Consider the two-dimensional vector Y = (U,V) which has components that are Gaussian-distributed, centered at zero, and independent.
f(x; \sigma) = \frac{1}{2\pi\sigma^2} \int_{-\infty}^\infty \int_{-\infty}^\infty e^{-u^2/\left(2\sigma^2\right)} e^{-v^2/\left(2\sigma^2\right)} \delta(x-\sqrt{u^2+v^2}) \,dv\,du.