Lomb-Scargle periodogram for complex exponential signal

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The discussion centers on the implementation of a generalized Lomb-Scargle periodogram for aperiodically sampled complex data, referencing Brethorst's paper. The user notes that their implementation aligns with a Schuster periodogram, raising questions about the validity of their approach. Key assumptions include setting the decay factor to zero and using equal lengths for the real and imaginary parts of the signal. The user calculates specific quantities and expresses uncertainty about whether their results are correct. They seek recommendations for a periodogram that offers lower side-lobe levels than the Schuster periodogram for their complex signal analysis.
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I found a paper by Brethorst where he developed a periodogram that is a generalized version of the Lomb-Scargle periodogram. You can find it here [1].

I tried to implement (22) from this paper to make a periodogram for an aperiodically sampled complex data that is stochastic. I observed that it is the same as a Schuster periodogram. I want to verify what I did. Please let me know if something is wrong.

In the paper, they added a decay factor in the model ## Z ##, which I set to [itex ] 0 [/itex ].
Second, they also have different lengths for the real and imaginary parts of the signal. However, for me, they are collected at the same time. ## N_R = N_I = N_d ##, and ## t_i = t_j ##.

I choose the $H$ as the basis ## H = 2 \pi f t ## as they do in (23).

If I go by these assumptions, the following quantities become:

$$\theta = \frac{1}{2} \tan^{-1}\left(\frac{0}{0}\right) = 0$$ From (20)
$$ C = N_d $$ from (17)
$$ S = N_d $$ from (18)

$$ R = \sum_{i = 1}^{N_d} d_R(t_i) \cos{(H(t_i))} - d_I(t_i) \sin{(H(t_i))} $$
$$ I = \sum_{i = 1}^{N_d} d_R(t_i) \sin{(H(t_i))} + d_I(t_i) \cos{(H(t_i))} $$

Here, ## d_R = \Re({z}) ##, and ## d_I = \Im({z}) ##. So, the final expression (22) becomes:

$$ \bar{h}^2 = \frac{1}{N_d} \times (R^2 + I^2) $$

I think this is the same as the Schuster periodogram. Am I correct? In that case, which periodogram should I use with lower side-lobe levels than the Schuster periodogram for the aperiodically sampled complex signal?

[1]: https://bayes.wustl.edu/glb/general.pdf
 
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