The unit step function, often denoted as $u(t)$, is a piecewise function that jumps from 0 to 1 at the origin, effectively acting as a signal that turns on at a specific point in time. This function is crucial in systems analysis, especially in the context of linear time-invariant systems, as it serves as an input that allows us to analyze the system's response from rest to a constant value. By applying the unit step function, we can explore how systems behave when transitioning from an inactive to an active state.
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