[time-nuts] Tbolt issues
kb8tq at n1k.org
Thu Sep 1 21:46:02 EDT 2016
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.
> On Sep 1, 2016, at 8:51 PM, Charles Steinmetz <csteinmetz at 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.
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