Kronecker vs. Dirac delta functions

Delta functions are used to sample time-domain signals in signal processing, but their type is often unstated or incorrectly specified. A Dirac delta function is a continuous function d(t-a) whose integral is exactly equal to 1 and is only nonzero at t=a. Hence, it is infinitesimal in width (its width approaches 0) and infinitely tall (its amplitude approaches infinity). The Dirac delta is applied to analog, continuous signals to return a continuous scaled impulse function at a specified instant of time (indicated by the domain of the Dirac delta). This is done by taking the integral over all values of the signal x(t) times the Dirac delta centered at a, d(t-a). The resulting function is a continuous function equal to x(a)*d(t-a).

The Kronecker delta function is similarly infinitesimally thin, but its amplitude is equal to 1, not its area. The function d[t-a] is equal to 1 when t=a and 0 otherwise, i.e., when t!=a. The Kronecker delta is a discrete function. It applies a discrete impulse to a continuous signal, and returns the original amplitude of the signal. Therefore, the resulting function is discrete, and equal to x[a]*d[t-a].

Since they can be written the same way (the square brackets to indicate that a function is discrete are not required), most texts will just call the delta function the Dirac type, if any. But if this were so, a discrete signal would have all infinite (positively and negatively) amplitudes! So, the difference is significant.

Wish I could have TeX’ed that but hopefully you found this helpful. The concept of a sampled function is really what is important here, and it represents the amplitude of a continuous signal at specific points of time. But a Dirac returns a continuous sampled function, and a Kronecker returns a discrete one.