Signals as sums of weighted delta functions

Any discrete-time signal x: Integers → Reals can be given as a sum of weighted Kronecker delta functions,

What this says is really trivial. Each term in the summation is of the form x(k)δ (n − k). This term, by itself, defines a signal that is zero everywhere except at n = k, where it has value x(k). This term is called a weighted delta function because it is a (time shifted) delta function with a specified weight. Thus, the above summation can be viewed as a way to describe a signal as a composition of weighted delta functions, much the way the Fourier series describes a signal as a composition of complex exponential functions.

The continuous-time version of this is similar, except that the summation becomes an integral (integration, after all, is just summation over a continuum). Given any signal x: Reals → Reals

Although this is mathematically much more subtle than the discrete-time case, it is very similar in structure. It describes a signal x as a sum (or more precisely, an integral) of weighted Dirac delta functions.

Week 11

Signals as sums of weighted delta functions