In comp.dsp, Steve <email@example.com> wrote:
>To all you DSP mavens out there.
>I am looking for some references to distortions introduced by sampling =
>especially sampling close to the Nyquist criterion. In all the =
>have seen, reconstruction of a waveform is guaranteed if the sampling =
>Nyquist criterion, that is sample rate > 2F. =20
>There are lots of cases where the Nyquist criterion doesn't seem to work =
>A classic case is there you have a single frequency (frequency F) sine =
>you sample at 4F starting at a phase angle of 0 degrees, you can =
>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 =
>you can reconstruct the wave shape, but not the amplitude because the =
>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
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
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:
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,
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):
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 =
>how does one compute the second order statistics?
>I have heard musicians criticize CD recordings, complaining about =
>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 =
Hope this helps. Have a nice day.