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In this note we study the conditions for convergence of recently introduceddynamic regressor extension and mixing (DREM) parameter estimator when theextended regressor is generated using LTI filters. In particular, we areinterested in relating these conditions with the ones required for convergenceof the classical gradient (or least squares), namely the well-known persistentexcitation (PE) requirement on the original regressor vector, $\phi(t) \in\mathbb{R}^q$, with $q \in \mathbb{N}$ the number of unknown parameters.Moreover, we study the case when only interval excitation (IE) is available,under which DREM, concurrent and composite learning schemes ensure globalconvergence, being the convergence for DREM in finite time. Regarding PE weprove that, under some mild technical assumptions, if $\phi(t)$ is PE then thescalar regressor of DREM, $\Delta(t) \in \mathbb{R}$, is also PE, ensuringexponential convergence. Concerning IE we prove that if $\phi(t)$ is IE then$\Delta(t)$ is also IE. All these results are established in the almost suresense, namely proving that the set of filter parameters for which the claims donot hold is of zero measure. The main technical tool used in our proof isinspired by a study of Luenberger observers for nonautonomous nonlinear systemsrecently reported in the literature.

Conditions for Convergence of Dynamic Regressor Extension and Mixing Parameter Estimator Using LTI Filters