# Dictionary:Impulse (δ(t))

(im’ puls)

**1**. The limit of a pulse of unit area as its width approaches zero and its height approaches infinity. Also called Dirac function and **delta function** and symbolized by δ(*t*).

The Dirac delta *function* is not mathematically
a *function* but is a different category of mathematical object called a *distribution.* The introduction of the delta function
is attributed to physicist Paul A. M. Dirac, but the necessity of such an object was alluded to in the discussion of
the cascade of forward and inverse Fourier transforms in *Theorie de Chaleur* by
Joseph Fourier. ^{[1]} The formal mathematical
theory was introduced by mathematician Laurent Schwartz.^{[2]}
Another direction of representing the Dirac delta function as a *generalized function,* as presented by Lighthill, 1964.^{[3]}

**2**. The fundamental properties of the Dirac delta function include the *sifting property*

implying that only has support at The notion of *support*
in this case is that the limits of integration must contain the point for the integral
to be nonzero.

The function
is called a *test function* and is any function such that this integral exists (i.e. doesn't blow up). The most general
class of these functions are called the which are the infinitely differentiable functions
that vanish smoothly at infinity.

**3**. The second property of is that it has a unit integral

Indeed, the definite integral

thus, the integral of the delta function is the Heaviside step function. Conversely, the derivative of a step function is a delta function. Hence, distributions extend our ability to define differentiation to cases for which the derivative is not defined classically.

**4**. While it appears in many engineering and less formal texts that the value of delta function has an infinite value
where its argument is zero, this is not correct, because the delta function has no intrinsic meaning outside of integration
with a test function.

What is more correct is to consider defining the delta function as the limit of a sequence of strongly peaked functions that have, in the limit, support at the desired value of

and the limit of the sequence must exhibit the sifting property

There are many functions that have this property, including functions for which the limit
of the function does not exist at

**5**. We may define, formally the derivative of a delta function (called *delta prime*) via integration by parts with a test function

which follows because at

Higher derivatives are defined by successive integration by parts.

See *impulse response*.

**6**. A pulse that is of sufficiently short time-duration that its waveshape is of no consequence.

**7**. A **complex impulse** δ^{*}(*t*) or complex delta function is defined as an analytic
signal, through the construction
of an imaginary part via the Hilbert transform as

**8**. See *Kroenecker delta*.

## References

- ↑ Fourier, Joseph. Theorie analytique de la chaleur, par M. Fourier. Chez Firmin Didot, père et fils, 1822.
- ↑ Schwartz, Laurent. "Theory des distributions." Hermann, Paris (1950).
- ↑ Lighthill, Michael J. An introduction to Fourier analysis and generalised functions. Cambridge University Press, 1964.