Fourier decay for homogeneous self-affine measures

We show that for Lebesgue almost all $d$-tuples $({\theta }_1,\dots ,{\theta }_d)$, with $\mathrm{\mid }{\theta }_j\mathrm{\mid }>1$, any self-affine measure for a homogeneous non-degenerate iterated function system $\{Ax+a_j{\}}_{j=1}^m$ in ${\mathbb{R}}^d$, where $A^{-1}$ is a diagonal matrix with the entries $({\theta }_1,\dots ,{\theta }_d)$, has power Fourier decay at infinity.