For stable reachable linear systems, the state is persistently exciting exactly when a window of input shifts or derivatives is persistently exciting, with weaker partial-excitation conditions needed for multi-input systems.
Lyapunov techniques for the exponential stability of linear difference equations with random coefficients,
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On Sufficient Richness for Linear Time-Invariant Systems
For stable reachable linear systems, the state is persistently exciting exactly when a window of input shifts or derivatives is persistently exciting, with weaker partial-excitation conditions needed for multi-input systems.