Does anyone have any reference to sampling distortion?

To all you DSP mavens out there.

I am looking for some references to distortions introduced by sampling =
errors,
especially sampling close to the Nyquist criterion.  In all the =
literature I
have seen, reconstruction of a waveform is guaranteed if the sampling =
meets the
Nyquist criterion, that is sample rate > 2F. =20

There are lots of cases where the Nyquist criterion doesn't seem to work =
well.
A classic case is there you have a single frequency (frequency F) sine =
wave, and
you sample at 4F starting at a phase angle of 0 degrees, you can =
perfectly
reconstruct the wave.  However, if you begin to sample at a lag of 45 =
degrees,
you can reconstruct the wave shape, but not the amplitude because the =
sampling
will miss the peaks of the wave.

Is the sampling really a statistical phenomena, where on average we can
reconstruct the waveform, or is it more deterministic?  If it is =
statistical,
how does one compute the second order statistics?

I have heard musicians criticize CD recordings, complaining about =
distortion at
the higher frequencies.  This seems to be part of the same phenomena.

Any help in understanding (including any references) this would be =
appreciated.

Steve

In article erd7lvct2b0sluh0ocqs8166jiotd7jlkq@4ax.com, Steve at
nospam@nowhere.org wrote on 09/01/2003 17:43:

> To all you DSP mavens out there.
> 
> I am looking for some references to distortions introduced by sampling errors,
> especially sampling close to the Nyquist criterion.  In all the literature I
> have seen, reconstruction of a waveform is guaranteed if the sampling meets
> the Nyquist criterion, that is sample rate > 2F.
> 
> There are lots of cases where the Nyquist criterion doesn't seem to work well.
> A classic case is there you have a single frequency (frequency F) sine wave,
> and you sample at 4F starting at a phase angle of 0 degrees, you can perfectly
> reconstruct the wave.  However, if you begin to sample at a lag of 45 degrees,
> you can reconstruct the wave shape, but not the amplitude because the sampling
> will miss the peaks of the wave.

no, 4F is sufficient.  there will be only one sine wave below 2F that will
go through the sample points and that will have the correct amplitude.

> 
> Is the sampling really a statistical phenomena, where on average we can
> reconstruct the waveform, or is it more deterministic?

it's deterministic, but it calls for an ideal brickwall low-pass filter,
something i haven't seen in the real world yet.

> I have heard musicians criticize CD recordings, complaining about distortion
> at the higher frequencies.  This seems to be part of the same phenomena.

Ah, the anal-retentive wimps!  someone should slap them on the side of their
head and force them to listen to 78s (or even "modern" LPs).  :-)

r b-j

In comp.dsp, Steve <nospam@nowhere.org> wrote:

>To all you DSP mavens out there.
>
>I am looking for some references to distortions introduced by sampling =
>errors,
>especially sampling close to the Nyquist criterion.  In all the =
>literature I
>have seen, reconstruction of a waveform is guaranteed if the sampling =
>meets the
>Nyquist criterion, that is sample rate > 2F. =20
>
>There are lots of cases where the Nyquist criterion doesn't seem to work =
>well.
>A classic case is there you have a single frequency (frequency F) sine =
>wave, and
>you sample at 4F starting at a phase angle of 0 degrees, you can =
>perfectly
>reconstruct the wave.  

   Assuming an amplitute range of -1 to 1, and the sine is just at
full scale, the samples should look like 0, +1, 0, and -1.

>However, if you begin to sample at a lag of 45 =
>degrees,
>you can reconstruct the wave shape, but not the amplitude because the =
>sampling
>will miss the peaks of the wave.

   In this case, the samples should be:
sin(45 degrees), sin(135 degrees), sin (225 degrees), sin (315
degrees).
Or:
sqrt (2), sqrt (2), -sqrt (2), -sqrt (2).
Or numerically to five significant figures:
0.70711, 0.70711, -0.70711, -0.70711.

   I don't know this for a 'theoretical fact,' but I think the
reconstruction filter (a brickwall [or practical near-brickwall]
low-pass filter at 1/2 the sample rate) will output a sine wave with
peaks of +1 and -1 for either of these two inputs. The filter
generates a sine wave (less than the Nyquist frequency) that passes
through all the points. There is only one such sine wave in either
case, and both sine waves have the same frequency and amplitude.

   I just proved to my own satisfaction that this is true. I generated
an 8-bit mono 6ksps [the lowest sample rate for standard PC
soundcards] .wav file in Cool Edit 2000. It starts with a 1500Hz
(giving four samples per cycle) sine wave (at -6dBFS), zero degrees
phase shift, for two seconds, followed by the same with a 45 degree
phase shift for two seconds. When played back there is a 'click'
transition between the two sections caused by the sudden phase shift,
but the loudness of both sections is identical to my ears. The level
indicator along the bottom shows -6dBFS for the first section, and
-9dBFS for the second section, which is actually correct because the
indicator does peak detection.
   Cool Edit 2000 has a statistics section that I used to read power
levels. Here are the stats for the first two seconds (these might not
be exact, as I just eyeballed the selection, I didn't try to be
sample-accurate):

	Mono
Min Sample Value:	-16640
Max Sample Value:	16384
Peak Amplitude:	-5.82 dB
Possibly Clipped:	0
DC Offset:	-.391 
Minimum RMS Power:	-9 dB
Maximum RMS Power:	-8.97 dB
Average RMS Power:	-8.99 dB
Total RMS Power:	-8.97 dB

Using RMS Window of 50 ms

Here are the stats for the last two seconds:

	Mono
Min Sample Value:	-11776
Max Sample Value:	11520
Peak Amplitude:	-8.79 dB
Possibly Clipped:	0
DC Offset:	-.394 
Minimum RMS Power:	-9.01 dB
Maximum RMS Power:	-9.01 dB
Average RMS Power:	-9.01 dB
Total RMS Power:	-8.99 dB

Using RMS Window of 50 ms


   This looks quite good, the min sample value and max sample value of
the second selection are very close to those of the first selection
times sin (45 degrees) or 1/sqrt(2), and the RMS power numbers of thw
two sections are almost identical.
   The proof to me is in 1) hearing that both sections play a 1.5kHz
tone at the same loudness, and 2) zooming in to the transition area to
see the data points for the first half are zero, +peak, zero, -peak
and the second half are 7/10ths peak, 7/10ths peak, -7/10ths peak,
-7/10ths peak.
   You could also do an FFT of the data, and see that both sections
give the same level of the same frequency.
   To further prove it, I generated a third section of two seconds of
a sine wave, this time starting at 45 degrees (the same phase angle as
the second section) but with a frequency of 1501 Hz instead of 1500
Hz. The file plays seamlessly from the second section to the third
section (at four seconds into the file), and I cannot hear any
variation in volume of the sound, as one might expect from looking at
the waveform: the data points vary from the .7, .7, -.7, -.7 scenario
to the 0, 1, 0, -1 scenario and back, four times per second. Zoomed to
display all six seconds (which shows peak values) you can see eight
'humps' on the right third of the waveform. I cannot hear any volume
variation nor the increase in frequency from 1500Hz to 1501Hz, but I
CAN hear some 'flying saucer' sounds in the background of the last two
seconds, apparently from the cheap, standard 'Soundblaster-compatible'
soundcard in this computer. I think it's 'birdies' (unwanted
modulation products, as sometimes heard in radio receivers) in the
converter circuitry. (It helps to close your eyes or otherwise not be
able to see the app's level indicator, to be sure you're not just
imagining you hear the loudness going up and down with the indicator.)

   I've saved the file here (36k bytes):

http://www.mindspring.com/~benbradley/1500.wav

   Play it in any .wav file player, look at it in any .wav editor.

>Is the sampling really a statistical phenomena, where on average we can
>reconstruct the waveform, or is it more deterministic?  If it is =
>statistical,
>how does one compute the second order statistics?
>
>I have heard musicians criticize CD recordings, complaining about =
>distortion at
>the higher frequencies.  This seems to be part of the same phenomena.

   I would have thought so too, but everything I've seen, read and
heard on this topic has led me to believe that any distortion is due
to practical real-world phenomena such as imperfect A/D/A converters,
especially imperfect anti-aliasing filters on the A/D side and
imperfect reconstruction filters on the D/A side, and not to any
theoretical distortion of frequencies near the Nyquist frequency.
(Jitter also plays a role as a source of distortion).
   Twenty or so years ago, the filters used to be physically separate
analog blocks (in the age of successive-approximation A/D's and R-2R
or equivalent-current-switched D/A's), but now they are highly
integrated into sigma-delta converters, and are now (digital) FIR
filters instead of analog.

>Any help in understanding (including any references) this would be =
>appreciated.

   Hope this helps. Have a nice day.

>Steve

Steve wrote:
> 
> To all you DSP mavens out there.
> 
> I am looking for some references to distortions introduced by sampling errors,
> especially sampling close to the Nyquist criterion.  In all the literature I
> have seen, reconstruction of a waveform is guaranteed if the sampling meets the
> Nyquist criterion, that is sample rate > 2F.
> 
> There are lots of cases where the Nyquist criterion doesn't seem to work well.
> A classic case is there you have a single frequency (frequency F) sine wave, and
> you sample at 4F starting at a phase angle of 0 degrees, you can perfectly
> reconstruct the wave.  However, if you begin to sample at a lag of 45 degrees,
> you can reconstruct the wave shape, but not the amplitude because the sampling
> will miss the peaks of the wave.
> 
> Is the sampling really a statistical phenomena, where on average we can
> reconstruct the waveform, or is it more deterministic?  If it is statistical,
> how does one compute the second order statistics?
> 
> I have heard musicians criticize CD recordings, complaining about distortion at
> the higher frequencies.  This seems to be part of the same phenomena.
> 
> Any help in understanding (including any references) this would be appreciated.
> 
> Steve

The two other responses I see do the job, but I'll add this:

Sample at 4F - .1, so that all phases are run through in a ten-second
interval. There will be no amplitude modulation; sampling wouldn't work
if there were. To sum all that up, realize that the waveform isn't
simply what you might imagine from a blurry picture of the dots, but
that single function which passes through all the points while
containing no frequencies as high as half the sampling frequency. For
any given set of samples, it is unique.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
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Jerry Avins <jya@ieee.org> wrote in message news:<3F54BB61.5EDF952E@ieee.org>...
> Sample at 4F - .1, so that all phases are run through in a ten-second
> interval. There will be no amplitude modulation; sampling wouldn't work
> if there were. To sum all that up, realize that the waveform isn't
> simply what you might imagine from a blurry picture of the dots, but
> that single function which passes through all the points while
> containing no frequencies as high as half the sampling frequency. For
> any given set of samples, it is unique.

Almost all confusion about the accuracy of sampling stems back to
people forgetting those magic words "while containing no frequencies
as high as half the sampling frequency". Any time you get confused
just think "am I properly band limited", and the confusion should go
away!

It can be fun to draw the sampled dots on a sheet of paper, and ask a
confused person to join them with all possible curves that are band
limited. If they draw more than one, you just tell them to iterate
until they correct their mistake :-) The really confused people, who
have some notion of square or triangular waves comming out of a
sampled system, can be real fun with this.

By the way, does anyone know a good web site to point people to, that
gets the sampling idea across well to the uninitiated?