Listening to an aperiodic inverse sound wave During the operation of noise canceling headphones, an inverse signal to the ambient noise is generated and is played through the speaker where every compression of the ambient noise is a rarefaction of the headphone signal and vis versa. If the ambient noise is an aperiodic signal (i.e. following the subsequent mathematical description) (let this signal's amplitude as a function of time be called $f(t)$), the headphone signal (call this signal $g(t)$ cannot be phase-shifted or time shifted in such a way that $f(t)=g(t)$ for all t. My question is: what would the headphone signal $g(t)$ sound like if it were isolated and played through a speaker independent of the ambient signal $f(t)$? 
 A: An inverted sound signal would sound no different because that is simply a shift of the time origin. Our brains don't distinguish time shifts between two separately played signals. For example, a square wave which starts at zero and first compresses, then rarifies and repeats, even at changing frequencies, will sound exactly like one which starts at zero, rarifies, then compresses.
This is easily demonstrated with a sound mixing board which allows one to phase invert an input signal. The only thing one might notice is a "click" when the phase button is pushed.
A: I suspect, based on experiments, that for any function $f(t)$, $-f(t)$ sounds the same in all but some very odd circumstances: in other words, it would sound the same.
I got interested in this because I'm interested in sound, and I am also deeply cynical about the golden-eared magic hifi stuff.  So I did some experiments, which I'll describe briefly below.
What I wanted to test was how sensitive, if at all, my ear was to the phase of sound, because golden-eared hifi people will claim it is but I don't believe them.  One reason that I don't believe them is that phase -- including the relative phase of different frequencies -- is aggressively dependent on your exact distance from the source of the sound (because the wavelengths are different), yet I don't observe any significant difference when I move my head by a few inches, but, say, at $2\,\mathrm{kHz}$, wavelength is about $17\,\mathrm{cm}$ so small movements will change relative phases hugely.
So here are the experiments I did.
The obvious test: wire speakers out of phase.  This is a famous mistake, and it is extremely obvious when you do it: if you wire two loudspeakers out of phase all the bass (low-frequency) goes away.  That's obviously because these have long wavelengths ($100\,\mathrm{Hz}$ is of the order of the size of many rooms), and these just cancel.  And you get the exact results you expect like this.  For both stereo and mono recordings you lose all the bass (it's easy to verify that low frequencies are strongly correlated between the two channels of a stereo recording).  For mono white-noise things sound a lot more hissy than they did, because the low frequencies in the noise are getting cancelled.  I did not try two uncorrelated sources of white noise: I should have, and I would expect them not to cancel in the same way (because uncorrelated).
Wire speakers in phase, but 'backwards'.  This means that, whatever $f(t)$ you were listening to before, you are now listening to $-f(t)$.  So I listened to a lot of music, tones, and various noise sources like this, and almost always this makes no difference.  But, interestingly, just occasionally it does seem to.  It never did for tones or noise, but for some musical sources it did, especiall when listening rather louder than I normally would.  It took me a long time to understand why, and I only have a theory backed by some slightly apocryphal evidence.
The theory is this: some sorts of music have big transients such as, for instance, bass drums.  These things are not anywhere near symmetric: think about what goes on when you hit a bass drum, for instance.  Well, speakers are pretty crap (even good speakers: essentially all the distortion in hifi systems is in the speakers, especially in the CD-and-later era) and become much crapper for large displacements as they fall off their linear range.  By hypothesis they are differently crap near the positive or negative limits of their displacement.  So, theory: big transient sounds like drums induce different distortion from the speakers when driven in different directions, and this is what you hear.
Well, the apocryphal evidence for this is that I never heard a difference for orchestral music, only for music with significant percussive content, and I only noticed it when it was played unpleasantly loud.
I like my theory, but it's clearly just a theory.  I can think of objective tests of it but they would all require much more equipment than I can muster.  A test would be to set up a very carefully positioned microphone, then play the same sound with the speaker wired each way, and see to what extent the waveforms were in fact the negative of each other, especially for high-volume clicks and bangs.  I don't have the equipment to do this, and nor am I willing to risk damage to my fairly expensive speakeers by doing these experiments (high-volume clicks and bangs are a good way to eat speakers).
However, understand that the effect, if there is one, is very small: I could, perhaps, sometimes, tell the difference, but I am sure I could not do so if I had not just previously listened to the same ten-second chunk of music many times over 'the other way up', and in many cases I could not tell.  I am absolutely sure I could not tell if you played it one way one day and the other way another day.  My experimental conditions were also awful: I had to grovel around behind the speakers to change the polarity. And finally and most importantly I knew the polarity had been changed which largely invalidates any result due to bias on my part.
There is a possible other phenomenon which is related to this: there is no reason to believe that your ears have identical responses to big transients with different polarities, either.  So really loud music with really loud percussive transients might actually sound different because of distortion coming from your ears.  An experiment to test this theory can't be ethically performed unfortunately.
So my conclusions from this were that I believe strongly that polarity does not make any difference to what you hear, but that to do the experiment properly would be beyond my ability (or interest).  If it does make a difference I think it is accounted for by distortions around transients as described above.

Notes: other than the caveats above, I'll repeat that my experimental conditions were awful, that I only tested my ears, that I have biasses.  I also didn't do any single-speaker tests, because I was interested in hifi aspects of it.
