thresholdcrossing

Threshold crossing is the event when a signal's amplitude passes through a specified threshold value, typically from below to above (up-crossing) or from above to below (down-crossing). They are used to detect occurrences, trigger actions, or characterize the dynamics of a system. Thresholds can be fixed or adaptive and may be defined for the entire signal or for components such as baseline or noise levels.

Mathematically, given a sequence x[n] or a function x(t) and a threshold T, a crossing occurs at a time t* where x(t*) = T and x(t) crosses the threshold with a change of sign in x(t) - T. In discrete data, crossings are often detected by examining consecutive samples: a crossing occurs when (x[n-1] - T) and (x[n] - T) have opposite signs. The crossing time is commonly estimated by linear interpolation between samples around the crossing.

Applications span multiple fields. In engineering and electronics, threshold crossing is used to trigger circuits or sampling events. In signal processing, it supports event detection and envelope or peak detection. In finance, crossing a price threshold can activate barrier options or stop orders. In neuroscience, threshold crossing is associated with the initiation of action potentials, and researchers study spike trains via crossing events.

Challenges include noise-induced false crossings, especially when the signal is noisy or has high-frequency components. Techniques such as hysteresis with two thresholds, smoothing, or enforcing a minimum dwell time help reduce false detections. In continuous-time analysis, concepts like first passage time and crossing rate quantify how often a process crosses a threshold.