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Is it defined by the source or by the nature of the waves?

Can one with some instrument make a reading like "In this room we have this level of interference and this level of noise"?

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Interference is generated by other signals (in other circuits or, more likely, in the same circuit), so it's artificial noise. Also the signal itself can generate interference, for instance if there is a conflict between subsequent symbols, or not perfect matching on a transmission line.

Noise is everything that is not useful signal, so can be due to interference, temperature, impurities, gamma rays, moon phase or whatever.

So interference is noise but the inverse is not true.

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