On 16-Jul-14 12:31, Olli Niemitalo wrote:
What does "O(B^N)" mean?
-olli
This is the so called "big O" notation.
f^(N)(t)=O(B^N) means (for a fixed t) that there is K such that
|f^(N)(t)|<K*B^N
where f^(N) is the Nth derivative. Intuitively, "f^(N)(t) doesn't grow
faster than B^N"
Regards,
Vadim
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
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Vadim Zavalishin
Reaktor Application Architect
Native Instruments GmbH
+49-30-611035-0
www.native-instruments.com
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