Once the signal is stored in the computer as a series of bits (now with reduced fidelity because of the limited word length per sample), this stored signal can be used to completely and faithfully reproduce the original analog signal, according to the sampling theorem (and in reality), taking into account again that the magnitude of the output signal may not be the exact value of the original signal (to repeat, because of the limit of the finite word-length representation). At this point, except for the "rounding error" (which is what I assume Ranjeetrain means in his post?) this signal contains ALL the information in the original analog signal. There is NO loss in this representation. I am repeating the same statement to drive home the point, so apologies to those who've already grasped it :|
Hi Ajinkya, first of all thank you for taking your time to write such a detailed post, it was very informative and a pleasure to read it. However, I disagree with the point you made here which I have highlighted.
There IS a very small amount of loss during signal approximation. It is not debatable. The only thing debatable is - HOW SIGNIFICANT is that "loss". Is that loss audible by ordinary humans?
Let me try to clarify my point. Take a graph paper (the one kids use in school to draw plots) and draw a sine wave, with the constraint that you must follow the lines on the graph paper (This constraint is the real world constrain digital imposes on us. And this constraint is what higher resolution media tries to minimize). You will see that the wave form doesn't look natural (free flowing). It has edges. How prominent these edges are will depend on how densely printed the graph paper is. If the paper has squares of a centimeter size, you will struggle very hard to create a proper wave form. When square sizes become small, the edges start becoming more and more round. As the squares approach a millimeter size you will almost see a perfect wave form (looked from a few feet away).
That's precisely what happens when an audio signal is sampled digitally (or video signal for that matter). Larger the sampling depth, less the edges, smoother the sound. To make is easily understandable I will take a visual example. On a B&W monitor you can see the bands easily if the monitor is capable of displaying only 16 shades. Situation will get better if it can display 64 shades, but our eyes can still see the banding. Unless it displays more than 256 shades of greys (limit of human eyes) we can see banding. With respect to color monitors the situation is even more serious. Human eyes can easily recongnize hundreds of thousand of colors (when seen in isolation) and even more when combined with varying levels of hues and saturations.
The same holds true for Audio. Human ear can easily hear signals as low as 5-6 dBs up to 120+ dB. I am mentioning this because some people claim on internet that human hearing range is only 40-60 dBs. I did a test on myself and I will provide details of that test if FMs like to know. But I was easily able to hear audio samples in the entire range of my amplifier used for test (about 100 dBs).
Why a longer word length (or the sampling size) is important is that - higher sampling size allows for capturing the analog signal with greater precision. As I concluded from my test, human ear is capable of about 140 dB range which is quite a lot. With the increased word length it doesn't only become possible to sample the analog signal with greater precision, but also stuff more dynamic range into the digital recording.
Finally, for the non believers in the word-length doesn't matter, please PM me your email, I will send you samples of same recording at various sampling depths, you will instantly hear the difference.
To conclude, digital does have approximation round-offs. With increased word length and increased sampling frequency these round-offs are attempted to be made as close to NIL as possible.
Only of you Ajinkya, since you are mathematically gifted: think of it in terms of Calculous. As the sampling frequency and sampling depth increases, approximation round-offs approach zero (and consequently the resultant signal is as close to the original signal as possible).
Disclaimer: I have typed all this stuff in a huff. Please ignore typos and point any factual errors politely. I didn't write it in manner it be read by editors at the Royal Society