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From: karn@petrus.UUCP
Newsgroups: net.audio
Subject: Re: CD Reflections - 44.1k?
Message-ID: <258@petrus.UUCP>
Date: Wed, 16-Jan-85 12:22:41 EST
Article-I.D.: petrus.258
Posted: Wed Jan 16 12:22:41 1985
Date-Received: Thu, 17-Jan-85 13:57:03 EST
References: <15100001@hpfcmp.UUCP> <3411@mit-eddie.UUCP> <1420@hplabs.UUCP> <755@clyde.UUCP>
Organization: Bell Communications Research, Inc
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> Now for the part that bothers me:  The Nyquist theorem always seems to
> be based on continuous signals.  I.e, the reason you can sample a 20k
> signal at just slightly more than 40k is that EVENTUALLY you will get
> some samples around the peak amplitude - you will also get some around
> zero.  But music isn't like that - notes start and end.  In effect, you
> are multiplying (or modulating) the sine wave by an envelope which also
> contributes to the spectrum.  HOW MUCH ADDED SAMPLING DOES THIS REQUIRE?
> I don't want to listen to a 20k sine wave, I want transients (anybody
> remember TIM distortion?).  How about it digital types? - any guesses?
> 

The Nyquist theorem is valid for ANY bandlimited signal, it doesn't matter
whether it is periodic or not. One way to visualize this intuitively is
that any signal which has been bandlimited (whether it be a transient
or a periodic waveform) is limited in how fast the signal voltage can
change with respect to time. This places a bound on how fast the signal
must be sampled in order to capture *all* of the information within it.
If the transient contains high frequency signals which cannot be sampled
fast enough, they must be removed by the pre-sampling low pass filter or
else aliasing distortion will result. But as long as the sampling rate is
at least twice as fast as the cutoff of the low pass filter, then the
sampling operation will retain ALL of the information coming OUT (not
necessarily going IN) to the low pass filter.  Check out your communications
theory textbook on this one.

In terms of multiplying the signal by a modulating waveform, yes, this
affects the spectrum of the result. The theory of sampling always begins
by using a train of Dirac delta functions as the sampling waveform;
Dirac delta functions are infintesimally short, impulses which do not
exist in the real world but have nice flat spectral properties. Real world
sampling waveforms are generally retangular because of the sample-and-hold
circuits used in real A/D converters. Since the fourier transform of a
rectangular pulse is a sinc (sin(x)/x) function, this has the effect of
giving a sinc shape to the resulting spectrum. This is corrected in
the player with an inverse sinc function which is added either to the CD low
pass reconstruction filter or to the pre-sampling anti-aliasing filter,
I'm not sure which one. Does anybody know?

Phil