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Search around on Google for deconvolution and 'blind deconvolution.' This is one of the holy grails of communications technology, since it would allow much cleaner signal to pass through less robust noisy networks.
Theoretically it should be possible, but as far as I know, no one has done it with the amount of success you're looking for.
Some things, like a distortion pedal, are pretty lossy from an information perspective. The waveforms are being clipped, flat-topped, so any 'data' that existed in that part of the waveform would be lost. Some high-frequency stuff, for instance.
If you know what the original signal 'probably' looked like, you could deconvolve the distorted signal and get something close to the original undistorted signal. That is, if you know that it was a sine wave, for instance, you can easily get the original waveform from one that's been heavily clipped. As the waveform gets more complex, it gets harder.
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