Dirac Delta Function: Definition & Samples

[EngWiPy]

Hello,

Dirac Delta Function is defined as the function that its amplitude is zero everywhere except at zero where its amplitude is infinitely large such that the area under the curve is unity.

Sometimes it is used to describe a function consists of a sequence of samples such as:

[tex]g_{\delta}(t)=\sum_{n=-\infty}^{\infty}g(nT)\,\delta(t-nT)[/tex]

How this weighting affect the amplitude? I mean what is the amplitude of [tex]0.4\,\delta(t)[/tex]?

Regards

 

[arildno]

Strictly speaking, the delta "function" can only be regarded as a proper function if we regard it as a function of two variables:

1. First variable:
its domain as a set of FUNCTIONS on R, rather than on R itself
2. Second variable:
An INTERVAL of R

The Dirac "function" thus defined is a functional, rather than a standard function.

Thus, given some function f(x), and an interval I lying within the domain of f, we have


D(f,I)=f(0), if 0 is in I
D(f,I)=0, if 0 is NOT in I

This definition makes D in what we call a distribution.

Note that for any f and I, it is utterly trivial to compute D's "values".


Unfortunately, this trivial sampling functional has gained notoriety by improper understanding of how it can be REPRESENTED in terms of an integral operator (typically, as the "limit" of spike functions)

To delve into these issues, you may look at my tutorial:
https://www.physicsforums.com/showthread.php?t=73447