[time-nuts] Tbolt issues

[jimlux]

On 9/1/16 5:51 PM, Charles Steinmetz 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.

in fact, there's a whole literature on how accurate (or more precisely, 
what's the uncertainty) of the frequency estimate is.

We often measure frequencies with less than a cycle - but making some 
assumptions - measuring orbital parameters is done using a lot less than 
a complete orbit's data, but we also make the assumption of the physics 
involved.


---

Instantaneous frequency does have a theoretical meaning, even if not 
measureable..

If I'm processing  a linear frequency chirp, I can say that the 
frequency at time t is some (f0 + t*slope).  the frequency at time 
t+epsilon is different, as is the frequency at time t-epsilon.