The Dirac-Delta Function - The Impulse

Fourier Transform Mathematics
Previous: Fourier Transform (Home)
Previous: Intro to Complex Math

The Dirac-Delta function, also commonly known as the impulse function, is described on this page. This function (technically a functional) is one of the most useful in all of applied mathematics. To understand this function, we will several alternative definitions of the impulse function, in varying degrees of rigor.

1. Dirac-Delta: The Limit of a Sequence of Functions

Consider the function, fn(t), shown described by equation [1], and plotted in Figure 1 for n=1 and n=5:

sequence that leads to the dirac delta       [1]

unit pulse function

Figure 1. This is a square pulse with amplitude n and duration 1/n. (a)n=1 (b)n=5

The area, or integral of fn(t)=1 for every value of n=1,2,3,.... The larger n gets, the narrower the pulse is in time, but the amplitude increases such that the total area is the same for all values of n. The Dirac-Delta function can be thought of as the limit as n gets very large for the fn sequence of functions:

definition for dirac delta function       [2]

2. Dirac-Delta: The Derivative of the Step Function

The unit step function is defined as:

unit step function      [3]

The unit step is plotted in Figure 2:

unit step function

Figure 2. The unit step function.

The dirac-delta function can also be thought of as the derivative of the unit step function:

dirac-delta is the derivative of the unit step function       [4]

From equation [4], the dirac-delta can be thought of as being zero everywhere except where t=0, in which case it is infinite. This is an acceptable viewpoint for the dirac-delta impulse function, but it is not very rigorous mathematically:

dirac-delta is infinite at 0 and 0 everywhere else       [5]

3. Dirac-Delta: The Sifting Functional

Probably the most useful property of the dirac-delta, and the most rigorous mathematical defintion is given in this section. Consider any function g(t), that is continuous (and finite) at t=0. Then the following relationship always holds:

the sifting property of the fourier transform       [6]

In [6], the integration region just needs to contain t=0, and doesn't necessarily need to go from -infinity to +infinity. This property is extremely useful in signal processing, communication systems theory, quantum physics, etc. This is known as the 'sifting property' of the impulse function. The more general version, assuming g(t) is continuous and finite at g(a) is:

the general sifting property of the fourier transform       [7]

This property is your friend whenever an integral is involved: it makes the integration extremely simple.

Finally, as a further note on notation, the impulse (shifted to the right by 1, given in equation [8]) is plotted as shown in Figure 3. It is graphed as a vertical arrow, since it is zero everywhere, except where t=1, when it is more or less infinite.

impulse shifted       [8]

graph of impulse function

Figure 3. The graph of the Dirac-Delta Impulse Function.

Previous: Intro to Complex Math

Fourier Transform Mathematical Topics

Fourier Transforms (Home)