http://en.wikipedia.org/wiki/Kronecker_delta

Kronecker delta

Inmathematics,theKroneckerdeltaorKronecker'sdelta, named afterLeopoldKronecker, is afunctionoftwovariables,usuallyintegers, which is 1 ifthey are equal and 0 otherwise. So, for example,

,but

It iswritten as the symbolδ_ij,and treated as a notational shorthand rather than as afunction.

[edit]Alternate notation

UsingtheIversonbracket:

Often, thenotationδ_iis used.

Inlinear algebra,it can be thought of as atensor, and iswritten.Sometimes the Kronecker delta is called the substitutiontensor.^[1]

[edit]Digital signal processing

Similarly,indigitalsignal processing, the same concept is represented as afunction on(theintegers):

The functionis referred to as animpulse,orunitimpulse. And when it stimulates a signal processing element,the output is called theimpulseresponseof theelement.

[edit]Properties of the delta function

TheKronecker delta has the so-calledsiftingpropertythat for:

and if theintegers are viewed as ameasure space,endowed with thecountingmeasure, then this property coincides with the definingproperty of theDirac deltafunction

and in factDirac's delta was named after the Kronecker delta because of thisanalogous property. In signal processing it is usually the context(discrete or continuous time) that distinguishes the Kronecker andDirac "functions". And by convention,generallyindicates continuous time (Dirac), whereas argumentslikei,j,k,l,m,andnareusually reserved for discrete time (Kronecker). Another commonpractice is to represent discrete sequences with square brackets;thus: .It is important to note that the Kronecker delta is not the resultof directly sampling the Dirac delta function.

TheKronecker delta is used in many areas of mathematics.

[edit]Linear algebra

Inlinear algebra,theidentitymatrixcan be writtenas.

If it isconsidered as atensor, the Kroneckertensor, it can be writtenwithacovariantindexjandcontravariantindexi.

This (1,1)tensor represents:

[edit]Relationship to theDirac deltafunction

Inprobabilitytheoryandstatistics, theKronecker delta andDirac deltafunctioncan both be usedto represent adiscretedistribution. If thesupportofa distribution consists of points,with corresponding probabilities,then theprobabilitymass functionofthe distribution overcanbe written, using the Kronecker delta, as

Equivalently, theprobabilitydensity functionofthe distribution can be written using theDirac deltafunctionas

Undercertain conditions, the Kronecker delta can arise from sampling aDirac delta function. For example, if a Dirac delta impulse occursexactly at a sampling point and is ideally lowpass-filtered (withcutoff at the critical frequency) per theNyquist–Shannonsampling theorem, the resulting discrete-time signal will be aKronecker delta function.

[edit]Extensions of the delta function

In the samefashion, we may define an analogous, multi-dimensional function ofmany variables

Thisfunction takes the value 1 if and only if all the upper indicesmatch the corresponding lower ones, and the value zerootherwise.

[edit]Integral representations

For anyintegern, using astandardresiduecalculationwe can write an integral representation for the Kronecker delta asthe integral below, where the contour of the integral goescounterclockwise around zero. This representation is alsoequivalent to a definite integral by a rotation in the complexplane.

[edit]The Kronecker comb

TheKronecker comb function with periodNisdefined (using digital notation) as:

whereNandnareintegers. The Kronecker comb thus consists of an infinite series ofunit impulsesNunitsapart, and includes the unit impulse at zero. It may be consideredto be the discrete analog of theDirac comb.

[edit]Kronecker Integral^[2]

TheKronecker delta is also called degree of mapping of one surfaceinto another. Suppose a mapping takes place fromsurfaceS_uvwtoS_xyzthatare boundaries of regions,R_uvwandR_xyzwhichis simply connected with one-to-one correspondence. In thisframework, if s and t are parameters forS_uvw,andS_uvwtoS_xyzareeach oriented by the outer normal n:

u=u(s,t),v=v(s,t),w=w(s,t),

while thenormal has the direction of:

Letx=x(u,v,w),y=y(u,v,w),z=z(u,v,w) be defined and smooth in a domaincontainingS_uvw, and letthese equations define the mapping ofS_uvwintoS_xyz. Then thedegreeδof mappingis1 / 4πtimes the solidangle of the image S ofS_uvwwithrespect to the interior point ofS_xyz, O. If Ois the origin of the region,R_xyz, then thedegree,δis given by theintegral:

[edit]References

1. ^Trowbridge, 1998.Journal of Atmospheric and Oceanic Technology. V15, 1 p291
2. ^Kaplan, Wilfred(2003),AdvancedCalculus, Pearson Education. Inc,p.364,ISBN0-201-79937-5

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