How to create an ideal sine wave that will be the best fit for given sine wave with noise and distortion

Question

I have a sine waveform that is a result of simulation.

This is always single tone with a constant offset, but with distortion and noise and may have some jitter:

$$s[k] = A\sin\left(2\pi \frac{f}{f_\text{s}} k + \phi\right)+ \alpha+N[k].$$

(since SNR after distortion is better than 40 dB, and jitter probably not dominant, let us ignore distortion and jitter for now.)

I need to create an ideal sine wave that will be the best fit for the given waveform. I am using Cadence calculator functions so, I will explain my current approach as pseudocode.

My input is waveform sampled using coherent sampling - sourceWave

dft1=dft(sourceWave t1 t2 256 "Rectangular")    #DFT with 256 points and rectangular window
dc=mag(value(sourceDFT 0))                      #Zero frequency DFT result to get sign wave offset
sourceWaveNorm  =sourceWave-dc                  #Remove DC
dft2=dft(sourceWaveNorm t1 t2 256 "Rectangular")#DFT of waveform without DC
sineFreq =maxamplfreq(magnitude(dft2))               #Get frequency with maximum amplitude as it should be frequenxy of the input sine wave
sineAmp=magnitude(value(dft2 sineFreq ))             #Amplitude 
sinePhase=phaseRad(value(dft2 sineFreq ))            #Phase
finalPhase      =(sinePhase+ (pi/ 2)) - (sineFreq * 2 * pi * t1)
ideslSineWave=sineAmp*sin(sineFreq*2*pi+finalPhase)+dc

This approach seems to work fine for input waveforms without DC. But when there is DC in the input waveform fitting not good, seems like the DC of ideal waveform is little bit wrong.

Additional clarification: Basically, I need to calculate values for

$$ \hat s(t) = \hat A\sin(2\pi \hat ft + \hat \phi) + \hat\alpha$$

So, I need to estimate $A,f,\phi$ and the offset $\alpha$ from the input waveform.

How can I improve this, or maybe a totally different approach is needed?

Answer 1

The standard way to do this would be to:

1. Remove / estimate the DC offset ($\hat{\alpha}$)
2. Estimate the frequency $\hat{f}$.
3. Change the polar estimation problem for $\hat{A}$ and $\hat{\phi}$ into a rectangular form by using $$\sin(X + Y) = \sin X \cos Y + \cos X \sin Y$$ and estimating $\hat{A}\cos(\hat{\phi})$ and $\hat{A}\sin(\hat{\phi})$.

Item 1. can just be done using: $$ \hat{\alpha} = \frac{1}{N} \sum_{n=0}^{N-1} s[n] $$

Item 2. can be done using the Quinn-Fernandes frequency estimator. There is a Python implementation here.

Item 3. can be done using the formulae from Kay that I transcribed here.