# Probability density function of A = B + C via Joint Characteristic function of B and C

This problem is actually a subproblem of a longer derivation that I am trying to understand. I hope that I striped away all the unnecessary stuff that is not relevant. Please correct me if the explanation below is not sufficient.

Assume that the final goal is to have an expression for the probability density function (PDF) of a random variable (RV) $\phi$. This random variable is a phase angle, implying that its PDF is actually periodic. The PDF can therefore be expressed as a Fourier series $$f_{\Phi}(\phi) = \sum_k \Psi_{\tilde{\Phi}}(k) e^{i k \phi},$$ where the Fourier coefficients $\Psi_{\tilde{\Phi}}(\nu)$ can be interpreted as the characteristic function (CF) of the "unwrapped" phase RV $\tilde{\Phi}$ evaluated at "integer angular frequencies". The problem is that this CF is not given. Instead, what is given is $\Psi_{\Phi_1, \Phi_2}(\omega, f)$ - the joint CF of $\Phi_1$ and $\Phi_2$, where $\Phi = \Phi_1 + \Phi_2$. For some reason, it seems to be possible to write now instead $$f_{\Phi}(\phi) = \sum_k \Psi_{\Phi_1, \Phi_2}(k, k) e^{i k \phi} .$$ However, I don't understand why the PDF can be written like this.

-