# Phase and Amplitude estimation

Context:

I have used FFT many times, but for real, non-periodic signals I consider it a poor estimator.

For most of my applications I am only interested on the power spectrum, so I use the Welch's method. I really like its approach to take overlapping segments, apply a window to each of them, calculate the FFT and finally average the result to get the power spectrum estimation. The method is easy to understand and its results are smooth and stable.

However I have one application that I am also interested in the phase of the signal. In fact am not interested in the phase estimation of particular signal segment, I want to see how the phases of very similar spread spectrum signals vary over time.

My signals spectrum are spread, but they always have a peak in the PSD, however this peak frequency varies over time. I will probably only need to trend the phase related to the frequency of this peak.

Questions:

1) Does it make sense to use Welch's method for magnitude and phase, instead of power? I mean average magnitude and phase, instead of power.

2) Is there any magnitude and phase spectrum estimator that is smooth/stable like Welch's method?

answer to 1, No, the phase of an tone isn't constant from one FFT to the next, so averaging phase provides no reduction of error unless there is reason to think there is like in a coherent algorithm like SAR.

Answer to 2 is a bit more complicated but is yes, because you need to do carrier and timing recovery in a communications systems

For something less structured There are several frequency estimation algorithms that work on FFT data like:

Macleod, Malcolm D. "Fast nearly ML estimation of the parameters of real or complex single tones or resolved multiple tones." IEEE Transactions on Signal processing 46.1 (1998): 141-148.

and this just a personal favorite. There are 270 papers that cite that paper and many before that. This method is nearly ML, so not Welch but not shabby.

La Scala, Barbara F., and Robert R. Bitmead. "Design of an extended Kalman filter frequency tracker." IEEE Transactions on Signal Processing 44.3 (1996): 739-742.

Is more along the lines of tracking frequency and there are 170 papers that cite this paper.

a good book is:

Quinn, Barry G., and Edward James Hannan. The estimation and tracking of frequency. Vol. 9. Cambridge University Press, 2001.

A recent tutorial paper

Vila-Valls, Jordi, et al. "Are PLLs dead? A tutorial on Kalman filter-based techniques for digital carrier synchronization." IEEE Aerospace and Electronic Systems Magazine 32.7 (2017): 28-45.

might be useful.

Phase introduces complications because phase is relative to some reference and having the same reference for multiple tones may or may not be meaningful.

One of the regular contributors here, Peter K has a web page (which I don't recall where) that has a lot of material that may be of interest.