This paper explores the concept of persistency of excitation in the context of linear time-invariant (LTI) systems, focusing on the behavioral language. The main result demonstrates that if a component of the response signal is persistently exciting of sufficiently high order, then the observed windows of the signal span the full system behavior. Specifically, if the input component is persistently exciting of order \( L + n \), where \( n \) is the dimension of the state space, then the left kernel of the Hankel matrix formed from these windows contains the annihilators of the system, and the row span of the Hankel matrix equals the space of all possible windows of the system. The paper also discusses the conditions under which the state trajectory of a state representation spans the whole state space, and the importance of the matrix formed from a state sequence having linearly independent rows compared to an input sequence and its shifts in subspace system identification. Key concepts include the variable cardinality, input cardinality, output cardinality, state cardinality, lag, and shortest lag, which are crucial for understanding the system's behavior and the conditions for successful system identification. Corollaries are provided to illustrate the implications of the main theorem, including conditions for recovering the system's matrices from input/state/output trajectories and the number of data points required for system identification. The research is supported by various grants and projects from the Belgian Federal Government and other institutions.This paper explores the concept of persistency of excitation in the context of linear time-invariant (LTI) systems, focusing on the behavioral language. The main result demonstrates that if a component of the response signal is persistently exciting of sufficiently high order, then the observed windows of the signal span the full system behavior. Specifically, if the input component is persistently exciting of order \( L + n \), where \( n \) is the dimension of the state space, then the left kernel of the Hankel matrix formed from these windows contains the annihilators of the system, and the row span of the Hankel matrix equals the space of all possible windows of the system. The paper also discusses the conditions under which the state trajectory of a state representation spans the whole state space, and the importance of the matrix formed from a state sequence having linearly independent rows compared to an input sequence and its shifts in subspace system identification. Key concepts include the variable cardinality, input cardinality, output cardinality, state cardinality, lag, and shortest lag, which are crucial for understanding the system's behavior and the conditions for successful system identification. Corollaries are provided to illustrate the implications of the main theorem, including conditions for recovering the system's matrices from input/state/output trajectories and the number of data points required for system identification. The research is supported by various grants and projects from the Belgian Federal Government and other institutions.