I would like to estimate the frequency of a discrete-time (real) sinusoidal tone - with unknown amplitude, frequency and phase - in noise
An LMS-type technique looks promising, for example:
Pradhan, A.K.; Routray, A.; Basak, A., "Power system frequency estimation using least mean square technique," in Power Delivery, IEEE Transactions on , vol.20, no.3, pp.1812-1816, July 2005
Can anybody comment on the strengths and weaknesses of this type of approach, in practical applications (convergence, noise immunity, interference immunity, impulse response, etc)?
I imagine a Kalman filter could also be used for this type of thing (comments?).
Any other similar approaches that might do the job?
I am trying to avoid methods that sample the spectrum using a bank of frequency analyzers (as in the link below).
Article Digital Filter Designs for Recursive Frequency Analysis
I've found a few more good papers on this ...
Ramos, C.J.; Martins, A.P.; Carvalho, A.S., "Frequency and Phase-Angle Estimation Using Ordinary Least Squares," in Industrial Electronics, IEEE Transactions on , vol.62, no.9, pp.5677-5688, Sept. 2015
Golestan, S.; Monfared, M.; Freijedo, F.D.; Guerrero, J.M., "Performance Improvement of a Prefiltered Synchronous-Reference-Frame PLL by Using a PID-Type Loop Filter," in Industrial Electronics, IEEE Transactions on , vol.61, no.7, pp.3469-3479, July 2014
Kusljevic, M.D.; Tomic, J.J.; Jovanovic, L.D., "Frequency Estimation of Three-Phase Power System Using Weighted-Least-Square Algorithm and Adaptive FIR Filtering," in Instrumentation and Measurement, IEEE Transactions on , vol.59, no.2, pp.322-329, Feb. 2010
Kusljevic, M.D., "On LS-Based Power Frequency Estimation Algorithms," in Instrumentation and Measurement, IEEE Transactions on , vol.62, no.7, pp.2020-2028, July 2013
... so I have some reading to do.
... OK, got it.:
Use a Hilbert transform filter to get an I/Q signal, compute the instantaneous phase, instantaneous frequency is the time derivative of the unwrapped instantaneous phase.
Sorry everyone, I've been (not) thinking out loud here!
Hi Hugh,
I was just on the way to type in "Hilbert...". Anyway, I would like to comment on the approach. Please take a look also in the "Hilbert-Huang-Transform / Empirical Mode decomposition" approach. We a using this approach (modified) for multi-mode-electro-mechanical systems. If you take a look into our papers you will find some sources, e.g.:
https://www.researchgate.net/publication/278848686_Structural_Health_Monitoring_of_Plastic_Components_with_Piezoelectric_Sensors
https://www.researchgate.net/publication/261282496_Sensorless_drive_diagnosis_using_automated_feature_extraction_significance_ranking_and_reduction
Conference Paper Structural Health Monitoring of Plastic Components with Piez...
Conference Paper Sensorless drive diagnosis using automated feature extractio...
Hi Hugh,
Some years ago I wrote a paper comparing frequency estimators and proposing another fast estimation technique, hope it can be helpful:
An algebraic frequency estimator for a biased and noisy sinusoidal signalJR Trapero, H Sira-Ramírez, VF Batlle
Signal Processing 87 (6), 1188-1201
All the best, JR.
I think it would be useful to share some details about the application you have in mind.
An important aspect is wether your frequency is limited to a certain range. In the cases where the range is relatively narrow, or there is at least a starting point (e.g. a coarse estimation from where to start), and if frequency is dynamically variable, PLLs are probably the most widely used tool.
PLLs have long (and are still) been used in RF communications, but also in the control of power systems (grid-connected devices synchronization) and motor control (position estimation). They have several advantages, among them the fact that their dynamics (and hence noise immunity) can be easily tuned by means of the gain of a PI regulator.
Once frequency has been determined and is constant across a certain time, one can detect phase and amplitude by means of Fourier series. This is a special case of a Hilbert space projection, and IIRC it gives the least squares solution.
From the practical (implementation) point of view, Fourier series is simple, and has the advantage it only requires the calculation (online update) of two sine (/cosine) variables and two integral variables (no memorization of samples) along multiples of half a period.
I was thinking about a digital control loop for rotating mechanical components. The sampling rate will be many times the rotation rate. So I would expect the digitized sinusoidal signal to have a wavelength of no less than say 10-20 samples, but typically much more, say 100-200. I know there are lots of different ways of approaching this control problem, but I would specifically like to try out a frequency-based approach (initially).
In this application, I think the existence of a single tone is reasonable, noise levels will not be high, but transient response (i.e. convergence of the estimates) will need to be rapid, and the latency/delay of any filters used must be low.
I was thinking that the crux of this problem is in the design of the digital Hilbert filter to make the analytical (I&Q) signal. This aspect of the problem seems to be overlooked in a lot of the stuff I have read on this, where the starting point is usually a complex signal or a 3-phase signal in the case of the power systems literature.
Right now I am thinking that designing an appropriate low-pass filter to make the analytical signal would be an interesting activity. But now I am thinking that a PLL might be also appropriate (thanks Sandro). I'll look into it as well. Could anybody recommend a reference with a block diagram that might be appropriate here?
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