Overview
The Dirac Delta Function is defined to be a function {% \delta(x) %} such that
{% \delta(x) = 0 \, for \, x \neq 0 %}
{% \delta(x) = \infty \, for \, x = 0 %}
such that
{% {\displaystyle \int_{- \infty}^{\infty} } \delta(x) = 1 %}
The definition given shows that the Dirac Delta Function is not a function in the traditional sense.
Properties
The Dirac Delta function has the following property, which is sometimes taken as the defintion of the Dirac Function.
{% {\displaystyle \int_{- \infty}^{\infty} } dx f(x) \delta(x-x') = f(x') %}
This property makes it clear that the Dirac function is not a function in the traditional sense. It can be utilized by assuming that
whenever it appears, what is meant is a limit as shown below. A more formal definition and treatment was presented by
Schwarz with the concept of
generalized functions
(or distributions)
Limit Definition
One interpretation of the Dirac Delta Function is that it is the result of taking the limit of a sequence of functions.
{% \delta_n(x) = \frac{1}{\pi} \frac{sin(Nx)}{x} %}
Then, take
{% \delta(x) = \lim_{N \to \infty} \frac{1}{\pi} \frac{sin(Nx)}{x} %}
Discrete Time Definition
The delta function is used in discrete signal processing, and has a discrete time definition that also becomes the continuous time version in the limit.