kronecker delta 与 dirac delta 关系

ref:http://en.wikipedia.org/wiki/Dirac_delta_function

Dirac delta :

The Dirac delta can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite,

and which is also constrained to satisfy the identity

[18]

This is merely aheuristiccharacterization. The Dirac delta is not a function in the traditional sense as no function defined on the real numbers has these properties.[17]The Dirac delta function can be rigorously defined either as adistributionor as ameasure.

Kronecker Delta:

is afunctionof twovariables, usuallyintegers. The function is 1 if the variables are equal, and 0 otherwise:

where Kronecker delta δijis apiecewisefunction of variables

and

. 这个representation可以表示为，

theta_ij==0 if i-j !=0;

theta_ij==1 if i-j ==0;

i.e. kronecker is the discrete analog of dirac!!!, kronecker usually consider integers!

Besides,

Inprobability theoryandstatistics, the Kronecker delta andDirac delta functioncan both be used to represent adiscrete distribution. If thesupportof a distribution consists of points

, with corresponding probabilities

, then theprobability mass function

of the distribution over

can be written, using the Kronecker delta, as

Equivalently, theprobability density function

of the distribution can be written using theDirac delta functionas

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 theNyquist–Shannon sampling theorem, the resulting discrete-time signal will be a Kronecker delta function.

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