Zp 24/96

By the way, a link from someone that isn't selling DACs:
http://documentation.apple.com/en/soundtrackpro/usermanual/index.html#chapter=B%26section=2%26tasks=true
And a quote from there:
The sample rate is the number of times an analog signal is measured—or sampled—per second. You can also think of the sample rate as the number of electronic snapshots made of the sound wave per second. Higher sample rates result in higher sound quality because the analog waveform is more closely approximated by the discrete samples.
Unquote...
Again, no mention of the Nyquist–Shannon sampling theorem and the limits of human hearing. It's the 'more is better' mantra.

Kumar, find a way to take a look at that video I linked. The practical demonstrations, especially of a 20kHz sinewave being perfectly reconstructed (give or take a bit of inherent low level noise in the analog test equipment), should reassure you.

Kumar,
A lot of the confusion and misrepresentation arises because of the oscilloscope model of the waveforms that is presented. It has it's uses but it actually misrepresents audio because it only loosly approximates to what we hear.

If you spend to much time looking at, and trying to analyse, oscilloscope waveforms then you go off on the wrong track.

A more appropriate view of the waveform would be the spectrogram because our ears and brains tend to resolve frequencies, not instantaneous sound level.

By the way, sound waves in air exist purely as areas of air compression and decompression (rarefaction). It's all relatively straightforward and there isn't the same duality as their is with light: there is no "sound particle".

A lot of the maths involved in audio (Nyquist, etc.) involves transforming various waveforms (such as oscilloscope-style time/amplitude waveforms) into the frequency domain so we can understand how it sounds.

If you take, as an example, a classic square wave:

http://mathworld.wolfram.com/images/eps-gif/SquareWave_700.gif

If you look at the oscilloscope display it doesn't look as if there is much audio information in there at all. Intuition suggests that there wouldn't be much to hear.

Intuition is misleading.

The reality is that square waves produce a huge (theoretically infinte) range of harmonics creating a very full "fat" sound.
http://ptgmedia.pearsoncmg.com/images/chap2_9780132349796/elementLinks/2-6.jpg

Rather than looking at oscilloscope-style displays, start thinking of it in terms of sine-waves. Sine waves are the building block of audio. Every sound can, ultimately, be considered to be constructed from combinations of sine waves of different frequencies and amplitudes.

When you see the jagged stepped sine waveform so often shown when talking about digital audio, your intuition tells you there is something missing, but that is incorrect. Your intuition is misleading you.

In actual fact the opposite is true. Our hearing lives in a world where everything has to be created from (or resolve to) a pure, complete sine-wave these steps simply cannot exist.There simply cannot be bits of the sine wave missing, because the smallest possible representation of sound is a complete sine-wave.

In fact, the stepped sine-wave diagram actually corresponds to a a sine wave with bits added in the form of higher-frequency harmonics.

But it's important to realise that the steps don't exist in the first place. The stepped waveform diagram is a misrepresentation of how sampling actually works. It's not an Engineering diagram, it's a marketing one used, primarily, to deceive people. It is a lie.

Because the liars, crooks, and con-men in the industry know how to manipulate our intuition in order to make us think that their $7k DAC would be a good thing.

If you want to investigate this further, I really recommend Monty's videos:

Digital Media Primer
Digital Show and Tell

Cheers,

Keith

Again, no mention of the Nyquist–Shannon sampling theorem and the limits of human hearing. It's the 'more is better' mantra.

Kumar, find a way to take a look at that video I linked. The practical demonstrations, especially of a 20kHz sinewave being perfectly reconstructed (give or take a bit of inherent low level noise in the analog test equipment), should reassure you.
There are plenty of places that subscribe to the more is better thing - places that aren't selling anything.
I will look at the videos too. More for understanding, less for reassurance. I was and remain convinced about the audible sound quality from my digital kit - regardless of the engineering reasons for that.

By the way, sound waves in air exist purely as areas of air compression and decompression (rarefaction).

Understanding just the quoted part, described also as vibrating air, was key for me to get my head around this. The sine wave representation of the sound wave actually misled me into thinking there needs to be information data points about every point on the sine wave line. Whereas this approach led to the realisation that all the information that exists is the amplitude of the vibrations and their frequencies. Sampling 40000 times captures all the information content about the amplitudes of all the vibrations up to 20000 times a second, i.e. 20khz. It is a 100% sample till there.

Because the liars, crooks, and con-men in the industry know how to manipulate our intuition in order to make us think that their $7k DAC would be a good thing.

$7k DAC? That is the bottom end model. I went to the products page only after reading about the $ 70K DAC here.
Who buys this stuff? And it doesn't even seem to have the fancy/heavy/milled out of one ingot - box that is the usual clothing for such gear.

http://en.wikipedia.org/wiki/Greater_fool_theory
Right, but one important difference.
Quote from above:
A price can be justified by a rational buyer under the belief that another party is willing to pay an even higher price.[2][3] Or one may rationally have the expectation that the item can be resold to a "greater fool" later.[4]
Unquote
Still a rational reason for buying.
But no one buys silly priced audio gear for this reason!

Sampling 40000 times captures all the information content about the amplitudes of all the vibrations up to 20000 times a second, i.e. 20khz. It is a 100% sample till there.

By the way, sampling at 40kHz does actually capture frequency information above 20kHz as well. The problem is those frequencies aren't usable because of a form of distortion called "aliasing" that naturally occurs (and cannot be prevented).

Aliasing is complex and it's not very intuitive, but you basically get a mirrored version of the original spectrum extending downwards from the sample frequency. When you hit the halfway point, (e.g. 20kHz at a 40kHz sample rate) they start to overlap.

http://www.nathaneadam.com/OLD_SITE/4200/Images/4200/alias_spectrum.jpg

By the way, the site I grabbed this image from looks to have a useful discussion of how sampling works:

http://www.nathaneadam.com/OLD_SITE/4200/Study_Notes/3-3_Conversion_Sampling.html

Note how there is no stepped waveform. The sampling is at discrete points. Even then, be careful how much you analyse this sort of representation.

Cheers,

Keith

In the version I heard, there were three parties: the scientist, the engineer and the salesman. They were told that every N seconds they could halve the distance to the girl.

As noted, the scientist gave up in disgust and the engineer was soon close enough to engage with the girl.

The salesman spent all his time trying to negotiate for a greater reduction of the distance on each cycle, so never even started.
Again the perspective of the Science student.

The tale as told by the Humanities student was probably from the girl's perspective, and related how she dismissed both of the boffins, as the Salesman knew how to listen, had a bigger pay cheque, and better personal hygene. 😃

By the way, sampling at 40kHz does actually capture frequency information above 20kHz as well. The problem is those frequencies aren't usable because of a form of distortion called "aliasing" that naturally occurs (and cannot be prevented).

Aliasing is complex and it's not very intuitive, but you basically get a mirrored version of the original spectrum extending downwards from the sample frequency. When you hit the halfway point, (e.g. 20kHz at a 40kHz sample rate) they start to overlap.

http://www.nathaneadam.com/OLD_SITE/4200/Images/4200/alias_spectrum.jpg

By the way, the site I grabbed this image from looks to have a useful discussion of how sampling works:

http://www.nathaneadam.com/OLD_SITE/4200/Study_Notes/3-3_Conversion_Sampling.html

Note how there is no stepped waveform. The sampling is at discrete points. Even then, be careful how much you analyse this sort of representation.

Cheers,

Keith
Interesting, thanks. It also covers how the aliasing you refer to is handled by a filter.

Kumar,

While there may be some secondary, practical considerations for picking a higher sampling rate, once the underlying theory is understood, some simple math is all that is required to pick the minimum sample rate. After decades of human studies, there was no compelling evidence demonstrating that system response beyond 20KHz is required.

The number of bits is similar to the number of decimal places and this requirement depends on the application. For example, it might make someone feel warm and fuzzy to know that the inch marks on that child's ruler in the book bag are 1.000000 inches apart, but this sort of accuracy is not required for any child application and preserving this level of accuracy would require us to track the temperature of the ruler and object being measured and worry about ruler damage as it wrestles with other items in the bag.

With respect to the number of bits required for our audio application, we need to consider the characteristics of the human's auditory perception system in addition to any practical hardware points. In terms of cost and convenience for computer storage and manipulation of data, it would be convenient if we limited the bit depth to 16 bits. Earlier, when the CD format was laid out, 18 or 20 bit samples would have increased the cost significantly and limited the playing time to the point where format success in the mass marketplace was unlikely. We would have needed to wait for a Blu-ray like technology to develop.

But, is 16 bits enough? We needed well managed trials with human subjects to make this determination. After these trials it was clear that 16 bits was enough. Unfortunately, due to implementation issues caused by inexperienced analog audio designers taking their first steps in the digital domain or digital engineers taking their first stab at analog design, there were early issues with CD audio. Some of those early CD players were rather nasty sounding. And, one could measure some really ugly things that were never discussed by reviewers, regardless of their audiophile pedigree.

The audiophiles claimed that these difficulties were caused by "digital". When, in fact, the difficulties were caused by misunderstanding of the math and unfortunate circuit board layout. At this point any recent graduate should understand the math and, while circuit board layout still has an element of "art", if one follows well established rules, the results are generally satisfactory and one can prove their results after a few careful measurements. There are plenty of consultants who can fix a board layout if necessary.

In my opinion, if the audiophiles want to make a valid claim that 16 bits is inadequate, they need to demonstrate this with some well controlled studies. To date, these studies have not surfaced. If the difference between 16 bits and something higher is so obvious, this should be very easy to demonstrate.

Unlike some of the audiophiles, science welcomes a challenge. This is the heart of science. Witness the current discussions about "the big bang".