On 13/04/2012, George Brock-Nannestad wrote:

> The so-called Nyquist criterion states that if you want to be able to
> reconstruct a signal from a sampled representation of it, you have to
> decide the highest frequency you want represented and then you have to
> sample at a rate that is at least twice the highest frequency. 

Note, _at least_. It won't do any harm to go higher.

> When
> you then reconstruct your signal by providing voltages at the sample
> rate according to the table of values that represents the audio
> signal, you will have your full bandwidth and dynamic range back,
> provided the resolution or bit-depth has been sufficient. It has been
> claimed that this stepwise presented waveform makes the signal
> unlistenable, but that is not the case, because you invariably smooth
> it by filtering, so nothing above your defined highest frequency gets
> out in the analog domain again. You will note that you are in full
> control: define the maximum frequency and define the resolution. If
> they are not sufficient to your purpose, go higher.

> Now, we have a problem with pure sampling of a waveform: if it has a
> frequency that is more than half of the sampling rate, that too will
> be sampled, but in this case under-sampled, which means that the
> result appears to be at a completely different and inharmonic and
> jarring frequency, an alias. Once we have adopted a sampling frequency
> we must simply ensure that no signal above half the sampling rate is
> available for sampling.
It is interesting to see the digital photography community wrestling
with these concepts. (Most cameras use optical anti-aliasing filters.)

> This is done by filtering, so-called anti-alias filtering. With 44.1
> kHz sampling rate, no signal above 22.05 kHz is permissible. On the
> other hand, we do want our 20 kHz bandwidth - this is what a young,
> pre-earbud ear can mostly hear. 

There seems to be good evidence that we can hear timing differences on
impulse signals corresponding to much higher frequencies than the
continuous sine waves normally used for testing hearing.

> So, our filter has to go from full
> transmission at 20kHz to zero transmission at 22.05 kHz *). No
> problem, our telephone engineers have done this kind of exercise for
> 90 years. However, such sharp cut-off filters come with some frightful
> time delay distortion (phase to some), very audible down to 2 kHz.
> That was the situation for about 10 years in CD audio, until someone
> came up with the idea to correct the time response in the digital
> domain by means of a digital filter. The currency to pay for this is
> total delay time, but for something recorded a year ago, some
> microseconds do not matter.
> In other words, many of our problems come from the filter. If we
> increase the sampling rate we can make use of our more frequent
> samples in two ways: we can do a gentler filtering that does not have
> the delay effect at low frequencies, or we can increase our highest
> frequency. If we do the latter, we shall be able to increase the time
> resolution of our digital representation. With 20 kHz the maximum
> slope of the waveform is only one quarter of that of an 80 kHz
> bandwidth - reachable by a 192 kHz sampling frequency and a gentler
> anti-aliasing filter.

> A couple of other items came up while we were at 78rpm reproduction:
> - the reason why we need an elevated bandwidth for recordings on
> rough surfaces is because that is where the noise signal is. 

This is the crucial point.

> - one might consider that the pickup would be encountering a
> formidable vertical wall when it met a square wave. However, the
> recording is usually a velocity recording, that is the square
> waveshape is differentiated (1st derivative), which means it turns
> into a triangular wave. The problem is that with a high bandwidth, the
> corners of this triangular wave (where the square shifts from constant
> positive to constant negative and vice versa) are very sharp, and even
> that may be difficult to trace. George Alexandrovich made a 7" test
> record with a "square wave" - there are virtually no wiggles at the
> corners of the triangles. It is for testing pickups, but I have not
> dared to use it on anything but the ELP.
Back in the days when The Gramophone carried technical reviews of
equipment, under John Borwick especially, oscilloscope traces of pickup
cartridges playing "square waves" were shown in all reviews. I assume
the actual waveforms on the LP were triangular.

> - for the same reason a tick, modelled by a steep rise followed by
> steep fall becomes two ticks, one positive and one negative.
> - even if you remove these two peaks and interpolate or draw a
> waveform connecting the ends, the low frequency excitation of the
> whole cartridge-tonearm system (stylus, cantilever, bearing, cartridge
> mass, tonearm mass and resonances) still remains as a low-level thud -
> a tail. CEDAR started back in 1988 with a program developed by Peter
> Rayner while still at the British Library to remove not only the
> ticks, but also the tail.

Don Cox
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