Dirac Delta Function - Fourier Transform

Fourier Transform

The delta function is a tempered distribution, and therefore it has a well-defined Fourier transform. Formally, one finds

Properly speaking, the Fourier transform of a distribution is defined by imposing self-adjointness of the Fourier transform under the duality pairing of tempered distributions with Schwartz functions. Thus is defined as the unique tempered distribution satisfying

for all Schwartz functions φ. And indeed it follows from this that

As a result of this identity, the convolution of the delta function with any other tempered distribution S is simply S:

That is to say that δ is an identity element for the convolution on tempered distributions, and in fact the space of compactly supported distributions under convolution is an associative algebra with identity the delta function. This property is fundamental in signal processing, as convolution with a tempered distribution is a linear time-invariant system, and applying the linear time-invariant system measures its impulse response. The impulse response can be computed to any desired degree of accuracy by choosing a suitable approximation for δ, and once it is known, it characterizes the system completely. See LTI system theory:Impulse response and convolution.

The inverse Fourier transform of the tempered distribution ƒ(ξ) = 1 is the delta function. Formally, this is expressed

and more rigorously, it follows since

for all Schwartz functions ƒ.

In these terms, the delta function provides a suggestive statement of the orthogonality property of the Fourier kernel on R. Formally, one has

This is, of course, shorthand for the assertion that the Fourier transform of the tempered distribution

is

which again follows by imposing self-adjointness of the Fourier transform.

By analytic continuation of the Fourier transform, the Laplace transform of the delta function is found to be