Hi Unfortunately if you read a typical text on FM modulation, "instantaneous frequency" comes up pretty fast. In that context it has a valid meaning. Once out of context, it gets you in trouble. That point is never made when the term is introduced.
Bob > On Sep 1, 2016, at 8:51 PM, Charles Steinmetz <csteinm...@yandex.com> wrote: > > Nick wrote: > >> On a theoretical basis, can one speak of the limit of the frequency observed >> as tau approaches zero? >> Might that in some way be the "instantaneous frequency" which people often >> think of? > > That is (or is "something like") what it *would* be, but a little thought > experiment will show that (and why) the linguistic construction is > meaningless. > > The period of a 10MHz sine wave is 100nS. Think about observing it over > shorter and shorter (but still finite) time intervals. > > When the time interval is 100nS, we see one complete cycle (360 degrees, 2 pi > radians) of the wave. At this point we still have *some* shot at deducing > its frequency, because no matter at what phase we start, we are guaranteed to > observe two peaks (one high, one low) and at least one midpoint (e.g., > zero-cross). Our deduction (inference) will be less accurate as the noise > and distortion (harmonic content) increases, and it won't be all that good > under the best of circumstances. > > Now shorten the observation time to 20nS. We see 1/5 of a complete cycle (72 > degrees, 0.4 pi radians) of the wave. No matter which particular 72 degrees > we see, we simply don't have enough information to reliably deduce the > frequency. By sampling very fast (say, every 100fS), we at least know pretty > well the trajectory of that little snippet of signal, and using heroic > measures we can make an educated guess about the frequency -- but we really > couldn't say we "knew" what the frequency was. Our error bars are growing, > growing.... > > Now consider a 1nS sample. Nothing we can do now will give us even a bad > guess as to the frequency. And finally, consider a genuine "instant" sample > (one mathematical point of the wave form). We have now reached the point > where there is literally NO information about the frequency. One > time-voltage point could be part of a literally infinite number of signals, > each one of a different frequency from DC to infinity. > > Thus we see that the well-formed English phrase, "instantaneous frequency," > is, literally, meaningless. It denotes absolutely nothing. > > Best regards, > > Charles > > > _______________________________________________ > time-nuts mailing list -- time-nuts@febo.com > To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts > and follow the instructions there. _______________________________________________ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
