Log-likelihood function for a filtered Fourier spectrum I have time series data from which I am trying to infer parameters using MCMC. I normally infer parameters about the data in the time domain, using a Normal log-likelihood. However, I now have to infer parameters using a filtered FFT of the time-series. The data looks like this, with the current plotted against time on the right, and the current plotted against the input potential on the left

This is because the "noise" process has a linear effect on the time-series whilst the signal process is non-linear. Below is the FFT of the current data

Each peak in the spectrum represents a harmonic of the input voltage frequency. There is more "signal" in the higher harmonics than noise (which is apparent if you individually inverse FFT each peak, like so). The harmonic number is on the right of the plot

I do not think the log-likelihood function I am using is valid for complex numbers, as it is returning infinite values. Is there an alternative log-likelihood that is valid for complex numbers?
 A: When I mentioned the graph of Fourier coefficients, I meant a graph where you have the harmonic on the x-axis and the coefficient (real and imaginary, or--better--amplitude and phase) on the y-axis. Never mind, I'll make some guesses based on what you've provided and try to give a meaningful answer.
From your figures it looks like the current is trailing the voltage by a more-or-less constant time delay (is it some kind of an inductive circuit?). If the time delay were perfectly constant, your curve would be an ellipse. If it were exactly 1/4 of the period, the curve would be a circle.
In the frequency domain, that means that the spectra of your voltage and current are shifted by an angle $\varphi$ relative to each other. If you represent your complex coefficients in polar notation:
$$
Z_n = A_n \cdot e^{i \varphi_n}
$$
then you can model the noise separately for the amplitude ($A$) and phase ($\varphi$). In the first approximation, both noises can be probably well modelled as Gaussian. If you want to be more exact, the amplitude noise is maybe better modelled as log-normal (since the amplitude cannot be < 0) and the phase noise by von Mises distribution.
Now, in contrast to ordinary time series, where you'd probably assume constant variance over time (your "independent variable"), this assumption seems to be severely violated in the frequency domain. It looks like your noise is much more pronounced in the 3rd and the 7th harmonic than in the remaining ones.
If you have enough data, you can try to model the noise for each harmonic separately. If, not, try modelling it only for the 7th harmonic (the one with the largest variance.
