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.
Read more about this topic: Kronecker Delta
Famous quotes containing the words relationship to, relationship and/or function:
“Sometimes in our relationship to another human being the proper balance of friendship is restored when we put a few grains of impropriety onto our own side of the scale.”
—Friedrich Nietzsche (1844–1900)
“Henry David Thoreau, who never earned much of a living or sustained a relationship with any woman that wasn’t brotherly—who lived mostly under his parents’ roof ... who advocated one day’s work and six days “off” as the weekly round and was considered a bit of a fool in his hometown ... is probably the American writer who tells us best how to live comfortably with our most constant companion, ourselves.”
—Edward Hoagland (b. 1932)
“Nobody seriously questions the principle that it is the function of mass culture to maintain public morale, and certainly nobody in the mass audience objects to having his morale maintained.”
—Robert Warshow (1917–1955)