Instantaneous frequency vs time for a piecewise signal

Question

I sample a signal that consists of a train of pulses which are amplitude and frequency modulated. I would like to generate a plot that is instantaneous frequency vs time. I compute the Hilbert transform from a down-converted signal to obtain the instantaneous phase and the envelope. Then I use the envelope to gate the instantaneous phase.

Therefore, I end up with two plots showing (a) the envelope and (b) the instantaneous phase. To obtain the frequency vs time trace, I need to differentiate the instantaneous phase trace. However, when I do so, the massive edges e.g. at the start of each pulse are dominating the numerical derivative.

What is a viable approach to obtain a frequency vs time plot from this data? Just to detect where pulses are and doing this piecewise did not really work as I have sometimes phase jumps during a pulse (here during the second pulse and the third pulse). For the above signal, I would get a trace that is non-zero only for the first and the last pulse as during the middle two ones the instantaneous phase is not changing.

Answer 1

Measurements of instantaneous frequency are highly susceptible to noise given the derivative operation which is a high pass filter. The results ultimately for a signal with noise present (as it appears here from the OP's plots) are an estimate of "truth", and such estimates are dependent on the SNR of the signal. That said, a rapid change in phase if it were to occur would result in very large frequency excursions up to the bandwidth limits of the system (a step change in phase is an impulse in frequency which is not physically realizable), so we would expect to see very short and large frequency changes for the transitions in the middle of the pulses. If this result is not meaningful (meaning if the OP was only interested in the instantaneous frequency once the envelope has settled from changing magnitude), then such excursions can easily be detected and blanked with a threshold detector.

That said, the results for the positions in the curve where the signal is nearly off (the envelope is very low) will not be useful as estimates of frequency and should be omitted. It appears the OP has already done this in the phase information, as the phase itself would not be zero as shown in the plot. For this same reason, I would blank the results associated with transitioning from nearly off to on.