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2 Fundamentals of Information, Signal and System Theory
Another important signal form in signal processing is the so-called delta-distribution
(also called δ-function, Dirac-function, Dirac-pulse⁸ and unit impulse ). Although for
the function value of this distribution holds: δ(t = 0) = ∞, in discrete signal pro-
cessing it is defined for numerical reasons as follows:
δ(t) =
{
{
{
1
if t = 0
0
otherwise t
̸= 0
,
t ⊂ℝ.
(2.20)
This distribution, like all other distributions, can be understood as the limit of a func-
tion series such as the Dirac series. In the following, two common approximations for
the delta-distribution δϵ(t) are given. In the limit ϵ →0 the continuously differenti-
able normal distribution
δϵ(t) =
1
√2πϵ
e(−t2
2ϵ )
(2.21)
produces functions with very narrow and high maxima at t = 0. The area under the
functions always has the value one which is a conservation variable for all ϵ. Con-
sequently, the mean width √ϵ →0 becomes narrower and narrower in the limit trans-
ition, while the height 1/√ϵ →∞of the function conversely increases strongly. For
ϵ →0, this results in an infinitely narrow and infinitely high momentum, the so-
called Dirac momentum (cf. Figure 2.14). Since the values "infinitely narrow" and "in-
finitely high" are not usable in discrete signal processing for numerical reasons, one
has agreed on the representation of the weight . The weight corresponds exactly to the
area of the pulse and thus has the value one.
Another common approximation results from the only piecewise continuously dif-
ferentiable function of the rectangular pulse
δϵ(t) = rect(t/ϵ)
ϵ
=
{
{
{
1
ϵ
|t| ≤ϵ
2
0
otherwise
.
(2.22)
In this case, the limit value consideration ϵ →0 leads to an infinitely narrow and
infinitely high impulse with an area of one. Regardless of the Dirac series used, the
delta-distribution has special properties that play an important role in the digitisation
of signals. To be mentioned is the so-called equation property or sieve property
⟨δ, f⟩=
∞
∫
−∞
δ(t) f(t) dt = f(0) ,
(2.23)
which, when multiplying a function f(t) by the delta-distribution, hides all function
values for t
̸= 0, i.e. only the product at the point t = 0 is different from zero and has
8 After the physicist Paul Dirac.