# 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.