Question about the likelihood function in GW parameter estimation

Hi, I learned in the Open data Workshop 2023 that the likelihood function of noise model for GW parameter estimation is given by the Equation~(8) in…91d2003V/abstract. However, to my knowledge, the Fourier transformation of data, d(t), must be a complex, and the likelihood function should include both the Real part and the Imaginary part, i.e., the likelihood function should be a two-dimensional Gaussian function. So, I am confused about the form of adopted one-dimensional Gaussian likelihood for the real data analysis. Does it simply assume that the real data in frequency domain are only adopted as the Real part of Fourier transformation of d(t)? Also, why this assumption can be made (or any conditions)?

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@duss Thank you for the question.

The likelihood function should always return a real-valued result. In the equation you mention, the data di appear only in a squared magnitude: |di|2

So, the data are complex valued, but the likelihood value will be real.

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@jonah Thank you for your kind explanation!