From: jim bell <jimbell@pacifier.com>
Date: Wed, 24 Jul 1996 17:32:23 +0800
To: cypherpunks@toad.com
Subject: Re: Digital Watermarks for copy protection in recent Billbo
Message-ID: <199607232200.PAA25339@mail.pacifier.com>
MIME-Version: 1.0
Content-Type: text/plain


At 05:03 AM 7/24/96 -0700, Timothy C. May wrote:


>>Not familiar with the Nyquist limit w/ regards to sampling rate vs
>>frequency :(
>
>Check any textbook, or even a good dictionary. Basically, it says that one
>must sample at more than twice the frequency of the highest frequency to be
>reconstructed. Thus, a 20 KHz top frequency needs at least 40 K samples per
>second. The exact number is, I think, about 2.2x the freqency, which is why
>CDs were standardized at 44 K samples per second per channel.

No, Tim, the minimum Nyquist sample frequency _is_ precisely 2.000 times the 
highest frequency to be recovered.  However, what is somewhat less 
well-known is the fact that in order to keep higher frequencies from being 
"aliased" (reflected to lower frequencies by heterodyne processes) it is 
necessary to remove (by filtering) any frequency content above that maximum, 
before sampling is done.  Real-world filters (at least, _economical_ ones) 
do not have instantaneous cutoffs, so it is necessary to provide a little 
margin, in this case about 10%.

Fortunately, a play-only CD only needs output filters, and the input data 
(the CD disk itself) has already been limited to about 20 kilohertz, so its 
requirements are not so stringent.

The reason so-called "oversampling" started to be done on CD players is that 
interpolating digitally between samples results in a far higher aliased noise 
frequencies, allowing  either better performance with the same-quality of 
filters, or equal quality with lower-cost filters, or a combination of them 
both.


Jim Bell
jimbell@pacifier.com