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I am working on the analytic signal concept for observing the frequency variation in the faulted phase current waveform in MATLAB/Simulink model.

In MATLAB/Simulink model, I have used analytic signal block and hilbert's transform method to calculate the real and imaginary components of the faulted phase current.

After that I have used the data for real and imaginary components of the faulted phase current to calculate the magnitude, phase, and instantaneous frequency.

When I plotted the curve between the instantaneous frequency variation vs time for the faulted phase current, I observed that there are high frequency spikes in the curve.

What can be the reason and what is this phenomena called ?

Peter K.
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Siddharth Kamila
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1 Answers1

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The instantaneous frequency is the time derivative of phase. The time derivative operator is a high pass filter, so it on its own will enhance high frequency noise and attenuate lower frequencies. For example, a step change in time would become an impulse after taking a derivative.

It is possible that the spikes are from "steps" that are incorrectly introduced into the phase during processing; for example if the phase is wrapping at the $2\pi$ boundaries. I suggest reviewing a plot of the phase versus time used prior to taking the derivative and confirm if this phase should be "unwrapped" prior to the derivative operation. (unwrap is a function for doing this in Matlab, Octave and Python scipy.signal).

As a simple example consider the case below of a constant frequency, which would therefore be a ramp in phase. A small amount of noise was added to the phase which is barely visible in the plot of phase versus time, but clear in the demodulated frequency (which was done as a simple difference of phase from sample to sample).

FM Demoduation

If the phase was incorrectly "wrapped" to stay within the range of $\pm pi$, the same demodulator would produce the result as given by the plots below, with very significant negative spikes where the phase wrapping occurred. Even if we did the demodulation with a improved digital differentiator that also filtered the high frequency noise, a significant amount of these spikes would still feed through:

phase wrapping

Not to conclude that phase wrapping is the source of the spikes in the OP's specific case as this is just one obvious / common source for this to occur; any other steps in phase (from actual noise sources) would also lead to spikes in the demodulated signal.

As a side-comment: this high frequency noise enhancement is a common 'feature' in FM demodulation, and is the reason for the use of pre-emphasis in some FM modulation implementations.

Dan Boschen
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  • Can you share a research paper (journal or conference paper) where this answer is written. I have to study more about it. – Siddharth Kamila Jan 09 '23 at 08:11
  • @SiddharthKamila These are some basic signal processing concepts so not sure I can located a research paper about this specifically- but what part confuses you? The phase wrapping or that instantaneous frequency is the time derivative of phase? Or that a time derivative is a high pass filter? – Dan Boschen Jan 09 '23 at 12:48
  • This may help you: https://en.wikipedia.org/wiki/Instantaneous_phase_and_frequency and to see how the derivative is a high pass filter, review the Laplace Transform of the time derivative: The Laplace of $d/dt x(t)$ is $sX(s)$. To get the frequency response replace $s$ with $j\omega$, and thus the result is $\omega X(\omega)$. – Dan Boschen Jan 09 '23 at 12:51
  • As far as the phase wrapping itself, phase wrapping produces steps in phase (as demonstrated in my answer). A derivative of a step is an impulse: consider the case of a discrete time differentiator which estimates a derivative given as $y[n]=x[n]-x[n-1]$ and what you would get for the output of that if there was a big step in time. – Dan Boschen Jan 09 '23 at 12:59