On self dual bent functions

Here, it is proved that a Boolean function $f$ in $n$ variables is self-dual bent if and only if the Hamming weight of the function $F_y(x)=f(x)\oplus f(y)\oplus x\cdot y$ is equal to $2^{n-1}-2^{n/2-1}$ for any $y\in\mathbb F_2^n$.