  Let $f$ be a full-level cusp form for $GL_m(\mathbb Z)$ with Fourier
coefficients $A_f(n_1,...,n_{m-1})$. In this paper an asymptotic expansion of
Voronoi's summation formula for $f$ is established. As applications of this
formula, a smoothly weighted average of $A_f(n,1,...,1)$ against
$e(\alpha|n|^\beta)$ is proved to be rapidly decayed when $0<\beta<1/m$. When
$\beta=1/m$ and $\alpha$ equals or approaches $\pm mq^{1/m}$ for a positive
integer $q$, this smooth average has a main term of the size of
$|A_f(1,...,1,q)+A_f(1,...,1,-q)|X^{1/(2m)+1/2}$, which is a manifestation of
resonance of oscillation exhibited by the Fourier coefficients
$A_f(n,1,...,1)$. Similar estimate is also proved for a sharp-cut sum.
