[time-nuts] Tbolt issues

[Charles Steinmetz]

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