 2015, at 14:31, Vadim Zavalishin
vadim.zavalis...@native-instruments.de 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
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
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
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. The
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
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 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 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
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
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
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
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
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