I have the time domain signal $$ u_o(t) = u(t)e^{-t/\tau}\eta(t) + \sigma(t) $$ where $\tau$ is known, $\eta$ is non-Gaussian noise, and $\sigma$ is Gaussian noise. The distribution of $\eta(t)$ is known, but only numerically. I also have prior knowledge that $u(t)$ is a sum of a small number of sinusoids. How can I recover $u(t)$ from $u_o(t)$?

In the case where $\eta$ is not present, I can Fourier transform to obtain: $$ \hat{u}_o(\xi) = \hat{u}(\xi)*l(\xi) + \sigma(\xi) $$ where $l$ is a Lorentzian. The deconvolution is easy to solve with basis pursuit: $$ argmin |u|_1 \; subject \; to \; \|l*u - \hat{u_o} \|^2 \leq \mu $$ This ignores $\eta$ as well as our statistical knowledge of $\eta$. Are there ideas on how I can incorporate $\eta$ into my denoising model? Is there a different model I should look into?

edit: looks like I need to set up a MAP estimate for $f = u*\eta + \sigma$. I think I can sort it out when it's just $\eta$ or just $\sigma$.