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 wrote:
The sampling theorem applies only to the class signals which do
have Fourier transform, separating this class further into
bandlimited and non-bandlimited. However, it doesn't say anything
about the signals which do not have Fourier transform.
Correct. But the class of signals which do have a Fourier theory is
much, *much* larger than you appear to think. Usually continuous time
Fourier theory is set in the class of tempered distributions, which
does include such things as Dirac impulses, trains of them with
suitably regular supports, and of course shifts, sums, scalings and
derivatives of all finite orders, and so on. Not everything you could
possibly ever want -- products are iffy, for instance -- but
everything you need in order to bandlimit the things and get the
most usual forms of converence in norm we're interested in.
Oh, I didn't know there's a rigorous framework behind Dirac deltas.
i think they call 'em "generalized distributions".
i remember the first time i had a conceptual problem with this about 38
years ago when i took a course is Real Analysis (using the Royden text).
we were learning about Lebesgue integration and then had a theorem
(dunno what it's named) that if f(x)=g(x) "almost everywhere" (that is
everywhere except on a set of real numbers of measure zero or a
countable number of discrete points of x) then the integral of f(x)
equals the integral of g(x) over the same interval. like if f(x) = 1
for rational x and 0 for irrational x, the integral is zero. but what
immediately came to my mind (and to the math profs attention via my
lips) was the Dirac impulse and the way we electrical engineers treat it
(which does not conform to the above theorem).
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 dimensionless, which means the
dimension of delta(t) is 1/time) and a width of 1 Planck time. that's
close enough for me regarding the physical reality and should keep the
mathematicians happy. it's not rigorous, but it's good enough and then
i can integrate it and say the integral is 1.
This can be further extrapolated to the polynomial signal case
(which are implicitly assumed bandlimited in the BLEP, BLAMP and
higher order derivative discontinuity approaches).
They are not implicitly assumed anything, but instead a) the part of
them which lands outside of your bandlimit are carefully bounded to
where aliasing is perceptually harmless, or b) they're used so that
they're fed bandlimited signals, which via the theory of Chebyshev
polynomials and the usual properties of the full Fourier transform
lead to exactly as many fold band expansion as the order of the
polynomial used has. That stuff then works perfectly fine with
oversampling.
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?
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.
Other signals (such as exponential function, FM'ed sine etc) are
also interesting.
Sure. And all that is well understood. The exponential function is
easy because of its connection via the Euler equation to sinusoids,
and via the usual energy norm, to the second order theory of the
Gaussian.
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))
where j^2 = -1.
what is the value of f(0)? (it's 1.)
what is the derivative of f(x)? (it's 0 for all x.)
that means the numerator and denominator must be equal.
there are several other derivations. Euler's original derivation used
Taylor or Maclauren series. i always thought that one was more messy
than necessary.
another good way to derive the relationship is with the 2nd-order diff eq:
(d^2/dx^2) f(x) + f(x) = 0
try the exponential f(x) = e^(alpha*x)
then try the sinusoidal f(x) = A*cos(x) + B*sin(x)
check out your initial conditions for f(0) and (d/dx)f(x) at x=0 and see
what your get for constants A and B.
FM, that's usually expanded in the terms of Bessel functions --
nasty, but in the narrow band, small modulation index limit, again
well understood from decade of radio work.
The question of FM rather popped in connection with my conjecture, which
I will write once again below in a more detailed form.
So, the sampling rate theorem doesn't work for this kind of
signals, in the sense of giving no answer whether they are
bandlimited or not.
The latter two aren't. But they do possess Fourier transforms and
once we have that, we can study which portion of their spectrum goes
outside of any given bandlimit.
Any hints how the spectrum of an exponential function looks like? How
does a bandlimited exponential look like?
depends what's in the exponent.
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.
Therefore we would like to have a generalization of the sampling
theorem, which includes these signals.
You can't really have any general version of such, because the class
of general signals is just too big to admit such a discrete basis.
But certain special cases can in fact be handled.
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)
Let f(t) be a signal such that the statement A(f) holds, let F[n]
be its naively sampled counterpart and and let {D_n} be a sequence
of DACs such that the statement B({D_n}) holds. Then the sequence
D_n(F) converges to f.
There is nor there can be such a general theory. In fact the idea
that you can just naïvely sample pretty much anything is *highly*
Well, I hope that I can naively sample no matter what.
yeah you can. unless you're writing a dissertation in a mathematics
department, don't let the mathheads tell you different.
The question is
what are the consequences of that sampling. Particularly, whether I can
restore the signal back, right? And this is exactly what the sampling
theorem is about.
I made some initial conjectures in this regard in the mentioned
thread, where A had to do with the rolloff speed of the function's
derivatives and D_n was just a sequence of time-windowed sincs
with increasing window size.
Won't work. Not even close. If you have no upper limit to the order
of continuity, I can plug in one of the vast class of continuous,
nowhere differentiable functions, and integrate down to something
which fulfils your condition yet will fail any normal sampling
theory under the assumption of shift-invariance. If you don't have
such a limit, then you're essentially already back to the theory of
Schwartz spaces, which are used in the construction of the topology
of the space of tempered distributions I mentioned above.
oh dear. i'm gonna have to read what this is about.
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
bandlimited wavetable synthesis (perhaps interpolating between
wavetables as we move up the keyboard). easy, cheap (computationally,
not so cheap regarding memory but most of the time gobs of memory are
cheap), and clean.
See the above statement about the entire function. I just didn't mention
this in the context of the other thread, trying to be brief.
Anyway, here is my conjecture.
The signals in our class of interest are a sum of two parts:
- the "continuous part" a real function of real argument which is
simultaneously entire in the complex plane.
- the "discontinuous part" which is a sum of "non-bandlimited BLEPS (0th
order), BLAMPS (1st order bleps) and higher-order bleps", where the
discontinuity points are isolated but are allowed to have infinite
"multiplicity" (that is at a single isolated point there may be
discontinuities in all the derivatives).
We are interested in:
- whether the continuous part of our signal is bandlimited?
- what is the rolloff speed of the derivative discontinuity amplitudes
(as the order of the discontinuity grows) at each multiple-discontinuity
point?
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.
The convergence of BLEPs is defined by the rolloff speed of the
derivatives. My conjecture is that so is the bandlimitedness of the
entire function, and that the rolloff speed requirement is exactly the
same for both conditions!
If this conjecture is true, this should automatically imply that all
polynomials are bandlimited, since all their higher-order derivatives
are zeros.
Here is the older thread where I was asking the same question:
http://music.columbia.edu/pipermail/music-dsp/2014-June/072679.html
oh dear. don't have time for it.
--
r b-j r...@audioimagination.com
"Imagination is more important than knowledge."
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