FastSlowImpulse

FastSlowImpulse is a conceptual model used in signal processing and control theory to describe an impulse response that combines fast and slow dynamics. The model represents systems whose immediate reaction is followed by longer-lasting effects, enabling more accurate representation than a single-time-constant impulse response.

Structure and mathematical formulation: The fast component and the slow component are combined. A common formulation is h(t) = a_f δ(t) + a_s e^{-t/τ_s} u(t), where δ(t) is the Dirac delta, u(t) is the Heaviside step, a_f and a_s are amplitudes, and τ_s is the slow time constant. Variants may replace the delta with a short-duration pulse or include multiple slow modes (a sum of exponentials). This form captures an instantaneous spike followed by a decaying tail.

Estimation and usage: Parameters can be identified from measured impulse responses using regression, system identification, or sparse optimization. The model is useful in capturing both instantaneous effects and lingering responses in mechanical, electrical, or acoustic systems, and serves as a parsimonious alternative to higher-order pole-zero models.

Applications: Digital filters, actuator dynamics, room acoustics, seismology, and any domain where fast transients coexist with slower tails. The concept is often employed in model-based control to design compensators that target distinct time scales, improving response accuracy without excessive model complexity.

Limitations and extensions: The two-term form assumes linear time-invariant behavior and may misrepresent nonlinear dynamics or non-exponential tails. Extensions include multiple slow components, non-exponential decay, or frequency-domain representations. Related concepts include impulse response and multi-rate system identification.