  We prove that, after removing a zero Hausdorff dimension exceptional set of
parameters, all self-similar measures on the line have a power decay of the
Fourier transform at infinity. In the homogeneous case, when all contraction
ratios are equal, this is essentially due to Erd\H{o}s and Kahane. In the
non-homogeneous case the difficulty we have to overcome is the apparent lack of
convolution structure.
