Delta function
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
Or it can be extended to two variables as
[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
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
to
) 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
If the value required to be discriminated is away from zero,
, where
is the target value, is used.
The interesting thing about this particular delta is its frequency domain representation: if we take
as the Fourier transform of a Dirac delta function, we have
. 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
, where
represents time and
represents the wave's frequency in hertz; the approximation is done by making the frequency
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:
-
iff
[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
.
If
denotes the space of infinitely differentiable functions on the real line which are 0 outside of some bounded set, one can consider the space
of linear functionals
which are continuous in some appropriate sense. The set of ordinary functions can be regarded as a subset of
, by identifying a function f with the map
defined by
. Thus the space
can be thought of as an enlargement of the space of functions. The delta "function" is another element of this space, defined by
, and so it's sometimes called a "generalized function".
An element
is called a distribution on
, 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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