  In this note, we investigate $C^2$ differential images of the homogeneous
self-similar measure associated with an IFS $\mathcal{I}=\{\rho
x+a_j\}_{j=1}^m$ satisfying the strong separation condition and a positive
probability vector $\vec{p}$. It is shown that the Fourier transforms of such
image measures have power decay for any contractive ratio $\rho\in (0, 1/m)$,
any translation vector $\vec{a}=(a_1, \ldots, a_m)$ and probability vector
$\vec{p}$, which extends a result of Kaufman on Bernoulli convolutions. Our
proof relies on a key combinatorial lemma originated from Erd\H{o}s, which is
important in estimating the oscillatory integrals. An application to the
existence of normal numbers in fractals is also given.
