From: mpd@netcom.com (Mike Duvos)
Date: Fri, 26 Jul 1996 03:06:03 +0800
To: cypherpunks@toad.com
Subject: Re: Digital Watermarks for copy protection in recent Billbo (fwd)
In-Reply-To: <199607250621.BAA07021@einstein.ssz.com>
Message-ID: <199607251521.IAA06190@netcom18.netcom.com>
MIME-Version: 1.0
Content-Type: text/plain


Jim Choate <ravage@einstein.ssz.com> writes:

 >> The Sampling Theorem states that equally spaced
 >> instantaneous samples must be taken at a rate GREATER THAN
 >> twice the highest frequency present in the analog signal
 >> being sampled.  If this is done, the samples contain all the
 >> information in the signal, and faithful reconstruction is
 >> possible.

 > Actualy the sampling theorem states that if you want to
 > reproduce a signal reliably it must be sampled at a
 > frequency AT LEAST TWICE that of the highest frequency of
 > interest in the FFT of the signal. This means the signal
 > must be deformable into sine waves, not all signals qualify
 > for this particular limitation and in general are not good
 > signals for sampling.

You want a continuous Fourier transform, not a discrete one, to
determine the frequency spectrum of the waveform being sampled.
The FFT is simply an algorithm for computing the DFT without
redundant computation.  In general, any Lebesgue integrable
complex function will have a Fourier transform, even one with a
finite number of discontinuities. The reverse transform will
faithfully reproduce the function, modulo the usual caveats about
function spaces and sets of measure zero.

There is no meaningful difference between speaking of the highest
frequency in a signal, and the highest frequency present in its
Fourier transform.

 > An example is a step change from v1 to v2. since there is
 > no change in voltage except for a brief moment the output
 > of the FFT is pretty much flat.

The Fourier transforms of step functions, square wave functions,
delta functions, and other oddities are perfectly well defined.
Again, the FFT is not relevant here.  A step function does not
have a upper cutoff in the frequency domain, so you can never
reproduce it perfectly from its samples, although the faster you
sample, the sharper the edges of the reconstruction will become.

 > What comes out the other end looks like a spike (similar to
 > what happens when you feed a square wave to a transformer
 > and look at the output). If you reconstruct this via the
 > sampling theorem you get a sign wave of extremely low
 > amplitude and frequency.

I don't think so.

[Incomprehensible Deletia]

 > Back to the comb. Where a given signal has a componant leave
 > the teeth. Where there is no componant break them off. You
 > are left with a ratty looking comb. Now to each of the
 > remaining teeth assign an amplitude for that specific
 > frequency that is consistent with the FFT you calculated.
 > The way this FFT is implimented in practice is called a
 > 'Comb Filter' where it samples the signal (wide bandwidth)
 > over a set of very small bandwidth filters in parallel.

I must have been absent on "tooth day" in Functional Analysis
class.

 > One of the basic ideas of Algebra is that any given curve
 > can be explained by a arbitrary set of equations. The
 > Sampling Theorem just gives you a rationale for picking from
 > that set.

AIIIIIIIIEEEEEEEEEEEEE!  (I feel much better now :)

--
     Mike Duvos         $    PGP 2.6 Public Key available     $
     mpd@netcom.com     $    via Finger.                      $