  Let $\{s_n\}_{n\in\N}$ be a decreasing nonsummable sequence of positive
reals. In this paper, we investigate the weighted Birkhoff average
$\frac{1}{S_n}\sum_{k=0}^{n-1}s_k\phi(T^kx)$ on aperiodic irreducible subshift
of finite type $\Sigma_{\bf A}$ where $\phi: \Sigma_{\bf A}\mapsto \R$ is a
continuous potential. Firstly, we show the entropy spectrum of the weighed
Birkhoff averages remains the same as that of the Birkhoff averages. Then we
calculate the packing spectrum of the weighed Birkhoff averages. It turns out
that we can have two cases, either the packing dimension of every level set
equals to its Hausdorff dimension or for every nonempty level set it is equal
to the packing dimension of the whole space.
