PLL Response Time vs. the Uncertainty Principle

[rrumpf]

I am quite new to PLL design and analysis and have what I hope is a simple question to somebody.

I am developing a system that will use a PLL to track the resonant frequency of a small micro-beam around 26 kHz. I would like retrieve the frequency to within 0.01 Hz resolution. Using canned formulas in the literature and text books, my "lock in" time, or the response time due to a step change in frequency, should be less than one second.

This seems to violate the Uncertainty principle from signal theory which states the more we try to "locate" a signal in frequency, the less we can locate it in time. Stated another way, to resolve a frequency to 0.01 Hz resolution should take a system much more than one second. An approximate formula can be written:

dt * df > 0.5

where df is the frequency resolution and dt is the time resolution.

I am trying to reconcile the "lock in" time or "pull in" time formulasfor PLLs with the Uncertainty principle and they seem to contradict each other. Any guidance anyone could provide would be greatly appreciated. Remember I am a novice PLL person!!

Thanks!!
-Tip

 

[LvW]

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This seems to violate the Uncertainty principle from signal theory which states the more we try to "locate" a signal in frequency, the less we can locate it in time. Stated another way, to resolve a frequency to 0.01 Hz resolution should take a system much more than one second. An approximate formula can be written:
dt * df > 0.5
where df is the frequency resolution and dt is the time resolution.

Interesting question. However, I don´t think that the uncertainty principle is involved when discussing PLL lock behaviour. As far as I know, this would be the case if a circuitry (like PLL) has to reconstruct a signal from a certain (incomplete ?) set of frequency samples. But this is not what a PLL does. In principle, the PD act as a pure analog sensor (for a "digital" PD: in conjunction with a low pass) in a closed control loop called PLL adjusting a certain parameter. And there is in no way the task to "measure" time and frequency samples at the same time. I must admit, most probably my explanation is not very scientific - it´s more or less based on an "engineering feeling".
Regards

 

[FvM]

dt * df > 0.5

The limitation doesn't apply for the frequency determination of an (almost) stationary single frequency signal. It's valid for spectral analysis of arbitrary signals. It's no problem to measure the frequency of a periodical signal within a single period up to any precision, it's limited mainly by noise but not primarly measuring time.

There may be a problem, if the bandwidth is too high respectively the frequency changing too fast. A real circuit has to consider noise and it may be necessary to limit the PLL bandwidth to handle it. Thus I suggest to analyze the frequency tracking for an empirical signal.