Kronecker Delta - Relationship To The Dirac Delta Function

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, 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.