# Amplitude/phase recovery on noisy signal

I have a noisy signal that has an ac component of fixed frequency but variable amplitude and phase. I'd like to recover the ac component.

The signal (blue trace) is mostly smooth, but has a few larger discontinuities.

The phase and amplitude of the ac component vary with time, this is how the signal looks like (green). Noise not shown:

The sampling rate is not commensurate with the ac period, this causes the aliasing seen in the picture. Unfortunately I cannot change this.

From this data, I need to recover amplitude and phase of the ac component, if possible also the harmonics.

My current approach is to multiply the data with a sine and cosine and then use a LPF to recover the quadrature components. I am not too happy with that approach, as the aliasing requires a very low cutoff frequency that smears out sharp transitions in the phase.

Also, I'd like to explore the solution space with some global constraints on amplitude and phase, e.g. sparsity and piece-wise continuity of the phase.

What is the right approach for phase and amplitude recovery? Can this be treated as a constraint convex optimization problem, or are there better solutions?

Update:

I have taken some real measurement data. I didn't have any data with the ac signal, so I have added an ac component numerically. The ac peak due to the signal is the highest green peak close to DC. The other two blue peaks are due to a digitizer artifact. The green curve is pretty much the same as the blue one in front elsewhere. This curve did not have any modulation on the ac signal, for a real signal, the peak would be about 2/3 that high and have some sidebands. This is shown in the inset image, the red curve is for the amplitude and phase modulated signal.

# simulated raw data for first plot
d=[0] * 50 + [1]*50 + list(np.linspace(0.5,1,50)) + list(np.exp(np.linspace(0,-1,30)))
# with additional small signal ac overimposed. Normally amplitude and phase would change as well
ds=d + 0.1*np.sin(100*np.linspace(0,6,180))
plt.plot(ds)


This is what I meant with global constraints: The data is very noisy, and it is not realistic to measure amplitude and phase without any averaging. On the other hand, I have some hypotheses about amplitude and phase, and would like to make use of them for filtering the data. For example, I know that there are sharp transitions in the phase, with areas of constant phase in-between. I don't want to average out the transitions. So ideally, I would like to recover the instantaneous amplitude and phase of the ac signal and then use a LASSO-like method to filter.

• could you plot the fft of that to see how it looks? maybe that give us a clue. Also, to get the ac, your LPF approach sounds good to me. – Behind The Sciences Mar 29 '16 at 6:16
• Did you tried smooth? – Brethlosze Nov 14 '16 at 7:18
• I like your approach of doing a complex multiplication to circularly rotate your signal (in frequency) to 0 and then applying a LPF, which really is identical to band pass filtering around your signal of interest and leaving it at the carrier. I say that as to the extent you are able to isolate in the frequency domain, your approach is best and then a matter of filter design. If your "interference" shares spectrum with your signal of interest, then you are as far as you can go without additional info. Does your signal have transitions at a consistent rate or within a range? Any other info? – Dan Boschen Feb 24 '17 at 15:46