Biosignal Processing Fundamentals and Recent Applications with MATLAB (Stefan Bernhard, Andreas Brensing etc.)

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) =

{

if t = 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) =

√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⁾

{

|t| ≤^ϵ

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.