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.
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