[MRC01]

I've measured IM distortion on various devices using a digital
waveform consisting of 19 & 20 kHz at 0 dB. Even on high quality
devices I always get pretty high numbers - the highest IM artifacts in
a spectrum analysis usually peak at -40 to -60 dB below the original
two waves.

As an experiment, I changed the waveform so the 19 & 20 kHz are at -1
dB instead of 0 dB. On the same devices, the measured IM distortion
dropped to -80 to -100 dB and lower.

What's happening here? Is this expected? If so why? Or is it some kind
of flaw in my testing? If so, what?

[Don Pearce]

On Tue, 23 Oct 2007 10:38:35 -0700, MRC01 wrote:

I've measured IM distortion on various devices using a digital
waveform consisting of 19 & 20 kHz at 0 dB. Even on high quality
devices I always get pretty high numbers - the highest IM artifacts in
a spectrum analysis usually peak at -40 to -60 dB below the original
two waves.

As an experiment, I changed the waveform so the 19 & 20 kHz are at -1
dB instead of 0 dB. On the same devices, the measured IM distortion
dropped to -80 to -100 dB and lower.

What's happening here? Is this expected? If so why? Or is it some kind
of flaw in my testing? If so, what?

Your DAC just ran out of performance. You don't say what you are
actually testing, but on the assumption that it is simply being used
as the source of the test signals, and you aren't testing the DAC
itself, keep it at -1dB, or even a bit lower if it gets better still.

d

--
Pearce Consulting
http://www.pearce.uk.com

[[email protected]]

On Oct 23, 1:38 pm, MRC01 wrote:
I've measured IM distortion on various devices using a digital
waveform consisting of 19 & 20 kHz at 0 dB. Even on high quality
devices I always get pretty high numbers - the highest IM artifacts in
a spectrum analysis usually peak at -40 to -60 dB below the original
two waves.

As an experiment, I changed the waveform so the 19 & 20 kHz are at -1
dB instead of 0 dB. On the same devices, the measured IM distortion
dropped to -80 to -100 dB and lower.

"0 dB" and "-1 dB" relative to what?

What's happening here? Is this expected? If so why? Or is it some kind
of flaw in my testing? If so, what?

Without seeing more data, I'd bet you're right near the upper
limit of the dynamic range of your measurement system, and
the change from "-1" to "0" db is just enough to push you
into limiting the test system.

If, for example, by "0 dB" you mean the highest level
waveform the system can generate before clipping,
then consider what happens when you add a 0 dB
19 kHz and a 0 dB 20 kHz waveform: the result is
a combine waveform whose average level is +3 dB,
and whose peak level can be +6 dB: you could be clipping.

[Arny Krueger]

"MRC01" wrote in message
oups.com...

I've measured IM distortion on various devices using a digital
waveform consisting of 19 & 20 kHz at 0 dB. Even on high quality
devices I always get pretty high numbers - the highest IM artifacts in
a spectrum analysis usually peak at -40 to -60 dB below the original
two waves.

As an experiment, I changed the waveform so the 19 & 20 kHz are at -1
dB instead of 0 dB. On the same devices, the measured IM distortion
dropped to -80 to -100 dB and lower.

What's happening here?

It is not unusual for digital devices to be far less linear over the last dB
before FS.

Is this expected?

Yes.

If so why?

We live in an imperfect world.

Or is it some kind of flaw in my testing? If so, what?

Testing digital equipment at FS - 0 dB is sort of like testing amplifiers
right at clipping.

Most manufacturers spec amplifiers so that rated output is 0.5-2 dB below
actual clipping.

Some standards recommend testing digital equipment at FS -3 dB .

[MRC01]

On Oct 23, 11:03 am, wrote:
"0 dB" and "-1 dB" relative to what?

0 dB meaning the sample values at the peaks and troughs of the
waveform reach 32767 and -32767 respectively.

Because of this, I suspect this is a limitation of the DAC, not of the
analog outputs. I've tried with multiple devices all having different
DACs, and all exhibit this behavior to varying degrees. Because of
this, it seems to be a limitation common to DACs in general.

I suspect a more useful and realistic test waveform for measuring
whatever IM distortion might be audible during actual listening of
real music would be to use two tones in the more sensitive area of
human hearing - say 1 kHz and 2 kHz encoded at digital -6 dB.

[[email protected]]

On Oct 23, 2:43 pm, MRC01 wrote:
On Oct 23, 11:03 am, wrote:

"0 dB" and "-1 dB" relative to what?

0 dB meaning the sample values at the peaks and troughs of the
waveform reach 32767 and -32767 respectively.

That means, then, that it's quite possible on a regular
basis, when the peaks of each coincide, for the values
to WANT to be 65534 and -65534: clearly this WILL be
a problem for a 16-bit system.

Because of this, I suspect this is a limitation of the DAC, not of the
analog outputs. I've tried with multiple devices all having different
DACs, and all exhibit this behavior to varying degrees. Because of
this, it seems to be a limitation common to DACs in general.

No, it's not even a DAC problem. You're trying to represent
a number in the generation itself which is outside the realm
of the number system, assuming you're using 16-bit signed
integers. If I try, for example, in a language like C, to do
the following:

short int a, b, c;

a = 32767;
b = 32767;
c = a + b;

the program will generate, on most platforms, an overflow
error.

Do it in floats:

float a, b, c;

a = (float) 32767;
b = (float) 32767;
c = a + b;

And you get non error, but now try to stuff that into the 16-
bit input register of a DAC, and something is going to break.

If you're trying to do this on your typical soundcard, my only
suprise is that the results are not worse than you're reporting.

I suspect a more useful and realistic test waveform for measuring
whatever IM distortion might be audible during actual listening of
real music would be to use two tones in the more sensitive area of
human hearing - say 1 kHz and 2 kHz encoded at digital -6 dB.

No, it will be pretty useless, for a couple of reasons, but
the most important one is that if there is intermodulation,
the result will be sum and ifference frequencies, so you'll
get:

1 kHz + 2 kHz = 3 kHz

and

2 kHz - 1 kHz = 1 kHz

The 3 kHz difference, unless it's fairly high, is not going
to be audible because of the masking of the 2 kHz, and
the 1 kHz difference? How are you going to distinguish
that 1 kHz from the input 1 kHz?

The reasons tone like 19 kHz and 20 kHz are used a

1. The difference frequency is way down where things
are audible, at 1 kHz.

2. It's not likely that any products are going to be masked
by the input frequencies, because they are so widely
separated,

3. None of the sum and difference products will be
mistaken for the input tones,

4. If you're going to listen, the ear isn't very sensitive to
the input frequen cies to begin with

5. Many devices are more NONlinear at higher frequencies
and are thus easier to send into nonlinearity.

But, generating them digitally, you run the risk of generating
aliasing products unless both your generation algorithm and
your DAC is properly implemented. Not only will 19 kHz and
20 kHz generate a difference of 1 kHz, they will also generate
a sum of 39 kHz, and that, if not done properly, can alias
down to 5.1 kHz (assuming a 44.1 kHz sample rate) or
9 kHz (assuming 48 kHz sample rate).

And example where your waveform generating algorrithm,
implemented in an obvious fashion, can go awry like this
many people attempt to generate a digital square wave
as simply a series of alternating values, e.g., 22 values
at, oh, 20000 followed by 22 values at -20000, and so on.
This will NOT sound very much like a square wave, because
the algorithm is ignorant of the fact that such a representation
has hamronics that extend far above the nyquist rate (
tom infinity, in fact)., and all thos harmonics WILL get
aliased back into the base band. Better instead to
simply compute the first 11 terms of the series and sum
them together: such is inherently band-pass limited.

[MRC01]

On Oct 23, 12:43 pm, wrote:
No, it's not even a DAC problem. You're trying to represent
a number in the generation itself which is outside the realm
of the number system, assuming you're using 16-bit signed
integers...

This assumes the waveform is incorrectly encoded. That may be true,
but it isn't necessarily true. What if it is correctly encoded, for
example the 19 kHz and 20 kHz components are reduced in level by 6 dB
so their sum peaks at zero with no overload? When I look at this raw
waveform in Adobe Audition it is perfectly smooth and symmetric with
no evident clipping.

No, it will be pretty useless, for a couple of reasons, but
the most important one is that if there is intermodulation,
the result will be sum and ifference frequencies...

OK that makes sense. Actually I knew that IM distortion was based on
difference tones but didn't think it through. Why not shift the test
tones up to 21 kHz and 22 kHz? This would make them inaudible for most
people, yet still below Nyquist. That makes a useful test: play it
back and whatever you hear is IM distortion. Just don't fry your
tweeters!

And example where your waveform generating algorrithm,
implemented in an obvious fashion, can go awry like this
many people attempt to generate a digital square wave
as simply a series of alternating values, e.g., 22 values
at, oh, 20000 followed by 22 values at -20000, and so on.
This will NOT sound very much like a square wave, because
the algorithm is ignorant of the fact that such a representation
has hamronics that extend far above the nyquist rate (
tom infinity, in fact)., and all thos harmonics WILL get
aliased back into the base band. Better instead to
simply compute the first 11 terms of the series and sum
them together: such is inherently band-pass limited.

It sounds like you're saying to low pass filter the raw data of the
waveform instead of relying on the playback digital filter to do the
same. If so, it seems the results would depend on the playback digital
filter, which could make it a good test of the digital filter. In
other words, you can pre-filter the raw data to achieve a reasonable
result on most systems - extraneous frequencies already removed so you
don't need the playback filter to do much of anything, or you can
encode it with no filtering to test how well the playback filter does.

[[email protected]]

On Oct 23, 4:18 pm, MRC01 wrote:
On Oct 23, 12:43 pm, wrote:

No, it's not even a DAC problem. You're trying to represent
a number in the generation itself which is outside the realm
of the number system, assuming you're using 16-bit signed
integers...

This assumes the waveform is incorrectly encoded.

No, it makes no such assumption. Again, consider
the code:

long in t;
float a, b, c, ampl = 32767.0;
short int r;

// big loop, where t is incremented and scaled
// accordingly, for each sample period

for (t = 0; T someBigNumber, t += 1)
{
a = ampl * sin(19000*t);
b = ampl * sin(20000*t);
c = a + b;

I think you would agree that, so far, c is being computed
correctly. The next step:

r = (short int) c;

breaks, because r will overflow if c 32767, for example,
which it WILL be when the two peaks coincide.

What if it is correctly encoded, for
example the 19 kHz and 20 kHz components are
reduced in level by 6 dB
so their sum peaks at zero with no overload? When I look at this raw
waveform in Adobe Audition it is perfectly smooth and symmetric with
no evident clipping.

Because what you see in Adobe Audition is a VISUAL
representation of the waveform, and there is no assurance
whatsoever it represents what the waveform itself is
actually doing.

But, just from the standpoint of the basic properties
of the algorithm, reducing both by 6 dB will ensure
proper encoding. Not reducing the levels invites
breakage. It has nothing to do with what the waveforms
LOOK like, it has everything to do with what they ARE.

No, it will be pretty useless, for a couple of reasons, but
the most important one is that if there is intermodulation,
the result will be sum and ifference frequencies...

OK that makes sense. Actually I knew that IM distortion was based on
difference tones but didn't think it through. Why not shift the test
tones up to 21 kHz and 22 kHz? This would make them inaudible for most
people, yet still below Nyquist.

Because both will probably be at or above the actual cutoff
frequency of the brickwall filter, and at 21 kHz, you're sailing
awfully close to the Nyquist wind anyway.

What I would do, if you were concerned about the audibility
of the input tones, is run your 19 kHz and 20 kHz through
whatever you're testing, take the output, put a 10 kHz low
pass filter in place, and get rid of the original stuff, leaving
only the difference tones.

And example where your waveform generating algorrithm,
implemented in an obvious fashion, can go awry like this
many people attempt to generate a digital square wave
as simply a series of alternating values, e.g., 22 values
at, oh, 20000 followed by 22 values at -20000, and so on.
This will NOT sound very much like a square wave, because
the algorithm is ignorant of the fact that such a representation
has hamronics that extend far above the nyquist rate (
tom infinity, in fact)., and all thos harmonics WILL get
aliased back into the base band. Better instead to
simply compute the first 11 terms of the series and sum
them together: such is inherently band-pass limited.

It sounds like you're saying to low pass filter the raw data
of the waveform instead of relying on the playback digital
filter to do the same.

No, not only that, but it's also relying on any algorithms that
follow that might be sensitive to aliasing products, The
playback reconstruction filter is only the very last thing you
have to worry about.

A very important principle of sampling, and one
that many people have a hard time grasping is that
a time-samples stream contains all the aliases from
infinity crammed into the baseband, and contains all
images of the baseband out to infinity. ANY process
must deal with that, whether its digital or analog.

That's one reason why, for example, sample rate conversion
algorithms ALWAYS start FIRST with an anti-imaging filter
(even though you might not think it necessary), it does
the conversion, and then it has an anti-aliasing filter,
even though everything is done in the digital domain.

[MRC01]

On Oct 23, 4:18 pm, MRC01 wrote:
This assumes the waveform is incorrectly encoded.

On Oct 23, 12:43 pm, wrote:

No, it makes no such assumption. Again, consider
the code:

I understand your code. I call it "incorrect" because it overflows.
What I'm saying is that this code is not necessarily the way the
waveform was created. If you take that same code and normalize the
final value to a range of 32767 to -32767, while it is still an int,
BEFORE converting it to a short int, then it doesn't overflow. But
that's the same thing I was suggesting - cutting it in half or scaling
it back -6 dB.

I suspect that since I only have to attenuate it by 1 dB for the
distortion to drop to levels around -100 dB, it seems that the
waveform is properly constructed without overflow. In other words, if
the overflow was there as you are suggesting, I would have to
attenuate it a lot more - at least 6 dB before the distortion would be
eliminated.

Because what you see in Adobe Audition is a VISUAL
representation of the waveform, and there is no assurance
whatsoever it represents what the waveform itself is
actually doing.

I can also look at the actual sample values in adobe audition. Though
it's true the curve it draws through the samples is likely not
filtered the same way the DAC works.

No, not only that, but it's also relying on any algorithms that
follow that might be sensitive to aliasing products, The
playback reconstruction filter is only the very last thing you
have to worry about.

A very important principle of sampling, and one
that many people have a hard time grasping is that
a time-samples stream contains all the aliases from
infinity crammed into the baseband, and contains all
images of the baseband out to infinity. ANY process
must deal with that, whether its digital or analog.

That's one reason why, for example, sample rate conversion
algorithms ALWAYS start FIRST with an anti-imaging filter
(even though you might not think it necessary), it does
the conversion, and then it has an anti-aliasing filter,
even though everything is done in the digital domain.- Hide quoted text -

It depends on the goal. If the goal is to reliably produce the closest
thing to a square wave that 44.1 kHz samples allow, then what you
describe makes sense. But if the goal is to test how well the DAC
interprets a difficult waveform then one *should* use the simpler
mathematically pure square wave that you described. Theoretically, an
ideal DAC should output a proper looking square wave when fed that
signal. It should be able to filter out everthing 22.5 kHz with
minimal passband distortion. Nothing in the real world is perfect; one
should expect to see some distortion, but the goal is to compare the
distortion generated by different DACs.

[[email protected]]

On Oct 23, 11:48 pm, MRC01 wrote:
That's one reason why, for example, sample rate conversion
algorithms ALWAYS start FIRST with an anti-imaging filter
(even though you might not think it necessary), it does
the conversion, and then it has an anti-aliasing filter,
even though everything is done in the digital domain.

It depends on the goal. If the goal is to reliably produce the closest
thing to a square wave that 44.1 kHz samples allow, then what you
describe makes sense. But if the goal is to test how well the DAC
interprets a difficult waveform then one *should* use the simpler
mathematically pure square wave that you described. Theoretically, an
ideal DAC should output a proper looking square wave when fed that
signal. It should be able to filter out everthing 22.5 kHz with
minimal passband distortion.

No, that's what you're not getting: The waveform I described is
NOT "mathematically pure;" It's already broken BEFORE it
gets to the DAC: because it is discrete time sampled, and
because it was NOT band-limited before it was generated,
it already contains all of the aliases folded down into the
baseband.

Nothing in the real world is perfect; one
should expect to see some distortion, but the goal is to compare the
distortion generated by different DACs.

Then the step-generated square-wave I described is NOT
the way to do it, because it is intrinsically distorted before
it hits the DAC.

Let's try a different approach: whatever code you are using
to generate the waveform can be viewed as THE analog-to-
digital conversion process for that waveform. The code IS
sampling a waveform. And to prevent ANY aliases from
finding there way into the sampled stream, the waveform
MUST be low-pass filtered to less than 1/2 the sample rate
BEFORE sampling.

Therefore, a sample sequence of the type I described first,
where you have some number of samples at some positive
level, followed by the same number of samples at the
same negative value, IS ALREADY BROKEN in that it
was not properly anti-alias filtered before sampling.

A 10 kHz waveform generated this way will have, in
it's "pure mathematical" form, harmonics at 30 kHz,
50 kHz, 70 kHz and so on. In that sampled stream,
those harmonics will already be folded back to 14.1 kHz,
5.9 kHz, 28.2 kHz and so on. Those aliases ARE ALREADY
IN THE SAMPLED STREAM. A PERFECT DAC could NEVER
filter them out: they're within the passband of a perfect
anti-imaging filter.

Now, what would that same 10 kHz square wave really look
like in the sampled stream if captured by the perfect ADC?
Well, since 30 kHz and everything else above the Nyquist
frequency gets filtered, the sampled stream would consist
ONLY of the sampling of a 10 kHz SINE wave.

Let's repeat, a step-generated sampled square wave is a
BAD test for a DAC, becuase that sampled stream is already
heavily distorted with the aliases, because the Nyquist
criteria was violated by the very sampling process used to
generate it. If a DAC playing such a waveform sounds
distorted, the DAC is doing its job correctly, because
the sampled waveform is distorted.

It's far from intuitive, to be sure. But it's another case
where intuition about things is simply wrong.

[MRC01]

On Oct 24, 5:41 am, wrote:
Let's repeat, a step-generated sampled square wave is a
BAD test for a DAC, becuase that sampled stream is already
heavily distorted with the aliases, because the Nyquist
criteria was violated by the very sampling process used to
generate it. If a DAC playing such a waveform sounds
distorted, the DAC is doing its job correctly, because
the sampled waveform is distorted.

Are you saying that certain combinations of samples are invalid
because there does not exist a unique curve that passes through them
which is properly bandwidth limited? If so that is an interesting
proposition. A "correct" set of sampling points is a set in which
there exists a single, unique curve that passes through all the points
and is bandwidth limited below Nyquist. Thus I can see two possible
classes of invalidity. One, where there doesn't exist *any* curve
(properly bandwidth limited) that passes through the points. Two,
where multiple different curves may pass through the same points. In
either case, the output of the DAC is undefined.

[Dave Platt]

In article .com,
wrote:

Let's try a different approach: whatever code you are using
to generate the waveform can be viewed as THE analog-to-
digital conversion process for that waveform. The code IS
sampling a waveform. And to prevent ANY aliases from
finding there way into the sampled stream, the waveform
MUST be low-pass filtered to less than 1/2 the sample rate
BEFORE sampling.

Therefore, a sample sequence of the type I described first,
where you have some number of samples at some positive
level, followed by the same number of samples at the
same negative value, IS ALREADY BROKEN in that it
was not properly anti-alias filtered before sampling.

A 10 kHz waveform generated this way will have, in
it's "pure mathematical" form, harmonics at 30 kHz,
50 kHz, 70 kHz and so on. In that sampled stream,
those harmonics will already be folded back to 14.1 kHz,
5.9 kHz, 28.2 kHz and so on. Those aliases ARE ALREADY
IN THE SAMPLED STREAM. A PERFECT DAC could NEVER
filter them out: they're within the passband of a perfect
anti-imaging filter.

Yup.

Another way of looking at it, is that the "mathematically pure" sample
sequence is *the* correct and legitimate sampled representation of a
valid (properly-bandlimited) analog waveform - a waveform which could
have been presented to an ADC for conversion to digital form. And,
that waveform is *not* a bandwidth-limited square wave.

If you feed a good ADC a bandlimited analog waveform with the
frequency components that Dick indicates (in the correct amplitudes
and phase relationships, of course), the sample sequence which comes
out of the ADC will be the "mathematically pure" pattern of values
originally discussed.

If you feed these samples into a properly-functioning DAC, it will
re-create this bandlimited analog waveform... *not* an attempted
recreation of a sharp-edged square wave with frequencies lying above
half of the sample rate.

I'm sure that it's possible to deliberately tweak the design of a
DAC's reconstruction filter in order to make the output "look better"
in this case and cases like this... i.e. to create an analog waveform
which looks more like a square wave. However, doing so *reduces* the
DAC's actual accuracy, and will introduce linear or nonlinear
distortion into every signal which goes through the DAC... it makes
the DAC worse rather than better. If I recall correctly, there have
been a couple of DAC designs which did this (Wadia, and the Pioneer
Legato Link design come to mind, but my memory could well be wrong
about this).

It's far from intuitive, to be sure. But it's another case
where intuition about things is simply wrong.

Perhaps not quite as counter-intuitive as quantum physics, but it does
share the same violation-of-apparent-reasonableness problem :-)

--
Dave Platt AE6EO
Friends of Jade Warrior home page: http://www.radagast.org/jade-warrior
I do _not_ wish to receive unsolicited commercial email, and I will
boycott any company which has the gall to send me such ads!

[Dave Platt]

In article . com,
MRC01 wrote:

Let's repeat, a step-generated sampled square wave is a
BAD test for a DAC, becuase that sampled stream is already
heavily distorted with the aliases, because the Nyquist
criteria was violated by the very sampling process used to
generate it. If a DAC playing such a waveform sounds
distorted, the DAC is doing its job correctly, because
the sampled waveform is distorted.

Are you saying that certain combinations of samples are invalid
because there does not exist a unique curve that passes through them
which is properly bandwidth limited?

Every combination of samples is valid, and corresponds to a unique
(within the system's quantization limits) bandwidth-limited analog
waveform.

The "mathematically pure" sample pattern at issue (N samples at one
value, followed by another N samples at a different value,
lather/rinse/repeat) doesn't correspond to a bandwidth-limited square
wave of period 2N samples, though.

It corresponds to a bandwidth-limited signal with the characteristics
that Dick describes... one which has a sinewave fundamental of period
2N, plus a whole bunch of non-harmonically-related components that
correspond to folded-back aliases of the harmonics of the fundamental.

Thus I can see two possible
classes of invalidity. One, where there doesn't exist *any* curve
(properly bandwidth limited) that passes through the points. Two,
where multiple different curves may pass through the same points. In
either case, the output of the DAC is undefined.

It's not a problem of the sample sequence being invalid. It isn't.

It's perfectly valid. It just doesn't happen to correspond with the
sort of analog signal that intuition suggests that it does.

--
Dave Platt AE6EO
Friends of Jade Warrior home page: http://www.radagast.org/jade-warrior
I do _not_ wish to receive unsolicited commercial email, and I will
boycott any company which has the gall to send me such ads!

[Arny Krueger]

"MRC01" wrote in message
ups.com...

On Oct 24, 5:41 am, wrote:

Let's repeat, a step-generated sampled square wave is a
BAD test for a DAC, becuase that sampled stream is already
heavily distorted with the aliases, because the Nyquist
criteria was violated by the very sampling process used to
generate it. If a DAC playing such a waveform sounds
distorted, the DAC is doing its job correctly, because
the sampled waveform is distorted.

Are you saying that certain combinations of samples are invalid
because there does not exist a unique curve that passes through them
which is properly bandwidth limited? If so that is an interesting
proposition. A "correct" set of sampling points is a set in which
there exists a single, unique curve that passes through all the points
and is bandwidth limited below Nyquist. Thus I can see two possible
classes of invalidity. One, where there doesn't exist *any* curve
(properly bandwidth limited) that passes through the points. Two,
where multiple different curves may pass through the same points. In
either case, the output of the DAC is undefined.

There's a third interesting relevant case that could be relevant to the OP.

There can sometimes be a unique curve that passes through the samples and
has a maximum amplitude that significantly exceeds FS.

Such cases arise from time to time in the real world, and can exceed 120% of
FS.

[Eeyore]

MRC01 wrote:

wrote:
"0 dB" and "-1 dB" relative to what?

0 dB meaning the sample values at the peaks and troughs of the
waveform reach 32767 and -32767 respectively.

Which is FSD.

So if you add 2 such waveforms you'll clip the DAC it would seem.

Are BOTH frequencies at '0dB' level ?

Graham

[MRC01]

On Oct 24, 11:38 am, (Dave Platt) wrote:
Every combination of samples is valid, and corresponds to a unique
(within the system's quantization limits) bandwidth-limited analog
waveform.

That's what I thought... which is why I was confused by Pierce's
description.

The "mathematically pure" sample pattern at issue (N samples at one
value, followed by another N samples at a different value,
lather/rinse/repeat) doesn't correspond to a bandwidth-limited square
wave of period 2N samples, though.

It corresponds to a bandwidth-limited signal with the characteristics
that Dick describes... one which has a sinewave fundamental of period
2N, plus a whole bunch of non-harmonically-related components that
correspond to folded-back aliases of the harmonics of the fundamental.

Here is where that is counterintuitive. A perfect square wave cannot
exist in nature - the first derivative of the waveform curve is
discontinuous, implying an infinite rate of change of air pressure,
which is impossible. But suppose you generate a very sharp "square"
wave with frequencies up to, say, 100 kHz. Then you pick it up with a
super fancy microphone with bandwidth to 100 kHz. That microphone
produces an analog signal which is a very sharp square wave. Now you
sample it digitally at 44.1 kHz. The sampling points are measuring the
amplitude every 22.7 microseconds. This square wave is so sharp,
containing 100 kHz elements, it jumps full scale in less time than
that. And it has such high frequencies, the overshoot and ringing is
practically invisible to sampling points spaced so far apart in time.
So how could the sampling points produced by the ADC *not* be this
simplisitic wave?

[Don Pearce]

On Wed, 24 Oct 2007 13:52:28 -0700, MRC01 wrote:

On Oct 24, 11:38 am, (Dave Platt) wrote:
Every combination of samples is valid, and corresponds to a unique
(within the system's quantization limits) bandwidth-limited analog
waveform.

That's what I thought... which is why I was confused by Pierce's
description.

The "mathematically pure" sample pattern at issue (N samples at one
value, followed by another N samples at a different value,
lather/rinse/repeat) doesn't correspond to a bandwidth-limited square
wave of period 2N samples, though.

It corresponds to a bandwidth-limited signal with the characteristics
that Dick describes... one which has a sinewave fundamental of period
2N, plus a whole bunch of non-harmonically-related components that
correspond to folded-back aliases of the harmonics of the fundamental.

Here is where that is counterintuitive. A perfect square wave cannot
exist in nature - the first derivative of the waveform curve is
discontinuous, implying an infinite rate of change of air pressure,
which is impossible. But suppose you generate a very sharp "square"
wave with frequencies up to, say, 100 kHz. Then you pick it up with a
super fancy microphone with bandwidth to 100 kHz. That microphone
produces an analog signal which is a very sharp square wave. Now you
sample it digitally at 44.1 kHz. The sampling points are measuring the
amplitude every 22.7 microseconds. This square wave is so sharp,
containing 100 kHz elements, it jumps full scale in less time than
that. And it has such high frequencies, the overshoot and ringing is
practically invisible to sampling points spaced so far apart in time.
So how could the sampling points produced by the ADC *not* be this
simplisitic wave?

Because you have neglected that vital component - the anti-aliasing
filter. That will turn your steep-sided 100kHz square wave into a
sloping sided, rounded cornered wave which can be successfully sampled
without producing alias signals.

I've written a paper on aliasing that should explain things.

http://www.pearce.uk.com/papers/index.htm

d

--
Pearce Consulting
http://www.pearce.uk.com

[Dave Platt]

In article .com,
MRC01 wrote:


Here is where that is counterintuitive. A perfect square wave cannot
exist in nature - the first derivative of the waveform curve is
discontinuous, implying an infinite rate of change of air pressure,
which is impossible. But suppose you generate a very sharp "square"
wave with frequencies up to, say, 100 kHz. Then you pick it up with a
super fancy microphone with bandwidth to 100 kHz. That microphone
produces an analog signal which is a very sharp square wave. Now you
sample it digitally at 44.1 kHz. The sampling points are measuring the
amplitude every 22.7 microseconds. This square wave is so sharp,
containing 100 kHz elements, it jumps full scale in less time than
that. And it has such high frequencies, the overshoot and ringing is
practically invisible to sampling points spaced so far apart in time.
So how could the sampling points produced by the ADC *not* be this
simplisitic wave?

Well, there are two answers to your question, based on whether this
process did, or did not "follow the rules".

By "follow the rules", I mean whether there was a step that you did
not specifically mention: properly band-limiting the signal just
prior to sampling.

If you didn't band-limit the signal - if you sampled it when it was
still at its full bandwidth - then I'd say:

- The sample values you see *will* be the "simplistic" sequence, and

- The sample values are not a *meaningful* representation of the
original square wave, within the constraints of a sampled system,
because you've broken the cardinal rule - you're sampling a signal
which contain frequency components outside of the system's
bandwidth limit. "Garbage in, garbage out".

If you did bandwidth-limit the signal, then it will no longer have the
extremely fast rise-time of the original square-wave. The
higher-order odd harmonics will be missing. With them absent, the
"square wave' will no longer have a flat top... it will exhibit
ripple, and thus the sampler won't produce the same uniform sequence
of values.

You can think of it in another way. Yes, feeding a non-bandwidth-
limited square wave to a sampler will produce a "mathematically pure
square wave" in sampled form. However, there are a literally infinite
number of *other* non-bandwidth-limited waveforms which will produce
the exact same sequence of samples. How can the DAC possibly decide
between them?

There's only one "legal" (properly bandwidth-limited) signal (within
quantization limits) which will produce this particular pattern of
samples... and that's the only one which the a properly-designed DAC
can reproduce.

--
Dave Platt AE6EO
Friends of Jade Warrior home page: http://www.radagast.org/jade-warrior
I do _not_ wish to receive unsolicited commercial email, and I will
boycott any company which has the gall to send me such ads!

[MRC01]

So the simplistic square wave sequence is a valid sequence of samples
that defines a unique properly bandwidth filtered wave. But that wave
isn't a square wave.

The interesting concept here is that I assumed that if you sample a
waveform, the spacing / frequency of the samples would simply skip
over anything that was moving too fast / too high frequency to be
seen. But that is not the case.

For example, suppose the waveform contains frequencies above Nyquist,
so your sampling points are far apart relative to changes in the
waveform. So you are skipping over a lot of information simply based
on the spacing. But each sampling point has to land *somewhere*, and
it will frequently happen that it lands on a certain bump in the
waveform that wouldn't exist except for frequencies above Nyquist.
These frequences have to be eliminated BEFORE sampling because once
sampled they MUST be interpreted by the playback DAC as frequencies
below Nyquist - which they aren't so that means the wave constructed
from them MUST be different from the one sampled.

Now *that* is an AHA experience. Thanks!

[[email protected]]

On Oct 24, 6:14 pm, MRC01 wrote:
So the simplistic square wave sequence is a valid
sequence of samples that defines a unique properly
bandwidth filtered wave.

If by "simplistic square wave" you mean the alternating
sequence of positive and negative values, then no, it
most assuredly IS NOT a "unique, properly bandwidth
filtered wave." It has, in essence, infinite bandwidth for
the purpose of this discussion,

But that wave isn't a square wave.

Yes, it is, it just has a bandwidth that well exceeds
the Nyquist criteria. And sampled in that fashion, it
violates said criteria. In doing so, it is "broken" in that
all the out-of-bandwith products are aliased down
into the baseband.

The interesting concept here is that I assumed that
if you sample a waveform, the spacing / frequency of
the samples would simply skip over anything that was
moving too fast / too high frequency to be seen.

But they ARE "seen", they are aliased down into the
baseband where they can be readily "seen" (heard,
measured) as spurious information.


Here's completely apt analogy. Say you have a movie
camera running at 24 frames per second. You concept
of "the spacing/frequency ... moving to fast to be seen"
can be shown to break down when you look at a seen
from an old Western where the bad guys are chasing
the good guys riding in wagons with spoked wheels.
They're racing along forward at a breakneck speed,
yet there are the wheels, quite visibly turning BACKWARDS
at a low speed.. You can see exactly the same phenomenon
in the rotor of a helicopter, where we know the blade
is rotating fast, but a movie of it shows the blades almost
stationary or very slowly.

Why? Because due to the discrete time-sampling of
the camera, and the fact that the blades or spokes are
moving at a frequency much higher than the sampling
frequency, the image we get is quite an incorrect picture
of physical reality. Why? Because of the aliasing caused
by things moving too fast in a discrete-time sampled
stream.

Let's see how this works. Take our helicopter, which we
will assume has two blades. Let's, for simplicity sake,
set our frame (sampling) rate at 25 fps. Now, let's assume
the blade is rotating at, oh, 13.9 revolutions per second.
I pick that number for a two reasons:

1. It's a plausible value for the rotational speed of the blade,

2. It's above the Nyquist frequency of 12.5 Hz (half the frame
rate of 25 Hz) and thus deliberately violates the Nyquist
criteria for the purpose of the demonstration.

Now, what happens? Frame 1 capture the blade at 0 deg.
Frame 2 capture it having rotated about 10% less than half
a revolution, frame 3 about 20% less, and so on.

String all those frames together and view them, and what do
we see? We DON'T see the blade moving forward at 12.5
RPM, we REALLY see it moving BACKWARDS at about
75 RPM. The forward rotating blade was ALIASED by the
sampling process to appear rotating backwards.

EXACTLY the same principle applies to discrete time-sampled
digital audio.

Think of your "mathematically pure" square wave, the
sequence of plus and minus samples, as a bunch of
those rotating helicopter blades all connected together
by gear trains so that 1 is rotating at, oh, 10 kHz,
another at 30, another at 50, and at 70, 90, 110, 130, 150
kHz and so on. Sample that at 44.1 kHz, which is EXACTLY
what you've done by simply alternating a sequence of plus
and minus values, and what do you get? All those "blades"
get aliased down:

10 kHz - 10 kHz
30 kHz - 14.1 kHz
50 kHz - 5.9 kHz
70 kHz - 18.2 kHz
90 kHz - 1.8 kHz
110 kHz - 21.8 kHz
130 kHz - 2.3 kHz
150 kHz - 17.7 kHz

And so on.

The problem is that those aliases ARE ALREADY BUILT IN
TO THE SAMPLED DATA STREAM as an intrinsic result
of sampling.

For example, suppose the waveform contains frequencies above Nyquist,
so your sampling points are far apart relative to changes in the
waveform. So you are skipping over a lot of information simply based
on the spacing. But each sampling point has to land *somewhere*, and
it will frequently happen that it lands on a certain bump in the
waveform that wouldn't exist except for frequencies above Nyquist.
These frequences have to be eliminated BEFORE sampling because once
sampled they MUST be interpreted by the playback DAC as frequencies
below Nyquist - which they aren't so that means the wave constructed
from them MUST be different from the one sampled.

Yup. you got it!

Now *that* is an AHA experience. Thanks!

Indeed.

[MRC01]

On Oct 25, 6:03 am, wrote:
On Oct 24, 6:14 pm, MRC01 wrote:

So the simplistic square wave sequence is a valid
sequence of samples that defines a unique properly
bandwidth filtered wave.

If by "simplistic square wave" you mean the alternating
sequence of positive and negative values, then no, it
most assuredly IS NOT a "unique, properly bandwidth
filtered wave." It has, in essence, infinite bandwidth for
the purpose of this discussion,

But that wave isn't a square wave.

Yes, it is, it just has a bandwidth that well exceeds
the Nyquist criteria. And sampled in that fashion, it
violates said criteria. In doing so, it is "broken" in that
all the out-of-bandwith products are aliased down
into the baseband.

No, that's not what I meant. Let me rephrase in more precise terms.

*IF* every possible sequence of samples is valid - meaning every
sequence has exactly 1 unique Nyquist bandwidth limited waveform that
passes through all the sample points,

*THEN*, the sequence that consists of 22 samples of, say, 30,000
followed by 22 samples of -30,000 lather rinse repeat, must be a valid
sequence - by definition there must exist *some* analog waveform that,
after being properly anti-aliased, produces these values from the ADC.

*BUT*, the analog waveform that produces this sequence is not a square
wave. It is something else.

Actually this leads to another question. Is the DAC limited in
amplitude as it is in bandwidth? That is, does the DAC necessarily
have to clip the analog wave it produces just because the samples
appear to be clipped?

Example: Suppose you digitize a pure sine wave but you overshoot the
levels so the peaks are clipped. But the frequency of the sine wave is
high enough - yet still below Nyquist - that none of the clipped
portions of the wave happened to be sampled. There may not even be any
samples at digital zero (full scale), but there will be some very
close to that. When the DAC reconstructs this wave, will it generate a
wave whose amplitude is greater than full scale, which matches the
original? Or will it be required to produce a wave whose amplitude
never exceeds full scale - which will be a different wave, thus
distorted?

[Peter Larsen[_2_]]

MRC01 wrote:

OK that makes sense. Actually I knew that IM distortion was based on
difference tones but didn't think it through. Why not shift the test
tones up to 21 kHz and 22 kHz? This would make them inaudible for most
people, yet still below Nyquist. That makes a useful test: play it
back and whatever you hear is IM distortion. Just don't fry your
tweeters!

You seem to assume that all can hear up to exactly 20000 Hz? - also,
assuming a 44.1 or even 48 kHz sampling rate I tend to assume that the
antialiasing filters will matter.


Kind regards

Peter Larsen

[[email protected]]

On Oct 25, 1:38 pm, MRC01 wrote:
On Oct 25, 6:03 am, wrote:



On Oct 24, 6:14 pm, MRC01 wrote:

So the simplistic square wave sequence is a valid
sequence of samples that defines a unique properly
bandwidth filtered wave.

If by "simplistic square wave" you mean the alternating
sequence of positive and negative values, then no, it
most assuredly IS NOT a "unique, properly bandwidth
filtered wave." It has, in essence, infinite bandwidth for
the purpose of this discussion,

But that wave isn't a square wave.

Yes, it is, it just has a bandwidth that well exceeds
the Nyquist criteria. And sampled in that fashion, it
violates said criteria. In doing so, it is "broken" in that
all the out-of-bandwith products are aliased down
into the baseband.

No, that's not what I meant. Let me rephrase in more precise terms.

*IF* every possible sequence of samples is valid - meaning every
sequence has exactly 1 unique Nyquist bandwidth limited waveform that
passes through all the sample points,

And that assumption itself is false: Every possible sequence
of sample values IS NOT valid, because a very large number
of them, the example you give below, is one example of a
an entire class of sequences that violates the Nyquist
criteria.

*THEN*, the sequence that consists of 22 samples of, say, 30,000
followed by 22 samples of -30,000 lather rinse repeat, must be a valid
sequence - by definition there must exist *some* analog waveform that,
after being properly anti-aliased, produces these values from the ADC.

Yes, there does exist SOME function that passes through
these points. The problem is there is not a UNIQUE
function that passes through these points, because the
waveform you describe HAS NOT BEEN ANTI-ALIS FILTERED
BEFORE IT WAS SAMPLED.


*BUT*, the analog waveform that produces this sequence is
not a square wave. It is something else.

It makes absolutely no difference what the original was:
the waveform YOU describe above itself is intrinsically NOT
a valid sequence of samples.

Your assumption that every sequence of samples is a valid
sequence itself is intrinsically flawed. We keep describing,
over and over, an example of a sequence WHICH IS NOT VALID
because it violates Nyquist.

Actually this leads to another question. Is the DAC limited in
amplitude as it is in bandwidth? That is, does the DAC necessarily
have to clip the analog wave it produces just because the samples
appear to be clipped?

Nope. You can have a valid sequence of samples whose
reconstructed waveform exceeds the output voltage
produced by, say, a constant DC value of 32767. And
that waveform would be valid.

Example: Suppose you digitize a pure sine wave but
you overshoot the levels so the peaks are clipped.
But the frequency of the sine wave is high enough -
yet still below Nyquist - that none of the clipped
portions of the wave happened to be sampled.

Well, you keep making the mistake of not filtering
before sampling. Let's say it's a 15 kHz sine wave.
And let's say full scale (+- 32767) represents a
a voltage of +-1 volt. If we put in a sine wave whose
amplitude is +- 1.2 volts, ANY clipping in the analog
domain is irrelevant, because the 20 kHz anti-aliasing
filter will prevent ANY harmonics from ever reaching
the sampler.

But, you could well have two tones that intermodulate
before the filter, and whose products are BELOW
Nyquist, and they will be sampled and digitized quite
nicely, thank you.

When the DAC reconstructs this wave, will it generate a
wave whose amplitude is greater than full scale, which matches the
original?

It's certainly possible.

Or will it be required to produce a wave whose amplitude
never exceeds full scale - which will be a different wave, thus
distorted?

It's also certainly possible that a DAC could produce a
waveform that has peak levels in excess of the DAC's
o dB level and have a downstream buffer or line driver clip,
but that's another issues.

The short answer is that it's quite possible for a system
sampled quantized encoding to encode waveforms whose
continuous representation exceeds, at peaks, the nominal
maximum output level. It's unusual simply because this
represents a signal with a LOT of energy near the Nyquist
limit, which is a relatively rare occurrence in the things we
generally like to record and listen to.

[MRC01]

On Oct 25, 5:36 pm, wrote:
... Every possible sequence
of sample values IS NOT valid, because a very large number
of them, the example you give below, is one example of a
an entire class of sequences that violates the Nyquist
criteria.

But on Oct 24, 11:38 am, Dave Platt wrote:
Every combination of samples is valid, and corresponds to a unique
(within the system's quantization limits) bandwidth-limited analog
waveform.

This is an interesting mathematical question. Do there exist some
combinations of samples that are invalid? Or does every combination of
samples define a unique anti-aliased Nyquist limited wave? Hmmm....

It may be tunnel vision to assume that because this particular
example: 22 samples of 30,000 followed by 22 samples of -30,000
doesn't represent a properly anti-aliased square wave, that it doesn't
represent any wave at all. I'm pondering the idea that *some* kind of
wave, after being anti-aliased, might just so happen to produce this
exact set of samples, even though that wave might not even resemble a
square wave.

If there *are* combinations of samples that are invalid - that no anti-
aliased waveform could produce - then what is the DAC supposed to do
when it encounters one? It's going to fit a waveform... if the
algorithm or function it uses has an inverse then that would show us
the waveform that would have produced that set of samples. Of course
it doesn't necessarily have an inverse.

In that sense, the mathematical question about whether there exist
"invalid" sequences of samples, may be the same as asking whether the
ADC / DAC functions or algorithms are bijections.

[Dave Platt]

In article .com,
MRC01 wrote:

But on Oct 24, 11:38 am, Dave Platt wrote:
Every combination of samples is valid, and corresponds to a unique
(within the system's quantization limits) bandwidth-limited analog
waveform.

This is an interesting mathematical question. Do there exist some
combinations of samples that are invalid?

Nope.

Or does every combination of
samples define a unique anti-aliased Nyquist limited wave? Hmmm....

Yes (within the quantization resolution limit of the system).

It may be tunnel vision to assume that because this particular
example: 22 samples of 30,000 followed by 22 samples of -30,000
doesn't represent a properly anti-aliased square wave, that it doesn't
represent any wave at all.

This sample sequence *does* correspond a perfectly-legitimate, properly-
antialiased (Nyquist-limited) continuous waveform.

So, yeah, I'd say that you're suffering from intuitive tunnel vision.
The fact that the sample sequence does *not* correspond to the sort of
continuous signal that your intuition leads you to believe it should,
is causing you to (mistakenly) believe that it doesn't correspond to
*any* continuous waveform. That's a mis-conclusion, that you should
strive to let float away on the breeze :-)

I'm pondering the idea that *some* kind of
wave, after being anti-aliased, might just so happen to produce this
exact set of samples, even though that wave might not even resemble a
square wave.

Oh, it will. This sample sequence is *not* invalid. It's perfectly
legal.

If there *are* combinations of samples that are invalid - that no anti-
aliased waveform could produce - then what is the DAC supposed to do
when it encounters one?

Since the "If" you state isn't true, the "then what" is irrelevant.

It's going to fit a waveform... if the
algorithm or function it uses has an inverse then that would show us
the waveform that would have produced that set of samples. Of course
it doesn't necessarily have an inverse.

If you feed this set of samples to a properly-designed DAC, the DAC
will output a waveform.

Because this waveform is coming out of a properly-designed DAC with a
proper reconstruction filter, it will contain no frequency components
lying above the Nyquist limit. Hence, the output of the DAC has no
aliases to remove.

Now, feed this waveform into another sampler (an ADC).

If you lock the sampler's timing to that of the DAC accurately, so
that you sample at precisely the right moments... and if you've set
the gain correctly and have low-enough noise... then the sequence of
samples that you take will replicate the original ones which were fed
into the DAC!

Hence, you've just shown that this particular continuous waveform, and
the "square-wave-like" sequence of samples, have a 1-to-1
correspondence.

--
Dave Platt AE6EO
Friends of Jade Warrior home page: http://www.radagast.org/jade-warrior
I do _not_ wish to receive unsolicited commercial email, and I will
boycott any company which has the gall to send me such ads!

[Randy Yates]

writes:
[...]
And that assumption itself is false: Every possible sequence
of sample values IS NOT valid, because a very large number
of them, the example you give below, is one example of a
an entire class of sequences that violates the Nyquist
criteria.

I must be misreading you somewhere, Dick. It is senseless to speak of a
discrete-time signal as "violating the Nyquist criteria." The Nyquist
criteria is a test for an ANALOG (continuous-time) waveform. (By the
way, let's assume infinite resolution since I think what's relevent here
is the time quantization dimension.)

Each discrete-time waveform corresponds to some bandlimited
continuous-time waveform. At least mathematically. To produce that
continuous-time waveform, simply take the discrete-time waveform and run
it through a DAC. (I'm speaking of ideal ADCs and DACs here). If you
then took the DAC output and ran it into an ADC, you'd get your original
discrete-time sequence back.
--
% Randy Yates % "Midnight, on the water...
%% Fuquay-Varina, NC % I saw... the ocean's daughter."
%%% 919-577-9882 % 'Can't Get It Out Of My Head'
%%%% % *El Dorado*, Electric Light Orchestra
http://www.digitalsignallabs.com

[Randy Yates]

MRC01 writes:
[...]
In that sense, the mathematical question about whether there exist
"invalid" sequences of samples, may be the same as asking whether the
ADC / DAC functions or algorithms are bijections.

I think they are, but I can't come up with a proof quickly.
--
% Randy Yates % "With time with what you've learned,
%% Fuquay-Varina, NC % they'll kiss the ground you walk
%%% 919-577-9882 % upon."
%%%% % '21st Century Man', *Time*, ELO
http://www.digitalsignallabs.com

[[email protected]]

On Oct 25, 9:13 pm, (Dave Platt) wrote:
In article .com,

MRC01 wrote:
But on Oct 24, 11:38 am, Dave Platt wrote:
Every combination of samples is valid, and corresponds to a unique
(within the system's quantization limits) bandwidth-limited analog
waveform.

This is an interesting mathematical question. Do there exist some
combinations of samples that are invalid?

Nope.

Yes, there is. The most trivial example is an alternating
positiive and negative stream of constant values. It's a
waveform which is at precisely 1/2 the sample rate,
which violates the Nyquist criteria.

Show us an input function to a properly implemented
sampler that would result in such a waveform.

I'm willing to concede the point, but you'll have to explain
away cases such as the one illustrated above.

[Dave Platt]

In article .com,
wrote:

Nope.

Yes, there is. The most trivial example is an alternating
positiive and negative stream of constant values. It's a
waveform which is at precisely 1/2 the sample rate,
which violates the Nyquist criteria.

Erp. You're quite right... I'd forgotten that particular set of
boundary cases.

Shame on me :-(

Show us an input function to a properly implemented
sampler that would result in such a waveform.

Isn't one. You're right.

--
Dave Platt AE6EO
Friends of Jade Warrior home page: http://www.radagast.org/jade-warrior
I do _not_ wish to receive unsolicited commercial email, and I will
boycott any company which has the gall to send me such ads!

[MRC01]

On Oct 24, 1:13 pm, Eeyore
wrote:
MRC01 wrote:
wrote:
"0 dB" and "-1 dB" relative to what?

0 dB meaning the sample values at the peaks and troughs of the
waveform reach 32767 and -32767 respectively.

Which is FSD.

So if you add 2 such waveforms you'll clip the DAC it would seem.

Are BOTH frequencies at '0dB' level ?

No. I just double checked this waveform. A spectrum analysis shows the
19 kHz and 20 kHz components at -6 dB each. The overall waveform (sum)
samples peak at 0 dB.

Theoretically, this wave could be played back with no clipping or
distortion. But the DACs generate a good deal of distortion playing
them back - spurious IM frequencies peak at -50 to -60 dB relative to
the signal, depending on the device.

When I attenuate the waveform -1 db - the spectrum analysis shows the
components at -7 dB each - it plays with the IM frequencies at -80 to
-100 dB relative to the signal.

This IM distortion may go lower with even more attenuation, but -100
dB is low enough I'm not worried about it.

Based on this, it appears that this is a valid test waveform and it
shows that the DACs are going non-linear in the last 1 dB of
amplitude.

FWIW, one DAC is the Wolfson WM-8718, the other is a Burr Brown
PCM-1732, one is a Marantz CDR-630 (whatever DAC it uses, not sure)
and the 4th is from a portable CD player.

[MRC01]

On Oct 26, 1:45 pm, (Dave Platt) wrote:

wrote:
Yes, there is. The most trivial example is an alternating
positiive and negative stream of constant values. It's a
waveform which is at precisely 1/2 the sample rate,
which violates the Nyquist criteria.

Erp. You're quite right... I'd forgotten that particular set of
boundary cases.

Shame on me :-(

I wonder if there are other invalid sequences of samples. And what
does the DAC do if it encounters one of these invalid sequences? It
has to create *some* kind of waveform. Is it undefined? Different DACs
might generate totally different waves?

[Randy Yates]

writes:

On Oct 25, 9:13 pm, (Dave Platt) wrote:
In article .com,

MRC01 wrote:
But on Oct 24, 11:38 am, Dave Platt wrote:
Every combination of samples is valid, and corresponds to a unique
(within the system's quantization limits) bandwidth-limited analog
waveform.

This is an interesting mathematical question. Do there exist some
combinations of samples that are invalid?

Nope.

Yes, there is. The most trivial example is an alternating
positiive and negative stream of constant values. It's a
waveform which is at precisely 1/2 the sample rate,
which violates the Nyquist criteria.

I had forgotten this type of case.

The Nyquist criteria is not an "if and only if."
That is, it says that IF an input signal satisfies the
criteria, then it can be converted to digital without losing
any information. It does NOT say that if a digital signal
represents an input signal without losing information, it
satisfies the Nyquist criteria.

Show us an input function to a properly implemented
sampler that would result in such a waveform.

f(t) = sin(2*pi*(F_s / 2) * t),

assuming the sampler samples at times n*T_s, where n is integer and
T_s = 1 / F_s.
--
% Randy Yates % "Though you ride on the wheels of tomorrow,
%% Fuquay-Varina, NC % you still wander the fields of your
%%% 919-577-9882 % sorrow."
%%%% % '21st Century Man', *Time*, ELO
http://www.digitalsignallabs.com

[Randy Yates]

Randy Yates writes:

writes:

On Oct 25, 9:13 pm, (Dave Platt) wrote:
In article .com,

MRC01 wrote:
But on Oct 24, 11:38 am, Dave Platt wrote:
Every combination of samples is valid, and corresponds to a unique
(within the system's quantization limits) bandwidth-limited analog
waveform.

This is an interesting mathematical question. Do there exist some
combinations of samples that are invalid?

Nope.

Yes, there is. The most trivial example is an alternating
positiive and negative stream of constant values. It's a
waveform which is at precisely 1/2 the sample rate,
which violates the Nyquist criteria.

I had forgotten this type of case.

The Nyquist criteria is not an "if and only if."
That is, it says that IF an input signal satisfies the
criteria, then it can be converted to digital without losing
any information. It does NOT say that if a digital signal
represents an input signal without losing information, it
satisfies the Nyquist criteria.

Show us an input function to a properly implemented
sampler that would result in such a waveform.

f(t) = sin(2*pi*(F_s / 2) * t),

assuming the sampler samples at times n*T_s, where n is integer and
T_s = 1 / F_s.

Let me be quick to add that I know this signal violates the Nyquist
criteria because it is NOT Fs/2. However, the point I am attempting
to make is that all possible digital sequences produce valid analog
signals.
--
% Randy Yates % "Bird, on the wing,
%% Fuquay-Varina, NC % goes floating by
%%% 919-577-9882 % but there's a teardrop in his eye..."
%%%% % 'One Summer Dream', *Face The Music*, ELO
http://www.digitalsignallabs.com

[Chris Hornbeck]

On Fri, 26 Oct 2007 13:52:54 -0700, MRC01 wrote:

I wonder if there are other invalid sequences of samples. And what
does the DAC do if it encounters one of these invalid sequences? It
has to create *some* kind of waveform. Is it undefined? Different DACs
might generate totally different waves?

"DAC" is almost an undefined term, but in the rawest case
of a simple ladder with summed output, *any* data sequence
is valid, and an unambiguous output is generated. Aliasing,
in particular, doesn't apply here - it's just us chickens.

In modern usage the term "DAC" might be expected to be a
plastic package and to include sample rate conversion and
filtering, so your question is very deep indeed.


To the OP's observations: many other folks have reported
surprisingly large "abnormalies" in consumer-level DAC's
at peak outputs, especially in modern DVD players' audio.
Haven't heard any especially creditable explanations, but
there're lots of reasonable theories.

Thanks to all for a very interesting thread,

Chris Hornbeck

[MRC01]

On Nov 2, 7:10 pm, Chris Hornbeck
wrote:
To the OP's observations: many other folks have reported
surprisingly large "abnormalies" in consumer-level DAC's
at peak outputs, especially in modern DVD players' audio.
Haven't heard any especially creditable explanations, but
there're lots of reasonable theories.

Can you define "consumer level"? One of the devices that does this is
a Marantz CDR-630 which was sold by Marantz as "pro" gear. I bought
mine used from a recording studio in Chicago.

The DACs in most players - pro or consumer - often seem to be the same
DACs made by the same companies. They're not designing their own
chips; they're buying off the shelf from Burr Brown, Wolfson, or
whoever. Because of this I wouldn't expect to see any difference
between high end consumer versus pro gear.

[Arny Krueger]

"MRC01" wrote in message
oups.com...


The DACs in most players - pro or consumer - often seem to be the same
DACs made by the same companies.

Pretty much true today, but not always true. In the early days Sony CD
players had Sony-made converter chips, for example.

They're not designing their own chips; they're buying off the shelf from
Burr Brown, Wolfson, or
whoever.

Seems to be true. Price/performance has a lot to do with that.

Because of this I wouldn't expect to see any difference
between high end consumer versus pro gear.

The various vendor chip offerings do differ in terms of bandwidth and
dynamic range.

[Eeyore]

MRC01 wrote:

Chris Hornbeck wrote

To the OP's observations: many other folks have reported
surprisingly large "abnormalies" in consumer-level DAC's
at peak outputs, especially in modern DVD players' audio.
Haven't heard any especially creditable explanations, but
there're lots of reasonable theories.

Can you define "consumer level"? One of the devices that does this is
a Marantz CDR-630 which was sold by Marantz as "pro" gear. I bought
mine used from a recording studio in Chicago.

What exactly does it do ? Marantz are hardly a well-known 'pro' brand.


The DACs in most players - pro or consumer - often seem to be the same
DACs made by the same companies. They're not designing their own
chips; they're buying off the shelf from Burr Brown, Wolfson, or
whoever. Because of this I wouldn't expect to see any difference
between high end consumer versus pro gear.

The only audio DACs to be bought of any quality are indeed typically the ones
you mention plus AKM , Cirrus/Crystal and Anolog Devices.

It's simply not practical for many companies to make their own converters. Not
sure what Philips, Sony, Yamaha are currently doing.

Graham

[MRC01]

On Nov 5, 1:46 pm, Eeyore
wrote:
MRC01 wrote:
One of the devices that does this is
a Marantz CDR-630 ...

What exactly does it do ? Marantz are hardly a well-known 'pro' brand.

It's a rack mount CD burner. Inputs are coax & optical digital or XLR
or unbalanced RCA. It's a dated model but a solid reliable performer -
good specs and clean, natural sound. Having come from a recording
studio, mine in particular had already burned literally thousands of
CDs before I got it and I've done another thousand or so since then -
and still trucking along.

[Arny Krueger]

"Eeyore" wrote in message
...


MRC01 wrote:

Chris Hornbeck wrote

To the OP's observations: many other folks have reported
surprisingly large "abnormalies" in consumer-level DAC's
at peak outputs, especially in modern DVD players' audio.
Haven't heard any especially creditable explanations, but
there're lots of reasonable theories.

Can you define "consumer level"? One of the devices that does this is
a Marantz CDR-630 which was sold by Marantz as "pro" gear. I bought
mine used from a recording studio in Chicago.

What exactly does it do ? Marantz are hardly a well-known 'pro' brand.

In the US Marantz have largely gone underground, but they retain a
relatively large presence in pro audio. I think its the follow-on to their
old pro cassette recorder line.