What does "O(B^N)" mean? -olli
On Thu, Jul 10, 2014 at 4:02 PM, Vadim Zavalishin <vadim.zavalis...@native-instruments.de> wrote: > Hi all, > > a recent question to the list regarding the frequency analysis and my recent > posts concerning the BLEP led me to an idea, concerning the theoretical > possibility of instant recognition of the signal spectrum. > > The idea is very raw, and possibly not new (if so, I'd appreciate any > pointers). Just publishing it here for the sake of > discussion/brainstorming/etc. > > For simplicity I'm considering only continuous time signals. Even here the > idea is far from being ripe. In discrete time further complications will > arise. > > According to the Fourier theory we need to know the entire signal from > t=-inf to t=+inf in order to reconstruct its spectrum (even if we talk > Fourier series rather than Fourier transform, by stating the periodicity of > the signal we make it known at any t). OTOH, intuitively thinking, if I'm > having just a windowed sine tone, the intuitive idea of its spectrum would > be just the frequency of the underlying sine rather than the smeared peak > arising from the Fourier transform of the windowed sine. This has been > commonly the source of beginner's misconception in the frequency analysis, > but I hope you can agree, that that misconception has reasonable > foundations. > > Now, recall that in the recent BLEP discussion I conjectured the following > alternative "definition" of bandlimited signals: an entire complex function > is bandlimited (as a function of purely real argument t) if its derivatives > at any chosen point are O(B^N) for some B, where B is the band limit. > > Thinking along the same lines, an entire function is fully defined by its > derivatives at any given point and (therefore) so is its spectrum. So, we > could reconstruct the signal just from its derivatives at one chosen point > and apply Fourier transform to the reconstructed signal. > > In a more practical setting of a realtime input (the time is still > continuous, though), we could work under an assumption of the signal being > entire *until* proven otherwise. Particularly, if we get a mixture of > several static sinusoidal signals, they all will be properly restored from > an arbitrarily short fragment of the signal. > > Now suppose that instead of sinusoidal signals we get a sawtooth. In the > beginning we detect just a linear segment. This is an entire function, but > of a special class: its derivatives do not fall off smoothly as O(B^N), but > stop immediately at the 2nd derivative. From the BLEP discussion we know, > that so far this signal is just a generalized version of the DC offset, thus > containing only a zero frequency partial. As the sawtooth transition comes > we can detect the discontinuity in the signal, therefore dropping the > assumption of an entire signal and use some other (yet undeveloped) approach > for the short-time frequency detection. > > Any further thoughts? > > Regards, > Vadim > > -- > Vadim Zavalishin > Reaktor Application Architect > 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 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
