  In the theory of compressed sensing, restricted isometry analysis has become
a standard tool for studying how efficiently a measurement matrix acquires
information about sparse and compressible signals. Many recovery algorithms are
known to succeed when the restricted isometry constants of the sampling matrix
are small. Many potential applications of compressed sensing involve a
data-acquisition process that proceeds by convolution with a random pulse
followed by (nonrandom) subsampling. At present, the theoretical analysis of
this measurement technique is lacking. This paper demonstrates that the $s$th
order restricted isometry constant is small when the number $m$ of samples
satisfies $m \gtrsim (s \log n)^{3/2}$, where $n$ is the length of the pulse.
This bound improves on previous estimates, which exhibit quadratic scaling.
