  We consider the problem of recursively and causally reconstructing time
sequences of sparse signals (with unknown and time-varying sparsity patterns)
from a limited number of noisy linear measurements. The sparsity pattern is
assumed to change slowly with time. The idea of our proposed solution,
LS-CS-residual (LS-CS), is to replace compressed sensing (CS) on the
observation by CS on the least squares (LS) residual computed using the
previous estimate of the support. We bound CS-residual error and show that when
the number of available measurements is small, the bound is much smaller than
that on CS error if the sparsity pattern changes slowly enough. We also obtain
conditions for "stability" of LS-CS over time for a signal model that allows
support additions and removals, and that allows coefficients to gradually
increase (decrease) until they reach a constant value (become zero). By
"stability", we mean that the number of misses and extras in the support
estimate remain bounded by time-invariant values (in turn implying a
time-invariant bound on LS-CS error). The concept is meaningful only if the
bounds are small compared to the support size. Numerical experiments backing
our claims are shown.
