For $v, w \in \{0,1\}^n$, denote $v w = (v_1 w_1, \ldots, v_n w_n)$ and $|v|=\sum_{i} v_i$. 

Let $v_1, v_2 \in \{0,1\}^n$ and
\begin{align*}
f(x_1, x_2) = \sum_{d=0}^{|v_1 v_2|} \frac{1}{2^{|v_1|+|v_2|-|v_1 v_2|}} {|v_1| - |v_1 v_2| \choose x_1 - d} {|v_2| - |v_1 v_2| \choose x_2 - d}. 
\end{align*}

How to find the asymptotic formulas for $f(x_1, x_2)$? Any help would be greatly appreciated!