React or Predict? A Spectral Rule for Wireless Threshold Detection

Abstract

A wireless sensor must alert a remote monitor before a monitored process crosses a safety threshold; an alarm arriving afterward may be too late. The sensor can react to its current estimate or predict ahead and trigger earlier, but the value of such lookahead is not obvious. In some systems it creates an early-alarm opportunity unavailable to the current test, while in others it cannot cross the alarm boundary. This letter gives a practical three-stage rule for deciding when to predict. First, an algebraic spectral test decides at design time whether lookahead is structurally useful: it is redundant exactly when the threshold direction is a left-eigenvector of the dynamics with a non-negative eigenvalue. Second, a closed-form channel decomposition shows that deeper prediction becomes more valuable as the channel degrades, because longer lead windows permit more pre-crossing transmission attempts. Third, simulations show that large gains also require retained prediction magnitude; oscillatory dynamics amplify the benefit through rotation, and a two-sensor setting reveals a sensing-channel tradeoff.