Let $\phi(t)=\frac{1-b}{1+a}\frac{1+ae^{-it}}{1-be^{it}}$ be the characteristic function of a distribution where $a,b\in (0,1)$. I want to find the density function $f$ of the distribution.
$f$ is the Fourier transform of $\phi$ by definition.$$f(x)=\int_{-\infty}^{\infty} e^{- itx}\phi(t)dt$$
$$f(x)=\int_{-\infty}^{\infty} e^{- itx}\frac{1-b}{1+a}\frac{1+ae^{-it}}{1-be^{it}}dt$$ It looks like I should use contour integral to find $f$ but I cannot locate the poles of the integrand and the contour integral really depends on the value of $x$. Any suggestion is welcome.