  Time-domain simulation of wave phenomena on a finite computational domain
often requires a fictitious outer boundary. An important practical issue is the
specification of appropriate boundary conditions on this boundary, often
conditions of complete transparency. Attention to this issue has been paid
elsewhere, and here we consider a different, although related, issue: far-field
signal recovery. Namely, from smooth data recorded on the outer boundary we
wish to recover the far-field signal which would reach arbitrarily large
distances. These signals encode information about interior scatterers and often
correspond to actual measurements. This article expresses far-field signal
recovery in terms of time-domain convolutions, each between a solution
multipole moment recorded at the boundary and a sum-of-exponentials kernel.
Each exponential corresponds to a pole term in the Laplace transform of the
kernel, a finite sum of simple poles. Greengard, Hagstrom, and Jiang have
derived the large-$\ell$ (spherical-harmonic index) asymptotic expansion for
the pole residues, and their analysis shows that, when expressed in terms of
the exact sum-of-exponentials, large-$\ell$ signal recovery is plagued by
cancellation errors. Nevertheless, through an alternative integral
representation of the kernel and its subsequent approximation by a {\em
smaller} number of exponential terms (kernel compression), we are able to
alleviate these errors and achieve accurate signal recovery. We empirically
examine scaling relations between the parameters which determine a compressed
kernel, and perform numerical tests of signal "teleportation" from one radial
value $r_1$ to another $r_2$, including the case $r_2=\infty$. We conclude with
a brief discussion on application to other hyperbolic equations posed on
non-flat geometries where waves undergo backscatter.
