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 a heuristic characterization. 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 a distribution or as a measure.
Kronecker Delta:
is a function of two variables, usually integers. The function is 1 if the variables are equal, and 0 otherwise:
where Kronecker delta δij is a piecewise function of variables
and
. 这个representation可以表示成，
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,
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
, with corresponding probabilities
, then the probability mass function
of the distribution over
can be written, using the Kronecker delta, as
Equivalently, the probability density function
of the distribution can be written using the Dirac delta function as
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.
ref: http://en.wikipedia.org/wiki/Kronecker_delta
