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AIList Digest Volume 4 Issue 241
AIList Digest Thursday, 30 Oct 1986 Volume 4 : Issue 241
Today's Topics:
Philosophy & Physics - Analog/Digital Distinction
----------------------------------------------------------------------
Date: 27 Oct 86 06:08:33 GMT
From: rutgers!princeton!mind!harnad@titan.arc.nasa.gov
Subject: Re: Defining the Analog/Digital Distinction
Tom Dietterich (orstcs!tgd) responds as follows to my challenge to
define the A/D distinction:
> In any representation, certain properties of the representational
> medium are exploited to carry information. Digital representations
> tend to exploit fewer properties of the medium. For example, in
> digital electronics, a 0 could be defined as anything below .2 volts
> and a 1 as anything above 4 volts. This is a simple distinction.
> An analog representation of a signal (e.g., in an audio amplifier)
> requires a much finer grain of distinctions--it exploits the
> continuity of voltage to represent, for example, the loudness
> of a sound.
So far so good. Analog representations "exploit" more of the properties
(e.g., continuity) of the "representational" (physical?) medium to carry
information. But then is the difference between an A and a D representation
just that one is more (exploitative) and the other less? Is it not rather that
they carry information and/or represent in a DIFFERENT WAY? In what
does that difference consist? (And what does "exploit" mean? Exploit
for whom?)
> A related notion of digital and analog can be obtained by considering
> what kinds of transformations can be applied without losing
> information. Digital signals can generally be transformed in more
> ways--precisely because they do not exploit as many properties of the
> representational medium. Hence, if we add .1 volts to a digital 0 as
> defined above, the result will either still be 0 or else be undefined
> (and hence [un]detectable). A digital 1 remains unchanged under
> addition of .1 volts. However, the analog signal would be
> changed under ANY addition of voltage.
"Preserving information under transformations" also sounds like a good
candidate. But it seems to me that preservation-under-transformation
is (or ought to be) a two-way street. Digital representations may be
robust within their respective discrete boundaries, but it hardly
sounds information-preserving to lose all the information between .2
volts and 4 volts. I would think that the invertibility of analog
transformations might be a better instance of information preservation than
the irretrievable losses of A/D. And this still seems to side-step the
question of WHAT information is preserved, and in what way, by analog
and digital representations, respectively. And should we be focusing on
representations in this discussion, or on transformations (A/A, A/D,
D/D, D/A)? Finally, what is the relation between a digital
representation and a symbolic representation?
Please keep those definitions coming.
Stevan Harnad
{allegra, bellcore, seismo, packard} !princeton!mind!harnad
harnad%mind@princeton.csnet
(609)-921-7771
------------------------------
Date: 27 Oct 86 11:53:06 est
From: Walter Hamscher <hamscher@ht.ai.mit.edu>
Subject: The Analog/Digital Distinction
Date: 23 Oct 86 17:20:00 GMT
From: hp-pcd!orstcs!tgd@hplabs.hp.com (tgd)
Here is a rough try at defining the analog vs. digital distinction.
[ * * * ]
I don't read all the messages on AiList, so I may have missed
something here: but isn't ``analog vs digital'' the same thing as
``continuous vs discrete''? Continuous vs discrete, in turn, can be
defined in terms of infinite vs finite partitionability. It's a
property of the measuring system, not a property of the thing being
measured.
------------------------------
Date: 27 Oct 86 15:29:01 GMT
From: alice!jj@ucbvax.Berkeley.EDU
Subject: Re: Defining the Analog/Digital Distinction
> From allegra!princeton!mind!harnad Wed Dec 31 19:00:00 1969
>
>
> Tom Dietterich (orstcs!tgd) responds as follows to my challenge to
> define the A/D distinction:
>
> > In any representation, certain properties of the representational
> > ...
> > of a sound.
>
> So far so good. Analog representations "exploit" more of the properties
> ...
> for whom?)
>
> > A related notion of digital and analog can be obtained by considering
> > ...
> > changed under ANY addition of voltage.
>
> "Preserving information under transformations" also sounds like a good
> ...
> representation and a symbolic representation?
>
> Please keep those definitions coming.
>
> Stevan Harnad
What a pleasant little bit of sophistry. Mr. Harnad asks for a defination
of "digital" and "analog", both words used in a precise way in a particular
literature. He also asks that we do not use other words used in that literature
to write the defination.
In other words, we are asked to define something precisely, in a languange
that does not have precise values.
I suggest the first chapter of Rabiner and Gold, all of Wozencraft and Jacobs,
and perhaps a good general text on signal processing for starters. That will
define the language. Then the defination can be made.
Philosophy is wonderful, it doesn't have to have anything to do
with reality.
--
WOBEGON WON'T BE GONE, TEDDY BEAR PICNIC AT 11.
"If you love that Politician, use your Loo, use your Loo"
(ihnp4;allegra;research)!alice!jj
------------------------------
Date: 27 Oct 86 22:06:14 GMT
From: husc6!Diamond!aweinste@eddie.mit.edu (Anders Weinstein)
Subject: Re: The Analog/Digital Distinction: Soliciting Definitions
Philosopher Nelson Goodman has distinguishes analog from digital symbol systems
in his book _Languages_of_Art_. The context is a technical investigation into
the peculiar features of _notational_ systems in the arts; that is, systems
like musical notation which are used to DEFINE a work of art by dividing the
instances from the non-instances.
The following excerpts contain the relevant definitions: (Warning--I've left
out a lot of explanatory text and examples for brevity)
The second requirement upon a notational scheme, then, is that the
characters be _finitely_differentiated_, or _articulate_. It runs: For
every two characters K and K' and every mark m that does not belong to
both, determination that m does not belong to K or that m does not belong
to K' is theoretically possible. ...
A scheme is syntactically dense if it provides for infinitely many
characters so ordered that between each two there is a third. ... When no
insertion of other characters will thus destroy density, a scheme has no
gaps and may be called _dense_throughout_. In what follows, "throughout" is
often dropped as understood... [in footnote:] I shall call a scheme that
contains no dense subscheme "completely discontinuous" or "discontinuous
throughout". ...
The final requirement [including others not quoted here] for a notational
system is semantic finite differentiation; that is for every two characters
K and K' such that their compliance classes are not identical and every
object h that does not comply with both, determination that h does not
comply with K or that h does not comply with K' must be theoretically
possible.
[defines 'semantically dense throughout' and 'semantically discontinuous'
to parallel the syntactic definitions].
And his analog/digital distinction:
A symbol _scheme_ is analog if syntactically dense; a _system_ is analog if
syntactically and semantically dense. ... A digital scheme, in contrast, is
discontinuous throughout; and in a digital system the characters of such a
scheme are one-one correlated with compliance-classes of a similarly
discontinous set. But discontinuity, though implied by, does not imply
differentiation...To be digital, a system must be not merely discontinuous
but _differentiated_ throughout, syntactically and semantically...
If only thoroughly dense systems are analog, and only thoroughly
differentiated ones are digital, many systems are of neither type.
To summarize: when a dense language is used to represent a dense domain, the
system is analog; when a discrete (Goodman's "discontinuous") and articulate
language maps a discrete and articulate domain, the system is digital.
Note that not all discrete languages are "articulate" in Goodman's sense:
Consider a language with only two characters, one of which contains all
straight marks not longer than one inch and the other of which contains all
longer marks. This is discrete but not articulate, since no matter how
precise our tests become, there will always be a mark (infinitely many, in
fact) that cannot be judged to belong to one or the other character.
For more explanation, consult the source directly (and not me).
Anders Weinstein <aweinste@DIAMOND.BBN.COM>
------------------------------
Date: 28 Oct 86 04:20:07 GMT
From: allegra!princeton!mind!harnad@ucbvax.Berkeley.EDU
Subject: The Analog/Digital Distinction
Steven R. Jacobs (utah-cs!jacobs) of the University of Utah CS Dept
has given me permission to post his contribution to defining the A/D
distinction. It appears below, followed at the very end by some comments
from me.
[Will someone with access please post a copy to sci.electronics?]
>> One prima facie non-starter: "continuous" vs. "discrete" physical processes.
>I apologize if this was meant to avoid discussion of continuous/discrete
>issues relating to analog/digital representations. I find it difficult
>to avoid talking in terms of "continuous" and "discrete" processes when
>discussing the difference between analog and digital signals. I am
>approaching the question from a signal processing point of view, so I
>tend to assume that "real" signals are analog signals, and other methods
>of representing signals are used as approximations of analog signals (but
>see below about a physicist's perspective). Yes, I realize you asked for
>objective definitions. For my own non-objective convenience, I will use
>analog signals as a starting point for obtaining other types of signals.
>This will assist in discussing the operations used to derive non-analog
>signals from analog signals, and in discussing the effects of the operations
>on the mathematics involved when manipulating the various types of signals
>in the time and frequency domains.
>
>The distinction of continuous/discrete can be applied to both the amplitude
>and time axes of a signal, which allows four types of signals to be defined.
>So, some "loose" definitions:
>
>Analog signal -- one that is continuous both in time and amplitude, so that
> the amplitude of the signal may change to any amplitude at any time.
> This is what many electrical engineers might describe as a "signal".
>
>Sampled signal -- continuous in amplitude, discrete in time (usually with
> eqully-spaced sampling intervals). Signal may take on any amplitude,
> but the amplitude changes only at discrete times. Sampled signals
> are obtained (obviously?) by sampling analog signals. If sampling is
> done improperly, aliasing will occur, causing a loss of information.
> Some (most?) analog signals cannot be accurately represented by a
> sampled signal, since only band-limited signals can be sampled without
> aliasing. Sampled signals are the basis of Digital Signal Processing,
> although digital signals are invariably used as an approximation of
> the sampled signals.
>
>Quantized signal -- piece-wise continuous in time, discrete in amplitude.
> Amplitude may change at any time, but only to discrete levels. All
> changes in amplitude are steps.
>
>Digital signal -- one that is discrete both in time and amplitude, and may
> change in (discrete) amplitude only at certain (discrete, usually
> uniformly spaced) time intervals. This is obtained by quantizing
> a sampled signal.
>
>Other types of signals can be made by combining these "basic" types, but
>that topic is more appropriate for net.bizarre than for sci.electronics.
>
>The real distinction (in my mind) between these representations is the effect
>the representation has on the mathematics required to manipulate the signals.
>
>Although most engineers and computer scientists would think of analog signals
>as the most "correct" representations of signals, a physicist might argue that
>the "quantum signal" is the only signal which corresponds to the real world,
>and that analog signals are merely a convenient approximation used by
>mathematicians.
>
>One major distinction (from a mathematical point of view) between sampled
>signals and analog signals can be best visualized in the frequency domain.
>A band-limited analog signal has a Fourier transform which is finite. A
>sampled representation of the same signal will be periodic in the Fourier
>domain. Increasing the sampling frequency will "spread out" the identical
>"clumps" in the FT (fourier transform) of a sampled signal, but the FT
>of the sampled signal will ALWAYS remain periodic, so that in the limit as
>the sampling frequency approaches infinity, the sampled signal DOES NOT
>become a "better" approximation of the analog signal, they remain entirely
>distinct. Whenever the sampling frequency exceeds the Nyquist frequency,
>the original analog signal can be exactly recovered from the sampled signal,
>so that the two representations contain the equivalent information, but the
>two signals are not the same, and the sampled signal does not "approach"
>the analog signal as the sampling frequency is increased. For signals which
>are not band-limited, sampling causes a loss of information due to aliasing.
>As the sampling frequency is increased, less information is lost, so that the
>"goodness" of the approximation improves as the sampling frequency increases.
>Still, the sampled signal is fundamentally different from the analog signal.
>This fundamental difference applies also to digital signals, which are both
>quantized and sampled.
>
>Digital signals are usually used as an approximation to "sampled" signals.
>The mathematics used for digital signal processing is actually only correct
>when applied to sampled signals (maybe it should be called "Sampled Signal
>Processing" (SSP) instead). The approximation is usually handled mostly by
>ignoring the "quantization noise" which is introduced when converting a
>sampled analog signal into a digital signal. This is convenient because it
>avoids some messy "details" in the mathematics. To properly deal with
>quantized signals requires giving up some "nice" properties of signals and
>operators that are applied to signals. Mostly, operators which are applied
>to signals become non-commutative when the signals are discrete in amplitude.
>This is very much related to the "Heisenburg uncertainty principle" of
>quantum mechanics, and to me represents another "true" distinction between
>analog and digital signals. The quantization of signals represents a loss of
>information that is qualitatively different from any loss of information that
>occurs from sampling. This difference is usally glossed over or ignored in
>discussions of signal processing.
>
>Well, those are some half-baked ideas that come to my mind. They are probably
>not what you are looking for, so feel free to post them to /dev/null.
>
>Steve Jacobs
>
- - - - - - - - - - - - - - - - - - - - - - - -
REPLY:
> I apologize if this was meant to avoid discussion of continuous/discrete
> issues relating to analog/digital representations.
It wasn't meant to avoid discussion of continuous/discrete at all;
just to avoid a simple-minded equation of C/D with A/D, overlooking
all the attendant problems of that move. You certainly haven't done that
in your thoughtful and articulate review and analysis.
> I tend to assume that "real" signals are analog signals, and other
> methods of representing signals are used as approximations of analog
> signals.
That seems like the correct assumption. But if we shift for a moment
from considering the A or D signals themselves and consider instead
the transformation that generated them, the question arises: If "real"
signals are analog signals, then what are they analogs of? Let's
borrow some formal jargon and say that there are (real) "objects,"
and then there are "images" of them under various types of
transformations. One such transformation is an analog transformation.
In that case the image of the object under the (analog) transformation
can also be called an "analog" of the object. Is that an analog signal?
The approximation criterion also seems right on the mark. Using the
object/transformation/image terminology again, another kind of a
transformation is a "digital" transformation. The image of an object
(or of the analog image of an object) under a digital transformation
is "approximate" rather than "exact." What is the difference between
"approximate" and "exact"? Here I would like to interject a tentative
candidate criterion of my own: I think it may have something to do with
invertibility. A transformation from object to image is analog if (or
to the degree that) it is invertible. In a digital approximation, some
information or structure is irretrievably lost (the transformation
is not 1:1).
So, might invertibility/noninvertibility have something to do with the
distinction between an A and a D transformation? And do "images" of
these two kinds count as "representations" in the sense in which that
concept is used in AI, cognitive psychology and philosophy (not
necessarily univocally)? And, finally, where do "symbolic"
representations come in? If we take a continuous object and make a
discrete, approximate image of it, how do we get from that to a
symbolic representation?
> Analog signal -- one that is continuous both in time and amplitude.
> Sampled signal -- continuous in amplitude, discrete in time...
> If sampling is done improperly, aliasing will occur, causing a
> loss of information.
> Quantized signal -- piece-wise continuous in time, discrete in
> amplitude.
> Digital signal -- one that is discrete both in time and amplitude...
> This is obtained by quantizing a sampled signal.
Both directions of departure from the analog, it seems, lose
information, unless the interpolations of the gaps in either time or
amplitude can be accurately made somehow. Question: What if the
original "object" is discrete in the first place, both in space and
time? Does that make a digital transformation of it "analog"? I
realize that this is violating the "signal" terminology, but, after all,
signals have their origins too. Preservation and invertibility of
information or structure seem to be even more general features than
continuity/discreteness. Or perhaps we should be focusing on the
continuity/noncontinuity of the transformations rather than the
objects?
> a physicist might argue that the "quantum signal" is the only
> signal which corresponds to the real world, and that analog
> signals are merely a convenient approximation used by mathematicians.
This, of course, turns the continuous/discrete and the exact/approximate
criteria completely on their heads, as I think you recognize too. And
it's one of the things that makes continuity a less straightforward basis
for the A/D distinction.
> Mostly, operators which are applied to signals become
> non-commutative when the signals are discrete in amplitude.
> This is very much related to the "Heisenburg uncertainty principle"
> of quantum mechanics, and to me represents another "true" distinction
> between analog and digital signals. The quantization of signals
> represents a loss of information that is qualitatively different from
> any loss of information that occurs from sampling.
I'm not qualified to judge whether this is an anolgy or a true quantum
effect. If the latter, then of course the qualitative difference
resides in the fact that (on current theory) the information is
irretrievable in principle rather than merely in practice.
> Well, those are some half-baked ideas that come to my mind.
Many thanks for your thoughtful contribution. I hope the discussion
will continue "baking."
Stevan Harnad
{allegra, bellcore, seismo, rutgers, packard} !princeton!mind!harnad
harnad%mind@princeton.csnet
(609)-921-7771
------------------------------
Date: 27 Oct 86 03:29:00 GMT
From: uiucuxe!goldfain@uxc.cso.uiuc.edu
Subject: Re: The Analog/Digital Distinction: Sol
Analog devices/processes are best viewed as having a continuous possible
range of values. (An interval of the real line, for example.)
Digital devices/processes are best viewed as having an underlying granularity
of discrete possible values. (Representable by a subset of the integers.)
-----------------
This is a pretty good definition, whether you like it or not.
I am curious as to what kind of discussion you are hoping to get, when you
rule out the correct distinction at the outset ...
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End of AIList Digest
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