I do not understand the last bit of this message below at all.
There is no such thing as a signal that is limited
in bandwidth and in time--not if limited
means actually 0 outside a finite interval
in both cases. This is a basic result of Fourier
analysis.
This kind of signal does not exist, not mathematically
and of course not physically either.
Robert
On Fri, 28 Jun 2013, Fons Adriaensen wrote:
On Thu, Jun 27, 2013 at 04:33:46PM -0700, Eric Carmichel wrote:
I have to agree with Joern that the example miking demonstration
isn?t all that fair, and for another reason: How much low-frequency
energy can a 2-inch speaker provide?
I don't know what kind of signal was used for this test, but what
we hear as a 'click' doesn't have to be fully 'broadband'. A short
burst of e.g. octave band noise (or almost any signal) will sound
as a click.
but I?ll have to state that I don?t believe the ear works exactly
as math would predict. Let me explain...
Well, math doesn't try to predict how our ears work... We can use
some mathematical model to try an understand that, but that model
wouldn't be the canonical Fourier tranform defining the relation
between a pure time domain description of a signal and a pure
frequency domain one.
Clearly our perception of sound is a mix of those two domains.
The fuzzy border between the two seems to be at around 20 Hz -
anything that repeats at a lower frequency will have more chance
to be perceived as sequence of discrete events, if it repeats
faster we will hear it as a signal having some frequency.
A short time FT is a more appropriate model in that case. We can
also assume that the signal we are using is bandlimited, and thus
think in terms of sampled signals and use the discrete FT.
A DFT of lenght K samples will provide a spectrum consisting of
K/2 complex values [*]. The classical way to interpret this, found
in all textbooks, is to assume that the signal is cyclic with a
period of K samples, its spectrum will be discrete and consist
of frequencies that correspond exactly to the 'bins' of the DFT.
But what if we don't assume that the signal is cyclic, but a single
event preceded and followed by silence ? What does the DFT output
mean in that case ? That is actually quite simple: the K/2 values
are samples of the continuous spectrum. And they are all that is
needed in order to know that spectrum completely, just as the K
samples of the signal are all that is needed to reconstruct the
continuous signal.
Instead of the required condition for the latter - the signal being
limited in bandwidth - the condition for the discrete spectrum being
a complete (i.e. invertible) description of the signal is that the
signal is limited in time. This is just the dual of the Nyquist
sampling theorem. The factor of 2 that appears only in one case is
because we are using real-valued signals but complex-valued spectra.
What this shows is that you don't need any infinities to exactly
describe a signal that is limited both in bandwidth and time.
Ciao,
[*] More correctly, K/2-1 complex ones and two real-valued ones
at 0Hz and Fs/2.
--
FA
A world of exhaustive, reliable metadata would be an utopia.
It's also a pipe-dream, founded on self-delusion, nerd hubris
and hysterically inflated market opportunities. (Cory Doctorow)
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