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The graph of the Dirac delta is usually thought of as following the whole x-axis and the positive y-axis. [5]: 174 The Dirac delta is used to model a tall narrow spike function (an impulse), and other similar abstractions such as a point charge, point mass or electron point.
In classical mechanics, impulse (symbolized by J or Imp) is the change in momentum of an object. If the initial momentum of an object is p 1, and a subsequent momentum is p 2, the object has received an impulse J: =. Momentum is a vector quantity, so impulse is also a vector quantity.
The impulse can be modeled as a Dirac delta function for continuous-time systems, or as the discrete unit sample function for discrete-time systems. The Dirac delta represents the limiting case of a pulse made very short in time while maintaining its area or integral (thus giving an infinitely high peak).
In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if is a linear differential operator, then
The graph of the Dirac comb function is an infinite series of Dirac delta functions spaced at intervals of T. In mathematics, a Dirac comb (also known as sha function, impulse train or sampling function) is a periodic function with the formula := = for some given period . [1]
The graph on the right shows the impulse response of two similar systems. The green curve is the response of the system with impulse response () =, while the blue represents the system () = (). Although one response is oscillatory, both return to the original value of 0 over time.
First order LTI systems are characterized by the differential equation + = where τ represents the exponential decay constant and V is a function of time t = (). The right-hand side is the forcing function f(t) describing an external driving function of time, which can be regarded as the system input, to which V(t) is the response, or system output.
The impulse response and step response are transient responses to a specific input (an impulse and a step, respectively). In electrical engineering specifically, the transient response is the circuit’s temporary response that will die out with time. [ 1 ]