Stability (over time) of modified-CS for recursive causal sparse reconstruction

In this work, we obtain sufficient conditions for the “stability” of our recently proposed algorithm, modified-CS (for noisy measurements), designed for recursive reconstruction of sparse signal sequences from noisy measurements. By “stability” we mean that the number of misses from the current support estimate and the number of extras in it remain bounded by a time-invariant value at all times. The concept is meaningful only if the bound is small compared to the current signal support size. A direct corollary is that the reconstruction errors are also bounded by a time-invariant and small value. We show that “stability” holds under mild assumptions (bounded noise, high enough SNR and large enough measurements at every time) for a simple deterministic signal model with fixed signal power and support set size; support set changes allowed at every time; and gradual coefficient magnitude increase/decrease.