  This paper studies the stability of some reconstruction algorithms for
compressed sensing in terms of the bit precision. Considering the fact that
practical digital systems deal with discretized signals, we motivate the
importance of the total number of accurate bits needed from the measurement
outcomes in addition to the number of measurements. It is shown that if one
uses a $2k \times n$ Vandermonde matrix with roots on the unit circle as the
measurement matrix, $O(\ell + k \log(n/k))$ bits of precision per measurement
are sufficient to reconstruct a $k$-sparse signal $x \in \R^n$ with dynamic
range (i.e., the absolute ratio between the largest and the smallest nonzero
coefficients) at most $2^\ell$ within $\ell$ bits of precision, hence
identifying its correct support. Finally, we obtain an upper bound on the total
number of required bits when the measurement matrix satisfies a restricted
isometry property, which is in particular the case for random Fourier and
Gaussian matrices. For very sparse signals, the upper bound on the number of
required bits for Vandermonde matrices is shown to be better than this general
upper bound.
