  We use a multivariate version of Stein's method to establish a quantitative
Lindeberg CLT for the Fourier transforms of random $N$-vectors. We achieve this
by deducing a specific integral representation for the Hessian matrix of a
solution to the Stein equation with test function $e_t(x) = \exp(- i
\sum_{k=1}^N t_k x_k)$, where $t,x \in \mathbb{R}^N$.
