pulseintegral

Pulse integral refers to the integral of a pulse-shaped signal over time. In signal processing and physics, the pulse integral is often interpreted as the total amount carried by the pulse, such as charge, energy, or impulse, depending on the physical quantity represented by the signal.

For a time-domain signal x(t) that is nonzero only during a finite interval, the pulse integral is ∫_{-∞}^{∞} x(t) dt. If the pulse is rectangular with amplitude A on [t1, t2], the integral equals A(t2 − t1). More generally, the integral equals the area under the pulse curve.

Applications of the pulse integral span several fields. In electrical circuits, if x(t) is current i(t), the pulse integral gives total transferred charge Q = ∫ i(t) dt. In optics or photonics, if x(t) represents instantaneous power P(t), the integral yields pulse energy E = ∫ P(t) dt. In communications, the integral relates to the energy per symbol or to the impulse response of a channel when the pulse is used to excite it.

Relation to impulse: A sharp, short pulse can be treated as an approximation to an impulse with strength equal to its integral. The impulse delta δ(t) has unit integral ∫ δ(t) dt = 1. The pulse integral generalizes this idea to finite-width signals, capturing the total effect of the pulse.

Discrete setting and computation: when sampling, the integral is approximated by a sum: ∑ x[n] Δt, where Δt is the sampling interval. The operation is linear: the integral of a linear combination is the corresponding combination of integrals.