Linear and non-linear processes - Signals, Processes, and Systems - page 201

Note: Our eyes do this, for instance, with television. Instead of 50 (half-) pictures
a second we see a continuous „averaged“ sequence of images.

As a result of integration the curve of a signal is changed in a certain, mathematically very
important way. The simplest of all functions, the constant function u(t) = K becomes by
integration a linear function u(t) =K < t, as shown by Illustration 137. Is there a rule which
occurs in the case of repeated integration and which can be reversed by differentiation?
Illustration 139 shows the relationships involved:

• A constant function of the type u(t) = K

becomes by integration a linear function of the type u(t) = K < t

This becomes by integration a „quadratic“ function of the type u(t) = K 1 < t 2

(whereby K 1 = K/2 )

By integration this becomes a „cubic“ function of the type u(t) = K 2 <

t 3

whereby K 2 = K 1 /3) etc. This is only precisely true for the

integration carried out (blockwise) with signal curves („specific integral“).

• By (multiple) differentiation it is possible to reverse this (multiple) integration as
shown in Illustration 134.

Integration is in practice such an important operation - which everyone ought to be aware
of - that we almost forgot the most important question: what happens in the case of
integration to a sinusoidal signal?

This is shown by Illustration 140. It shows the behaviour expected if we regard integration
as the reversal of differentiation, i.e. differentiation of the integrated signal must result in
the original signal.

Malicious functions or signal curves
Not just for the sake of completeness we ought at this point to mention certain functions
or signal curves which can be integrated but not differentiated.

There is for instance the „chaotic“ noise signal“. If you differentiate this signal you will
obtain a result. But noise has a property which normal signals do not have - the successive
random values change in „steps“. The computer insists on measuring the gradient from
the difference between two neighbouring random results and the (constant) time interval.
The result is a different noise signal. This also shows that a computer-generated noise
signal is a simplified copy of natural noise processes. Natural noise does not produce
„clicks“ at constant time intervals.

In order to satisfy your curiosity here is another tip: try differentiating a periodic rectan-
gular signal or sawtooth signal. At what points are there problems or extreme values?

These are the „step“ points! This is why mathematicians have created a graphic term to
designate the precondition for differentiability - the continuity of a function. This can be
graphically defined as follows: