Dirac delta function

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The Dirac delta function, or δ function, is (informally) a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one.[1][2] It was introduced by theoretical physicist Paul Dirac. In the context of signal processing it is often referred to as the unit impulse function. It is a continuous analog of the Kronecker delta function which is usually defined on a finite domain, and takes values 0 and 1.

From a purely mathematical viewpoint, the Dirac delta is not strictly a function, because any real function that is equal to zero everywhere but a single point must have total integral zero.[3] While for many purposes the Dirac delta can be manipulated as a function, formally it can be defined as a distribution that is also a measure. In many applications, the Dirac delta is regarded as a kind of limit (a weak limit) of a sequence of functions having a tall spike at the origin. The approximating functions of the sequence are thus "approximate" or "nascent" delta functions.

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