Unanswerable Questions: differential threshold of sampled sound and light

I own a digital 8-track, and I've done a fair amount of recording with it. Now when you're acting as sound engineer of a recording session, you have vast expanses of time where your duty is simply to sit and wait for the performer to finish their piece of music, times during which I would often find my mind wandering. One question in particular would keep recurring to me, and interesting as it is, I'm fairly certain it's provably unanswerable:

How much sound would it take to alter a single bit of this recording?

Now the microphone I own is an Audio-Technica 4033a, a large-diaphragm condenser microphone, and it's far more sensitive than the Shure SM-58's I was used to before. Dynamic microphones like the SM58 have very light response past a foot or two, where a large-diaphragm condenser will clearly pick up sounds from across the room. But exactly how sensitive?

Would the sound of my breathing alter the recording (I was across the room from the mic)? Probably. But what about my heart beating? The chemical reactions that were breaking down my food? Probably not those. So where is the threshold, what is the smallest sound that would change at least one bit on the disk that was being recorded to?

There is a definite answer, because either a sound caused a bit to be changed or it didn't, but I'm convinced it's not possible to know, because there's no such thing as an ideal microphone. Even in complete silence, no microphone will record the same series of bits twice, so there's no way to know if a difference in any particular bit is a result of the microphone, electronic interference, the A/D converter, or the microscopic noise you were trying to detect.

You could apply the same idea to digital video. Would a raindrop at 10 feet away change a single bit? Probably. A spec of dust? A molecule? No? Then where is the threshold?

It'd sure be interesting to know.

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Whipster: Though the problem is not solvable (to the best of my knowledge, of course--I'm not an electrical engineer), I would argue that it is a well-defined question, and not a victim of the ambiguity that plagues the questions you pose.

You'll notice I asked "How much sound would it take to alter a single bit of this recording?" That is, in this specific, defined situation at this specific, defined moment, fork time into 2 alternate timelines: one where a specific sound occurs, and one where it doesn't. What is the smallest sound for which the resulting recordings will be different?

Or, to rephrase the questions you pose in this way:

• How many of these specific and identical angels can dance in exactly the way they are now on the head of this pin I am holding?
• If this specific butterfly were to flap it's wings right at this moment, would a tornado in Nebraska be a direct result?
• If I throw this penny right now with this velocity and trajectory, where exactly will it strike the ground?

...questions that are not so unreasonable. Well, except the angel one. :-)

There is no threshold, because before the interference, the sound level to be sampled can be arbitrarily close to the threshold, where it will change by one bit. So an arbitrarily small amount of interference can change it by one bit.

The quandary presented here arises more as a result of the way you have worded the question than as a matter of scientific principle.

There is indeed a threshold of detection, but it depends on the sensitivity of the microphone, the resolution of the analog to digital converter, and all the noise characterisitcs of every bit of circuitry in between the sound source and the ADC.

As an electrical engineer, all I can really say is that this is one of those areas where one has to make generalizations and assumptions in order to produce a meaningful answer.

As I said in my opening sentence, the question, as you have posed it, does not adequately define a solvable system. You have asked a question of this sort:

• How many angels can dance on the head of a pin?
• Can a butterfly in the Congo cause a tornado in Nebraska?
• If I throw a penny off the Empire State building, whose head will it hit?

The point being, while these are questions, their lack of solution has more to do with semantics than with any fundamental mystery or failure of analysis. In other words, this "puzzle" is a word game, not a problem of science.