Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.1 6/24/83; site decwrl.UUCP
Path: utzoo!watmath!clyde!bonnie!akgua!mcnc!decvax!decwrl!dec-rhea!dec-bergil!lauck
From: lauck@bergil.DEC
Newsgroups: net.audio
Subject: Re: CD Reflections
Message-ID: <246@decwrl.UUCP>
Date: Thu, 17-Jan-85 10:55:30 EST
Article-I.D.: decwrl.246
Posted: Thu Jan 17 10:55:30 1985
Date-Received: Thu, 24-Jan-85 08:29:37 EST
Sender: daemon@decwrl.UUCP
Organization: DEC Engineering Network
Lines: 43

>
>Re: a 2x sampling rate is not good enough to reconstruct a sine wave.
>
>I'm afraid it is.  The 2x rate is called the Nyquist Rate and is
>the minimum sampling rate necessary to prevent aliasing.  It has
>been proven rigorously that any waveform sampled at or above twice
>the highest frequency can be reconstructed.  This is part of the
>sampling theorem and is the basis of much of modern communication
>theory.  Since this is the accepted minimum, it is not the same
>2x rate that is usually referred to in the CD literature.  The
>highest audio frequencies are 22.1KHz, so the sampling rate must
>be 44.2KHz.  When CD manufacturers refer to a sampling rate of 2x,
>they usually mean 88.4KHz (that's what Yamaha says for my CD-2).

A perfect sine wave could be recovered if sampled at 2X plus epsilon,
but this would require a perfect filter.  (Sampling at exactly 2X won't 
work, for example, all the samples might be zero.)

The problem is that music doesn't consist of perfect sine waves and isn't
bandlimited to 22.1Khz, hence the need for anti-aliasing filters.  There
are two problems with these filters: 1) even if perfect they may filter out
musically significant information and 2) they can't be perfect.  Their
imperfections consist of amplitude and phase variation in the passband and
failure to completely eliminate the stop-band.  Similar problems exist in
the reconstruction filters on playback, but these filters have a less
difficult role, especially on players with two DACs where the digital
waveform before filtering is a staircase and not a collection of pulses.

The advantage of digital filters is primarily one of implementation;  the 
critical parameters of the steep filter can be implemented more precisely.  
In addition it is possible to build linear-phase digital filters, but these
may not be the answer since they have amplitude ripple in the passband.

As long as theoretical arguments are going to be applied where practical
arguments should be used, consider this:  the only causal signal which is 
bandlimited is the zero signal.  Such a signal could be processed perfectly 
by a digital recorder, but would only be useful to reproduce a single piece 
of music (by John Cage).

      Tony Lauck

               ...decvax!decwrl!rhea!bergil!lauck