Wide bandwidth = necessary?

I agree with Viridian. The main benefit of higher sample rates, which figures to be a very significant one, is that it allows for a gentler rolloff of the "anti-aliasing filter" that precedes the a/d converter in the recording chain.

The theoretical maximum frequency that a sampled data system can capture is 1/2 of the sample rate, or 44.1 x 1/2 = 22.05 kHz in the case of redbook cd. Anything above that frequency must be prevented by the filter from entering the a/d converter, or it would "fold down" to a much lower (and very audible) spurious frequency when reproduced.

So the anti-aliasing filter has to pass everything up to 20kHz, with flat frequency response, but attenuate everything at 22.05kHz and above to (hopefully) below what is the threshold of audibility at lower frequencies.

A filter with such an extremely sharp rolloff will inevitably introduce both phase distortions and frequency response ripple, within the 20 to 20kHz passband. 96 or 192kHz sampling would drastically reduce the sharpness of the rolloff, and minimize those effects correspondingly.

Regards,
-- Al

Your post makes sense *if* you are about to do an A-->D conversion

That's exactly what I was talking about. Please note that I said:

The main benefit of higher sample rates, which figures to be a very significant one, is that it allows for a gentler rolloff of the "anti-aliasing filter" that precedes the A/D converter in the RECORDING CHAIN.

I wasn't talking about DAC's or playback equipment at all.

-- Al

For those who haven't looked at it, I think that Stefanl's link is an excellent and very in-point medical/scientific paper. It documents a study in which "non-stationary" ultra-sonic sounds increased pleasurable brain activity in the test subjects, but only when sounds within the normal audio spectrum were simultaneously present. "We conclude, therefore, that inaudible high-frequency sounds with a nonstationary structure may cause non-negligible effects on the human brain when coexisting with audible low-frequency sounds."

So it would seem like ultra-sonic frequencies can be "audible," but only as a result of some intermodulation process with lower frequencies, or else, as the paper put it, "participation of nonauditory sensory systems such as somatosensory perception also needs to be considered in further investigations."

Still I can't understand the need for such a response (to 40kHz) while during recording the bandwidth is limited.

I'm not particularly familiar with the roll-off characteristics of professional recording microphones, but I would expect that although their bandwidth may only be specified to 20kHz in many cases, the roll-off would be gentle enough to capture significant content well above that frequency, assuming it is present.

So, if you are want to transport a 16-b word of music @ a 44.1KHz rate - this is what a CD laser mechanism does: every 1/44.1KHz seconds it spits out a 16-b word read off the physical spinning CD - you would have to transport each bit in 1/(16-b * 44.1KHz = 705.6KHz) seconds (so that you are ready to transport the next 16-b word that will arrive 1/44.1KHz seconds later.
So, your USB cable needs to have 705.6KHz bandwidth (& your DAC needs to run at 2*705.6KHz, as per Nyquist's criteria).

Keep in mind that two channels are present. That is the reason for the factor of 2 (which actually will be a little greater than 2 because additional non-data bits are needed to support the communication protocol).

Also, I'm not sure it's clear to everyone that the Nyquist criteria (sampled data systems being able to handle frequencies no higher than 1/2 the sample rate) does not relate to filtering at the output of a dac. Ultra-sonic spectral components that are present at the output of the dac chip itself, due to the 44.1kHz sample rate, will not alias, or "fold-down" to lower frequencies. A brick-wall anti-aliasing filter, which as I mentioned is needed at the input to an a/d, is not needed at the output of a d/a, even a non-oversampling d/a running at 44.1kHz. The spectral components associated with the sampling will be at frequencies of 44.1kHz and its harmonics (88.2, etc.), or at much higher frequencies if oversampling is used. They needn't necessarily even be filtered at all (consistent with Paulfolbrecht's comment), or if they are filtered the roll-off can be gentle.

Re Paul's comment about the desirability of no filtering, btw, I would expect that to be dependent on the specific components (preamp, power amp, speakers) that are being driven by the signal (particularly their bandwidths and their sensitivities to intermodulation distortion at high frequencies).

Regards,
-- Al

humans also have a type of wavefront detection mechanism that is independent of our freqency perception. The pre ripple trips this wavefront detector and is perceived as diminishing the detail in the music.

I have long believed that our sense of hearing includes a "waveform steepness" factor quite independent of frequency response.

Those are excellent points (and excellent posts), and I suspect that what is behind this is the Haas Effect, which causes our hearing mechanisms to "latch on" to the leading edge of closely spaced sound arrivals. We evolved that capability to aid localization of the source of sounds that may arrive at our ears via both a direct path and (slightly later) via reflections. See the following:

http://en.wikipedia.org/wiki/Haas_Effect

Also, a few comments re the Nyquist rate and the Sampling Theorem and why sampling at twice the highest frequency that may be present is valid in theory but not in practice.

Counter-intuitive though it may seem (as Viridian indicated, sampling at the minimum Nyquist rate provides only two samples per period of the highest input signal frequency), that sample rate (of twice the highest possible signal frequency) maintains 100% of the information in the original waveform, regardless of the complexity of the waveform (sinusoidal or not), provided that two things are true:

1)No out-of-band spectral components are present (which would alias down to lower frequencies following the sampling process).

2)The sample record is of infinite length. I believe that follows from the fact that any arbitrary waveform (in the time domain) is mathematically equivalent to a summation of sine waves at various frequencies and amplitudes, but determination of that equivalency requires that an entire sample history covering all time is available. The equation defining the Fourier Transform, which mathematically converts between the time domain and the frequency domain, involves an integration from -infinity to +infinity. The relevant distinction between a pure sine wave and a complex musical waveform, which ElDartford referred to, is that for the pure sine wave we essentially know its entire past and future history -- it's always the same.

That's in theory. In practice, we need an anti-aliasing filter in front of the a/d, to filter out-of-band frequencies, and the steeper the filter slope the worse its side-effects will be, everything else being equal. And of course, item 2 can only be satisfied in the real world to some approximation. The idea seems to be that the sample record need only be "long," relative to the changes that occur in the content.

All things considered, it's remarkable that redbook cd (sampling at just about 10% above the Nyquist rate) works as well as it does.

Regards,
-- Al