Re: [ARSCLIST] The circle analogy (was: Is recording to Reel-to-reel still the preferred preservation method?)

[Jon Noring <jon@xxxxxxxxxxx>]

A couple people, who may prefer to remain anonymous, privately replied
to the message I posted here on ARSC a few days ago regarding how to
understand Nyquist based on the "3 points define a circle" analogy.
One of them wrote:

> Thank you so much for providing this information. I have never heard
> expressed this way and never really understood how Nyquist's theory
> worked.
>
> Yet I still wonder, if the added information is that the wave is
> periodic, how would it know if three samples were representing a
> sine wave or a ramp wave or some other complex wave shape?

First of all, any periodic function of a finite duration (including
impulses which are like square waves) can be broken down into a
summation (finite or infinite) of sine functions. This is Fourier
analysis. Thus, any periodic signal can be represented by sine waves,
and Nyquist definitely applies even in this situation.

However, such non-harmonic functions, when broken down into the
series of sine waves, contain very high frequency components. With
finite sampling (e.g., 44.1K, 96K, 192K, etc.), one can only
represent those harmonic frequencies up to half the sampling rate
(this is Nyquist). Thus, such non-harmonic periodic signals cannot
be perfectly represented using digital sampling. But before the
analog media proponents rejoice, read on...

Although I know little about the physiology of human hearing
("psychoacoustics"), I speculate that the human ear itself breaks
down any periodic audio signal it hears into a summation of sine waves
via some sort of Fourier transform process -- the human auditory
system is a Fast Fourier Transform device (otherwise how could we
discern pitch?! -- the human ear has to translate variations in the
position of the eardrum at any instant in time into the frequency
domain.) Because of the known frequency limits (typically 15kHz) of
the human ear, we cannot hear the higher frequency components of
non-harmonic signals (e.g., impulses). Thus, human hearing is similar
to digital sampling in this regard -- it tries to to bring "sense" to
any audio signal by breaking it down into harmonics it can perceive,
and this to me is an important observation germane to this discussion
since it appears that human hearing itself is limited to discerning
non-harmonic signals.

And, more importantly, what about the analog media we are discussing
digitally preserving? Again, I assert that none of the analog media we
are considering (shellac, vinyl, tape, wire, etc.) somehow has
preserved the full color of any non-harmonic signals which might have
been in the original audio signal (the highest frequenices which may
have been preserved are in the 40kHz region for the rare quadrophonic
LPs from a few decades ago.) The recording and mastering processes to
produce these analog media have themselves filtered out the very
high-frequency harmonic components of non-harmonic signals (starting
with the microphone), just as the human ear does [it appears that all
recording devices employing "atoms" and "electrons" will "harmonize"
any audio signal it encounters.] So we simply won't *find* any of
these pure non-harmonic signals in the analog media -- they've already
been "harmonized" and filtered by the microphone, electronics (RLC
circuits), and mechanical and magnetic mastering to analog media steps.

>From this observation, I assert that analog media is inferior (in
theory at least) to very high sampling rate digital at preserving the
nuances of any non-harmonic stuff (including rapid transients of
audio power, another aspect to this issue) in the audio source. Thus
digital is not inferior, but superior to analog provided it is done
right (whatever that "right" is) because there are no mechanical/
physical limits to harmonic frequency that we see in analog media.
Digital is preferable (and there are other benefits as well), for both
the preservation backup and use of the preserved sound on existing
analog artifacts.

What this means to me is that there is no theoretical reason why
digital cannot fully and accurately represent any audio signal on any
analog artifact. It's a matter of determining what sampling rate and
bit depth is necessary (both in theory and in practice) to surpass any
analog media, and of course the answer to this is the $64,000 question
which I'll leave to the experts here with knowledge of the performance
and errors associated with *real-world* analog-to-digital converters
(and the reverse process of *real-world* digital-to-analog playback.)
It seems to me the issue is not "digital is inferior to analog", but
"digital is superior to analog when done right." It's the "what is
this 'right' for digital" that is in dispute, specifically to the
preservation of existing analog artifacts.

Jon Noring