Kronecker delta

The Kronecker delta appears naturally in many areas of mathematics, physics, engineering and computer science, as a means of compactly expressing its definition above.

and the last step is obtained by using the values of the Kronecker delta to reduce the summation over

In linear algebra, it can be thought of as a tensor, and is written

[1] In the study of digital signal processing (DSP), the unit sample function

represents a special case of a 2-dimensional Kronecker delta function

does not exist, and in fact, the Kronecker delta function and the unit sample function are different functions that overlap in the specific case where the indices include the number 0, the number of indices is 2, and one of the indices has the value of zero.

For the discrete unit sample function, it is more conventional to place a single integer index in square braces; in contrast the Kronecker delta can have any number of indexes.

In contrast, the typical purpose of the Kronecker delta function is for filtering terms from an Einstein summation convention.

The discrete unit sample function is more simply defined as:

does not have an integer index, it has a single continuous non-integer value t. To confuse matters more, the unit impulse function is sometimes used to refer to either the Dirac delta function

and if the integers are viewed as a measure space, endowed with the counting measure, then this property coincides with the defining property of the Dirac delta function

[2] In signal processing it is usually the context (discrete or continuous time) that distinguishes the Kronecker and Dirac "functions".

generally indicates continuous time (Dirac), whereas arguments like

Another common practice is to represent discrete sequences with square brackets; thus:

The Kronecker delta forms the multiplicative identity element of an incidence algebra.

For example, if a Dirac delta impulse occurs exactly at a sampling point and is ideally lowpass-filtered (with cutoff at the critical frequency) per the Nyquist–Shannon sampling theorem, the resulting discrete-time signal will be a Kronecker delta function.

Below, the version is presented has nonzero components scaled to be

in § Properties of the generalized Kronecker delta below disappearing.

[4] In terms of the indices, the generalized Kronecker delta is defined as:[5][6]

(the dimension of the vector space), in terms of the Levi-Civita symbol:

Kronecker Delta contractions depend on the dimension of the space.

The generalization of the preceding formulas is[citation needed]

which are the generalized version of formulae written in § Properties.

Reducing the order via summation of the indices may be expressed by the identity[9]

The 4D version of the last relation appears in Penrose's spinor approach to general relativity[10] that he later generalized, while he was developing Aitken's diagrams,[11] to become part of the technique of Penrose graphical notation.

[12] Also, this relation is extensively used in S-duality theories, especially when written in the language of differential forms and Hodge duals.

, the Kronecker delta can be written as a complex contour integral using a standard residue calculation.

The integral is taken over the unit circle in the complex plane, oriented counterclockwise.

An equivalent representation of the integral arises by parameterizing the contour by an angle around the origin.

It may be considered to be the discrete analog of the Dirac comb.