On 30-Jun-14 18:44, Stefan Stenzel wrote:
The tools of BLXX are DC, digital integrators/filters and your BLXX signals that
are bandlimited by definition, no sampling and no nonlinear operation involved.
So as there is no possible source for aliasing, there is no aliasing.
Okay. So in principle, we could construct a BLXX-ed signal by simply
integrating DC and bandlimited impulses a sufficient number of times, if
necessary correcting the DC to zero along the way. Convincing enough. (A
small remark: I belive that, contrarily to the BLIT method, BLXX
typically integrates in the continuous time domain. I fail to see any
reason why should we prefer discrete-time domain integration. The
continuous-time domain integration has the benefit of avoiding any
spectral distortion in the hi freq area. Also it can be performed
analytically.) This means that in principle "any" piecewise polynomial
signal with bandlimited discontinuities of the signal and its
derivatives is also bandlimited.
This addresses my original motivation to a large extent: applying BLXX
not only to waveforms of stable frequencies, but also to their modulated
versions. Although, not completely. Particularly, self-FM sawtooth
produces an exponential signal. I wonder, whether exponents (at least
those which are slow enough) are bandlimited. After all a good part of
the sine signals (which are kind of versions of exponentials) are
bandlimited.
So, can we apply the above reasoning to Taylor series expansions
(constructing Taylor series by repeated integration of signals)?
Especially, if we consider only exponential segments, rather than
infinitely long exponentials, then we could apply the above integration
scheme an infinite number of times to arrive at the result. But (!) so
we could do for the sine signals. This would mean that *any* sine is
bandlimited. So, there must be some flaw in that reasoning.
Besides, while Stefan provided an almost convincing justification of the
BLXX by integration of DC and impulses ("almost" is because there is
this unanswered question of inifinite integration in the construction of
the Taylor series, which is somewhat bothering). However, it would still
be interesting to get a consistent look at the same problem from the
virtual ADC point of view, if only for the sake of understanding.
Regards,
Vadim
--
Vadim Zavalishin
Reaktor Application Architect
Native Instruments GmbH
+49-30-611035-0
www.native-instruments.com
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