Klarinet Archive - Posting 000652.txt from 1997/11

From: "Jerry Korten" <jerryk@-----.com>
Subj: Re: Nyquist and analog
Date: Wed, 19 Nov 1997 13:00:57 -0500

This topic is a little off course for the Klarinet list. However, Mr.
Cohler, whose capabilities as a clarinetist I admire, is challenging some
basic foundations of the digitization process which should not be taken at
face value. I reply once more in this forum but will take the debate
offline to send data samples to Mr. Cohler in order to point out what
happens during the digitization process. So I apologize in advance.

I will take both of Jonathan's replies together...

>>
>>As you pointed out, there are harmonics in a triangle wave. What I was
busy
>>griping about was the harmonic structure of the original analog waveform
was
>>not truly represented by the digitization process.
>>
>This is not a true statement. And furthermore, you have presented not a
>single fact to back it up.
By definition, a sampling process that cannot represent a continous
process. This is a fact. It is why the process is called sampling. The high
frequency portion of the signal (which is filtered out to avoid aliasing)
is not present on the recorded material when digitization occurs. This is
the main reason that the harmonic structure on a CD does not match what the
ear hears from a live performance or an analog recording.

>>><<
>>> This is not correct. The Nyquist Theorem says that if an analog signal
is
>>> digitally sampled at a frequency F then using those digital samples one
can
>>> PERFECTLY reconstruct all frequency components of the original signal
up to
>>> a frequency of 1/2 * F. The statements about triangular waves above or
>>> just wrong. It is a mathematical fact.
>>>>>
>>
>>Wrong. The mathematical fact is that the frequency can be represented (or
>>discerned). The Nyquist theorem does not state that the original waveform
can
>>be constructed PERFECTLY if the sample rate is twice the recorded
frequency.
>>Spend a little time with an A/D converter and a signal generator on your
>>computer. This will be quite evident.
>>
>>In fact a 22KHz sinewave sampled at 44.1 KHz will look like a triangle
wave
>>with an amplitude that varies in amplitude (modulating larger and smaller
>>with a periodicity of 0.1 Hz). If you are interested I can send you some
>>sample programs to play with that will demonstrate this phenomenon.
>You are incorrect and clearly misunderstood the basic mathematics that
>underly the well-established fact which I have stated above. If you wish
>to see the theorem stated in a more formal fashion you can reference the
>following web site:
> http://www.its.bldrdoc.gov/fs-1037/dir-025/_3621.htm
>Here is the theorem in summary:
> An analog signal waveform may be uniquely reconstructed, without
> error from samples taken at equal time intervals. The sampling
> rate must be equal to, or greater than, twice the highest
> frequency component in the analog signal.
>
>Your misunderstanding is a fairly common one, however. Allow me to
>explain. If one samples a 22KHz sine wave at 44.1KHz one obtains a set of
>discreet digital "samples". If you were to draw straight lines between
>these samples, you may indeed obtain something resembling a triangle wave.
>But that is not what one does.
You appear to be applying the math in the case of a continuous periodic
waveform. Which music is not. Yes, in the way you describe sampling in the
above example, after many cycles of the measured waveform have been sampled
you can reconstruct its shape (still not entirely because we are talking
about a sampled system for which data points do not exist in time when the
continuous analog signal does). But this is not the case for discretely
sampled waveforms.

And, when we are talking about digitizing music, filled with transients
(piano, attacks etc.) this is where the application of this type of
reasoning fails. You cannot reconstruct the actual waveform.

The definition you provide above is not in agreement with my reference. In
fact it is wrong. You should refer to page 29 of Oppenheim and Schafer
"Digital Signal Processing" (Prentice hall), in which they describe how
interpolation must be used to reconstruct the original waveform when using
a discrete sampling system. And interpolation is an approximation, not
actual reconstruction.

This reference to a definition on the web is a big problem with the
internet, there is no peer review involved with posting facts in this
media. I always recommend a text with rigorous academic peer review.

>A consequence of the Fourier Theorem is that any periodic waveform (which
>musical sounds are, and which most sounds are over short periods of time)
>is that they can be contructed out of a sum of sine waves whose
frequencies
>are the integer multiples of the fundamental or lowest frequency
component.
>Without getting into the details of the alogrithmic process (with which I
>am intimately familiar, and have designed many computer systems for this
>purpose), suffice it to say that using basic digital signal processing,
>well known mathematics, one can take that set of sample points and
>PRECISELY reconstruct that 22KHz sine wave.
The fourier theorem as described above deals with continuous waveforms.
This means that in order to represent a waveform, the series of sinewaves
used must go up in frequency to infinity. In fact in a digital system, the
number of sinewaves used is limited. And therefore the waveform CANNOT be
PRECISELY reconstructed. This is also a fact as represented in Oppenheim
and Schaffer page 15.

There is nobody in the field of DSP who will agree that one can "precisely"
reconstruct an digital waveform from its FFT (or DFT). The DFT samples at
discrete finite number of freuqencies and as a result distorts the
waveform through a sampling process of its own. I can send you data that
show this.

>>
>><<
>> That is why the sampling of 44.1 KHz was chosen for CDs. The logic was
>> that human beings can only hear up to 20KHz in extreme cases.
Therefore,
>> by sampling any sound at 40KHz one can reproduce PERFECTLY any audible
>> sound. The extra 4.1KHz in the chosen rate was due to some
technicalities
>> that are beyond the scope of this discussion.
>> >>
>>
>>Wow have you been hypnotized by marketeers or what...
>>
>>Is this why they are considering going to a higher sampling rate right
now?
>>
>>
>Again, where are your facts? All I am talking about are well established
>mathematical and physical facts. Nothing about marketing here.

Please refer to any introductory text on DSP. Please tell me exactly what
type of DSP work do you do?

>><<
>>I hope this clarifies some of the mythology that is so fervently spread
by
>>the "audiophile" zealots.
>>
>>I believe and adhere to the principle that if some aspect of sound is
real,
>>then more than one person can hear it, and they hear it without being
told
>>that they are supposed to hear it! Bring out your double blind studies!
>>
>>Cheers.
>>
>>- -----------------------
>>Jonathan Cohler
>>cohler@-----.net
>>>
>>
>>To the contrary Jonathan, your post only revealed a bit of ignorance
>Ignorance about what?
Actual A/D converted waveform topology.

>>and does
>>not deal with the issue that there are few in the professional audio
field
>>who will not concede that CD sound is not as good as analog sound. In
fact
>>the audiophile zealots include a lot of professional recording engineers
and
>>musicians.
>I have yet to see a relevant fact from you. Again who concedes what to
>whom has nothing to do with facts.
Now you do!

>>SNIP..

>Jerry, I do not see a single factual statement in any of your long
>response, to my fact-based message.
Your hyperlink to an erroneous definition doesn't qualify either.

>The person or group of people who say something is true, does not make it
>any more true or false. People's opinions have no bearing on facts.
>People's feelings have no bearing on facts.
>Again, bring on your double blind studies!

The audio engineers are through with that stage (they have been convinced),
they are currently trying to find a way to ensure that a higher fidelity
recording medium than the current CD standard. So it is out of our hands
already.

>Cheers!
>- ------------------------
>Jonathan Cohler
>cohler@-----.net
>------------------------------
And later...

Date: Tue, 18 Nov 1997 23:33:45 -0500
From: Jonathan Cohler <cohler@-----.net>
Subject: Re: Digital Recordings.
>Jordan Selburn wrote:
>> Jerry -
>>
>> I'm puzzled by your post. Are you saying that analog recordings are
>> objectively superior to digital?
>>
>> The only comment that attempted to back this up was your comment on
>> the sampled 22KHz wave looking like a triangle wave. While this may
be
>> the case on the output of the DAC, there is always a filter
following
>> the DAC for exactly this reason. Put together the converter and
>> filter, and you get out what you put in: a 22KHz sine wave. Jonathon
>> was correct regarding Nyquist, and this holds in your example for
any
>> waveform under 1/2 the sampling frequency. As humans can only hear
up
>> to ~~20KHz, the "lost" data above 22.05 KHz is irrelevant; analog
>> recording, especially if reproduced on an LP, doesn't go even this
>> high.
>>

You are both wrong. Interpolation is required for reconstruction and this
is a process that makes "guesses" about what the waveform does between
sample points. It does not faithfully reconstruct the waveform. One can of
course reconstruct sinewaves as their trajectory is completely predictable.
But this is not the case for the overtones of a clarinet. I prefer to hear
the real thing rather than what a machine thinks I might have heard if the
original waveform were present.

>> Regarding the proposed higher resolution digital recording standards
>> (i.e., 24-bit/96KHz) - now here is an example of marketing hype.
While
>> there may be some incremental improvements in going to an 18- or
>> 20-bit recording (24 is overkill), are there any level-matched,
>> double-blind studies supporting an increased sampling rate?

Read the audiohpile press. The current problem with 24 bit digitization is
that the playback standards do not support a low enough jitter to enable
the benefit of 24 bit to be heard over 20 bit. (Indeterminacy in time
causing indeterminacy in amplitude when compared to the original waveform).

>Very succinctly and well put.

In your opinion! Its just not true.
>>
>> Certainly some people prefer analog recording to digital, and
>> everybody has their own tastes.
>Another good point. Everyone is entitled to their own subjective
opinions,
>of course. For example, some people prefer old clunky, noisy cars, to
>newer, sleeker, faster quieter ones. But this does not change the
>objective reality that the newer cars are faster, quieter, better gas
>mileage etc....

But when one sounds more like the real thing, I'll take it. I will take
digital when it sounds better too!

>>
>> Frankly, I'm not swayed by the comment
>> about how "there are few in the professional audio field
>> who will not concede that CD sound is not as good as analog sound"
(I
>> seriously doubt this is true in any case).
>Finally, someone who understands logic. Thanks, Jordan.
>> But I've never seen any
>> objective claim for analog superiority hold water; digital clearly
has
>> higher bandwidth and resolution (vs. ~13 bits max for an LP) and
lower
>> distortion.
>And no generational loss, and no tape hiss, and no loss of magnetic
>information over time, etc., etc.....
Wow, what about the facts? Who says 13 bits max? What about rise time? I
agree about hiss, but records do not use magnetic media. They are
mechanical (with their own problems!). There are plenty of objective claims
that have been published in "The Absolute Sound" in past years, however I
have thrown them all away and cannot respond with a reference.

>
>Even if one concedes that the ear may be more sensitive
> than a digital converter, analog recording fall still farther behind.
>Absolutely correct. But I still maintain that the ear is not more
>sensitive than the best digital converters on the market today (although
it
>is certainly better than cheapo bad ones). Again, I would challenge
anyone
>to produce even just one level-balanced double-blind study proving that
the
>ear is more sensitive than a high-end converter.
Gee, I'm beginning to notice a pattern, whenever someone makes a statement
that agrees with Jonathan, no facts are required. hmmmm...

I never made a claim that the record was as good as the ear. But that the
ear is better than the A/D converters currently in use was made evident
when people (dissapointed with the quality of CD sound) began exploring the
whole issue of jitter in the bit stream. Nobody claims that a system with
less jitter sounds worse. In fact double blind studies in the audio press
showed that people could discern a difference easily (even though the data
streams were analyzed to be identicle).

>Thank you for your succint and cogent remarks, Jordan.
>Best regards,
>Jonathan Cohler
>cohler@-----.net
But where are the facts? These are also statements of opinion, and a
mistaken understanding of the Nyquist Theorem.

Jerry Korten
NYC