Delta function

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This article/section deals with mathematical concepts appropriate for a student in late high school or early university.


Delta function is a Godsend to students as a way to discriminate numbers; it usually comes in two flavours.

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[edit] Kronecker's Delta function

Also known as the unit impulse, it is defined as a function that has value of 1 if the variables are 0, and 0 if they are not. As a single variable function, it is usually written as

\delta_{i} = \begin{cases} 1, & i = 0 \\ 0, & i \ne 0 \end{cases}

Or it can be extended to two variables as

\delta_{ij} = \left\{\begin{matrix} 
1, & \mbox{if } i=j   \\ 
0, & \mbox{if } i \ne j   \end{matrix}\right.


[edit] Dirac Delta function

Paul Dirac says, "You can't integrate that and end up with meaningful stuff!" and goes off to invent a function that works with integration. It is defined as

\delta(x) = \lim_{a \to 0}\frac{1}{a \sqrt{\pi}} \mathrm{e}^{-x^2/a^2}

At the limit, it has a nonzero value if the variables is 0, and 0 if they are not. Since the total area over the real line (from -\infty to \infty) is defined to be 1, the value at 0 would be infinite. That gives the nickname of infinite impulse and it is usually written as

\delta(x) = \begin{cases} \infty, & x = 0 \\ 0, & x \ne 0 \end{cases}

If the value required to be discriminated is away from zero, \delta(x-y), where y is the target value, is used.

The interesting thing about this particular delta is its frequency domain representation: if we take F(\omega) as the Fourier transform of a Dirac delta function, we have F(\omega) = 1. This means the following things:

  • A Dirac pulse function has constant amplitude in the entire frequency spectrum. This means we must be careful when transmitting a delta function over a transmission channel, because it will interfere with every single other transmission made in the medium's passband.
  • Dirac pulses have infinite energy, because their bandwidth is infinite. Therefore, they're impossible to transmit on a real system - to do so, the normal practice is to approximate the Dirac pulse with the cardinal sine function sinc(2 \pi f T) = sin(2 \pi f T) / 2 \pi f T, where T represents time and f represents the wave's frequency in hertz; the approximation is done by making the frequency f as ludicrously high as possible.

[edit] Working with integration

The way the Dirac delta function is used in integration is given by the following identity:

\int_{a}^b f(x) \, \delta(x-y)dx =  f(y) iff a<y<b

[edit] What is it, really?

Strictly speaking, the delta function isn't actually a function at all: a function must have a real value at every point, and delta doesn't. What it is is a "generalized function" or "distribution": it's a gadget that when fed a function, spits out a real number, in this case f(y).

If \mathcal D(\mathbb R) denotes the space of infinitely differentiable functions on the real line which are 0 outside of some bounded set, one can consider the space \mathcal D^\prime(\mathbb R) of linear functionals  T : \mathcal D(\mathbb R) \to \mathbb R which are continuous in some appropriate sense. The set of ordinary functions can be regarded as a subset of \mathcal D^\prime(\mathbb R), by identifying a function f with the map T_f \in \mathcal D^\prime(\mathbb R) defined by T_f(g) = \int_{-\infty}^\infty fg \, dx. Thus the space \mathcal D^\prime(\mathbb R) can be thought of as an enlargement of the space of functions. The delta "function" is another element of this space, defined by \delta(g) = g(0), and so it's sometimes called a "generalized function".

An element \mathcal D^\prime(\mathbb R) is called a distribution on \mathbb R, and it turns out that it's possible to carry out many familiar operations on these generalized functions. For example, the delta function can be regarded in a very rigorous sense as being the derivative of the unit step function! Fourier analysis too carries over to the space of distributions largely intact.



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