Stability of threshold-based sampling as metric problem

Threshold-based sampling schemes such send-on-delta, level-crossing with hysteresis and integrate-and-fire are studied as non-linear input-output systems that map Lipschitz continuous signals to event sequences with -1 and 1 entries. By arguing that stability requires an event sequence of alternating -1 and 1 entries to be close to the zero-sequence w.r.t. the given event metric, it is shown that stability is a metric problem. By introducing the transcription operator T, which cancels subsequent events of alternating signs, a necessary criterion for stability is derived. This criterion states that a stable event metric preserves boundedness of an input signal w.r.t to the uniform norm. As a byproduct of its proof a fundamental inequality is deduced that relates the operator T with Hermann Weyl´s discrepancy norm and the uniform norm of the input signal.