Detect a Frequency Change in a Step Wise Frequency Chirp

Question

I am driving a system with a step-wise frequency chirp from 50-400 hz with a frequency step of 0.01 hz. I am having difficulty identifying the points of frequency change as the data collected has an indeterminate amount of time at 50 Hz and an indeterminate amount of time at 400 Hz. Since the time between frequency changes is 0.1 seconds an fft calculation doesn't provide the proper frequency resolution to determine this change. I also tried calculating the gradient of the frequency chirp with inconclusive results as well as attempting a lock-in calculation. I wrote some sample code that has a randomly located frequency step followed by another 0.1 secs later that I have been trying to diagnose with varying success. Ultimately I am trying to identify the beginning and end of the frequency chirp since at that point the step changes are known but this change point identification was attempted to identify the point of change between 50 and 50.01, then backtracking to find the beginning of the series. Any suggestions would be very much appreciated.

import numpy as np
from scipy import signal
import matplotlib.pyplot as plt

fs = 10000
t = np.linspace(0,1,1*fs)
x = np.zeros(len(t))

step_loc = int(2500 + 5000*np.random.random(1))

x[:step_loc] = np.sin(50*2*pi*t[:step_loc])
x[step_loc:step_loc+1000] = np.sin(50.01*2*pi*t[step_loc:step_loc+1000])
x[step_loc+1000:] = np.sin(50.02*2*pi*t[step_loc+1000:])

grad = np.gradient(x,t[1])

lock = x*(np.cos(50*2*pi*t) + 1j*np.sin(50*2*pi*t))
b, a = signal.butter(5,30/10000, 'low')
lock_filt = signal.filtfilt(b, a, lock)

plt.plot(t,x)
plt.plot(t,grad*3e-3)     #this is offering very inconsistent identifiable results
plt.plot(t,lock_filt*1e1) #although this can identify the general area of the random frequency change i
                      # dont know how to extend it to identifying the actual point 

Answer 1

There are many way to tackle this:

1. Time - Frequency Analysis
   Classic choice would be a spectrogram but probably a Fourier Synchrosqueezed Transform would do a better job (Have a look at even more advanced approaches ssqueezepy - Synchrosqueezing, wavelet transforms, and time frequency analysis in Python by @OverLordGoldDragon).
2. Parameter Estimation
   Methods for optimization of the instant parameter. Usually by Non Linear Least Squares.
3. Bayesian / Prior Based Estimation
   Methods with some prior (Sparsity) for estimation of the parameters. See Estimation of Amplitude, Frequency and Phase of Linear Combination of Harmonic Signal Beyond the Leakage Resolution of DFT.

But, for creativity, I'd like even one more approach - Tracking.
Similar to what I did in Estimate and Track the Amplitude, Frequency and Phase of a Sine Signal Using a Kalman Filter.

We basically track the parameters of the signal and once we see a "Jump" we can say there is a change.

I recreated a similar signal to yours.
The basic tracking of the Extended Kalman Filter can be seen in the linked question. But the summary is given by:

With noise level for a good measurement of 33 [dB] we can see the tracker is estimating well the signal and smoothens the noise.

What about frequency? Let's see:

So we can easily see the jumps and with more tweaking of the parameters of the Kalman Filter it will be more visible.

The code is available at my StackExchange Signal Processing Q64772 GitHub Repository (Look at the SignalProcessing\Q64772 folder).

Answer 2

I suggest monitoring the phase versus time directly instead of frequency. Frequency is the derivative of phase so the slope of the phase would indicate the frequency. Detrend the phase slope for the starting frequency and then the point in time where the phase starts to ramp up should be easier to detect. The window in which to detect this change will be balanced with the SNR of the signal itself.

Since frequency is the derivative of phase, it will be more sensitive to high frequency noise (the derivative is a high pass function), which is why monitoring phase directly could be a more reliable approach.