# Dictionary:Impulse (δ(t))

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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

${\displaystyle \int _{-\infty }^{\infty }\phi (t)\delta (t-t_{0})\;dt=\phi (t_{0})}$

implying that ${\displaystyle \delta (t-t_{0})}$ only has support at ${\displaystyle t=t_{0}.}$ The notion of support in this case is that the limits of integration must contain the point ${\displaystyle t=t_{0}}$ for the integral to be nonzero.

The function ${\displaystyle \phi (t)}$ 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 ${\displaystyle C_{0}^{\infty }}$ which are the infinitely differentiable functions that vanish smoothly at infinity.

3. The second property of ${\displaystyle \delta (t-t_{0})}$ is that it has a unit integral

${\displaystyle \int _{-\infty }^{\infty }\delta (t-t_{0})\;dt=1.}$

Indeed, the definite integral

${\displaystyle \int _{-\infty }^{t}\delta (\tau -t_{0})\;d\tau ={\begin{cases}0&{\text{if }}tt_{0}\end{cases}}=H(t-t_{0})}$

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 ${\displaystyle t_{0}}$

${\displaystyle \lim _{n\rightarrow \infty }\int _{-\infty }^{\infty }S_{n}(t-t_{0})\;dt=1}$

and the limit of the sequence ${\displaystyle \left\{S_{n}(t)\right\}}$ must exhibit the sifting property

${\displaystyle \lim _{n\rightarrow \infty }\int _{-\infty }^{\infty }S_{n}(t-t_{0})\phi (t)\;dt=\phi (t_{0}).}$

There are many functions ${\displaystyle S_{n}(t-t_{0})}$ that have this property, including functions for which the limit of the function does not exist at ${\displaystyle t=t_{0}.}$

5. We may define, formally the derivative of a delta function (called delta prime) ${\displaystyle \delta ^{\prime }(t-t_{0})}$ via integration by parts with a test function

${\displaystyle \int _{-\infty }^{\infty }\phi (t)\delta ^{\prime }(t-t_{0})\;dt=\left.\phi (t)\right|_{-\infty }^{\infty }-\int _{-\infty }^{\infty }\phi ^{\prime }(t)\delta (t-t_{0})\;dt=-\phi ^{\prime }(t_{0}),}$

which follows because ${\displaystyle \phi (t)=0}$ at ${\displaystyle \pm \infty .}$

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

${\displaystyle \delta ^{*}(t)=\delta (t)+\left({\frac {i}{\pi }}\right)t}$.

8. See Kroenecker delta.

## References

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