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The restriction to positive integers is common, but there is no reason it cannot have negative integers as well as positive, or any discrete rational numbers. If i and j above take rational values, then for example
This latter case is for convenience. However, the Kronecker delta is not defined for complex numbers.
The following equations are satisfied:
Therefore, the matrix ? can be considered as an identity matrix.
Another useful representation is the following form:
Often, a single-argument notation ?i is used, which is equivalent to setting j = 0:
In linear algebra, it can be thought of as a tensor, and is written ?i j. Sometimes the Kronecker delta is called the substitution tensor.
Digital signal processing
Unit sample function
In the study of digital signal processing (DSP), the unit sample function represents a special case of a 2-dimensional Kronecker delta function where the kronecker indices include the number zero, and where one of the indices is zero. In this case:
Or more generally where:
However, this is only a very special case. In Tensor calculus, it is more common to number basis vectors in a particular dimension starting with index 1, rather than index 0. In this case, the relation doesn't exist, and in fact, the Kronecker delta function and the unit sample function are really different functions that by chance overlap in one 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.
While the discrete unit sample function and the Kronecker delta function use the same letter, they differ in the following ways. 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. Further, the purpose of the discrete unit sample function is different from the Kronecker delta function. In DSP, the discrete unit sample function is typically used as an input function to a discrete system for discovering the system function of the system which will be produced as an output of the system. 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:
In addition, DSP has a function called the Dirac delta function, that is often confused for both the Kronecker delta function and the unit sample function. The Dirac Delta is defined as:
Unlike the Kronecker delta function and the unit sample function , the Dirac Delta function doesn't have a integer index, it has a single continuous non-integer value t.
and in fact Dirac's delta was named after the Kronecker delta because of this analogous property. In signal processing it is usually the context (discrete or continuous time) that distinguishes the Kronecker and Dirac "functions". And by convention, ?(t) generally indicates continuous time (Dirac), whereas arguments like i, j, k, l, m, and n are usually reserved for discrete time (Kronecker). Another common practice is to represent discrete sequences with square brackets; thus: ?[n]. The Kronecker delta is not the result of directly sampling the Dirac delta function.
Under certain conditions, the Kronecker delta can arise from sampling a Dirac delta function. 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.
The map K -> V* ? V, representing scalar multiplication as a sum of outer products.
The generalized Kronecker delta or multi-index Kronecker delta of order 2p is a type (p,p) tensor that is a completely antisymmetric in its p upper indices, and also in its p lower indices.
Two definitions that differ by a factor of p! are in use. Below, the version is presented has nonzero components scaled to be ±1. The second version has nonzero components that are ±, with consequent changes scaling factors in formulae, such as the scaling factors of in § Properties of the generalized Kronecker delta below disappearing.
Definitions of the generalized Kronecker delta
In terms of the indices, the generalized Kronecker delta is defined as:
For any integer n, using a standard residue calculation we can write an integral representation for the Kronecker delta as the integral below, where the contour of the integral goes counterclockwise around zero. This representation is also equivalent to a definite integral by a rotation in the complex plane.
The Kronecker comb
The Kronecker comb function with period N is defined (using DSP notation) as:
where N and n are integers. The Kronecker comb thus consists of an infinite series of unit impulses N units apart, and includes the unit impulse at zero. It may be considered to be the discrete analog of the Dirac comb.
The Kronecker delta is also called degree of mapping of one surface into another. Suppose a mapping takes place from surface Suvw to Sxyz that are boundaries of regions, Ruvw and Rxyz which is simply connected with one-to-one correspondence. In this framework, if s and t are parameters for Suvw, and Suvw to Suvw are each oriented by the outer normal n:
while the normal has the direction of
Let x = x(u,v,w), y = y(u,v,w), z = z(u,v,w) be defined and smooth in a domain containing Suvw, and let these equations define the mapping of Suvw onto Sxyz. Then the degree ? of mapping is times the solid angle of the image S of Suvw with respect to the interior point of Sxyz, O. If O is the origin of the region, Rxyz, then the degree, ? is given by the integral: