Kronecker delta

Not to be confused with the Dirac delta function.

In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker, is a function of two variables, usually integers, which is 1 if they are equal and 0 otherwise. So, for example,

$\delta_{1 \, 2} = 0$ , but

$\delta_{3 \, 3} = 1.$

It is written as the symbol δ_ij, and treated as a notational shorthand rather than as a function.

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

1 Alternate notation
2 Digital signal processing
3 Properties of the delta function
- 3.1 Linear algebra
4 Relationship to the Dirac delta function
5 Extensions of the delta function
6 Integral representations
7 The Kronecker comb
8 References
9 See also

Alternate notation

Using the Iverson bracket:

$\delta_{ij} = [i=j ].\,$

Often, the notation $\delta_i$ is used.

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

In linear algebra, it can be thought of as a tensor, and is written $\delta^i_j$ . Sometimes the Kronecker delta is called the substitution tensor. ^[1]

Digital signal processing

Similarly, in digital signal processing, the same concept is represented as a function on $\mathbb{Z}$ (the integers):

$\delta[n] = \begin{cases} 1, & n = 0 \\ 0, & n \ne 0.\end{cases}$

The function is referred to as an impulse, or unit impulse. And when it stimulates a signal processing element, the output is called the impulse response of the element.

Properties of the delta function

The Kronecker delta has the so-called sifting property that for $j\in\mathbb Z$ :

$\sum_{i=-\infty}^\infty a_i \delta_{ij} =a_j.$

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

$\int_{-\infty}^\infty \delta(x-y)f(x) dx=f(y),$

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, $\delta(t)\,$ 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: $\delta[n]\,$ . It is important to note that the Kronecker delta is not the result of directly sampling the Dirac delta function.

The Kronecker delta is used in many areas of mathematics.

Linear algebra

In linear algebra, the identity matrix can be written as $(\delta_{ij})_{i,j=1}^n\,$ .

If it is considered as a tensor, the Kronecker tensor, it can be written $\delta^i_j$ with a covariant index j and contravariant index i.

This (1,1) tensor represents:

The identity matrix, considered as a linear mapping
The trace
The inner product $V^* \otimes V \to K$
The map $K \to V^* \otimes V$ , representing scalar multiplication as a sum of outer products.

Relationship to the Dirac delta function

In probability theory and statistics, the Kronecker delta and Dirac delta function can both be used to represent a discrete distribution. If the support of a distribution consists of points $\mathbf{x} = \{x_1,\dots,x_n\}$ , with corresponding probabilities $p_1,\dots,p_n\,$ , then the probability mass function $p(x)\,$ of the distribution over $\mathbf{x}$ can be written, using the Kronecker delta, as

$p(x) = \sum_{i=1}^n p_i \delta_{x x_i}.$

Equivalently, the probability density function $f(x)\,$ of the distribution can be written using the Dirac delta function as

$f(x) = \sum_{i=1}^n p_i \delta(x-x_i).$

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.

Extensions of the delta function

In the same fashion, we may define an analogous, multi-dimensional function of many variables

$\delta^{j_1 j_2 \dots j_n}_{i_1 i_2 \dots i_n} = \prod_{k=1}^n \delta_{i_k j_k}.$

This function takes the value 1 if and only if all the upper indices match the corresponding lower ones, and the value zero otherwise.

Integral representations

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.

$\delta_{x,n} = \frac1{2\pi i} \oint_{|z|=1} z^{x-n-1} dz=\frac1{2\pi} \int_0^{2\pi} e^{i(x-n)\varphi} d\varphi$

The Kronecker comb

The Kronecker comb function with period N is defined (using digital notation) as:

$\Delta_N[n]=\sum_{k=-\infty}^\infty \delta[n-kN]$

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.

References

^ Trowbridge, 1998. Journal of Atmospheric and Oceanic Technology. V15, 1 p291