Phase difference between two signals?

Question

I'm working on this embedded project where I have to resonate the transducer by calculating the phase difference between its Voltage and Current waveform and making it zero by changing its frequency. Where I(current) & V(Voltage) are the same frequency signals at any instant but not the fixed frequency signals approx.(47Khz - 52kHz). All I have to do is to calculate phase difference between these two signals. Which method will be most effective.

FFT of Two signals and then phase difference between the specific components Or cross-correlation of two signals? Or another if any ? Which method will give me most accurate result ? and with what resolution? Does sampling rate affects phase difference's resolution (minimum phase difference which can be sensed) ? I'm new to Digital signal processing, in case of any mistake, correct me.

ADDITIONAL DETAILS:-

• Noise In my system can be white/Gaussian Noise(Not significant) & Harmonics of Fundamental (Which might be significant one in resonant mismatch case).

• Yes 4046 can be a good alternative with switching regulators. I'm working with (NCO/DDS) where I can scale/ reshape sinusoidal on ongoing basis.

• Implementation of Analog filter will be very complex as I will require higher order filter with high roll-off rate for harmonic removal , so I'm choosing DSP based filter and its easy to work with MATLAB DSP Processors.

• What sampling rate would you suggest for a ~50 KHz (47Khz-52KHz) system for achieving result in FFT or Goertzel with phase resolution of preferably =<0.1 degrees or less and frequency steps will vary from as small as ~1 to 2Hz . to 50 Hz-200Hz.

• My frequency is variable 45KHz - 55Khz ... But will be known to my system... Knowing phase error for the last fed frequency is more desirable. After FFT AND DIGITAL FILTERING , IFFT can be performed for more noise free samples which can be used for further processing. So i guess FFT do both the tasks ...

But I'm wondering about the Phase difference accuracy cause thats the crucial part.

Answer 1

Not to detract from chip's answer (I upvoted it!) but some other options are:

• Cross correlation. Off the top of my head, I am not sure what the performance difference between that and the Goertzel algorithm will be, but both should be doable on an embedded system.
• Ad-hoc methods. For example, I would try something like this: bandpass the signals to eliminate noise, find the peaks and measure the time difference between the peaks. This will probably be more efficient, and, provided you do a reasonable job throwing out outliers and handling wrap-around, should be extremely robust. The bandpass filters will, themselves, alter the phase, so you'll have to make sure you apply exactly the same filter to both signals.

Answer 2

If the input signal-to-noise ratios are not too bad, a computually efficient solution can be built based on zero crossing detection. Also, have a look at http://www.metrology.pg.gda.pl/full/2005/M&MS_2005_427.pdf for a nice comparison of phase difference detection algorithms, including zero-crossing ones.

Answer 3

Computing 1-bin of a DFT (or using the similar complex Goertzel block filter) will work if the signal frequency is accurately known. (Set the DFT bin or the Goertzel to exactly that frequency).

If the frequency isn't exactly known, you could try using an FFT with an FFTshift to interpolate the frequency magnitude peak, and then interpolate the phase at that frequency for each of the two signals. An FFT will also allow you to window the data, which may improve phase estimation accuracy if the frequency isn't exactly bin centered (or exactly the Goertzel filter frequency). Different windows may improve the phase estimation accuracy for frequencies "between bins". A Blackman-Nutall window will be better than a rectangular window, but there may be better window choices.

The phase measurement accuracy will depend on the S/N ratio, the length of time one samples the two (assumed stationary) signals, and possibly the window used.

Answer 4

If you have a Phase Locked Loop (PLL) that tracks each input, then you can subtract the phase coefficients (of the generator components) to determine offset between the phases. This would also be robust against noise.