Linear and non-linear processes - Signals, Processes, and Systems - page 196

Time domain

Noise LP fil ter Deriv/Integr Integr/Deriv

2,0
1,5
1,0
0,5
0,0
-0,5
-1,0
-1,5
-2,0
-2,5
-0,025
-0,050
-0,075
-0,100
-0,125
-0,150
-0,175
400
300
200
100
0
-100
-200
-300
-400
2,0
1,5
1,0
0,5
0,0
-0,5
-1,0
-1,5
-2,0
-2,5
2,0
1,5
1,0
0,5
0,0
-0,5
-1,0
-1,5
-2,0
-2,5

Input si gnal : fi l ter ed noi se

Integrated input signal

D i ffer enti ated i nput si gnal

Di ffer enti ati on of the i ntegr ated i nput si gnal

Integration of the differentiated input signal

50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950

ms

Illustration 134: Changing the order is possible

As was proved experimentally the order of the processes of differentiation (derivative) and integration can
be changed round. „First differentiation and then integration“ leads back to the original signal exactly as
„first integration and then differentiation“ does.

On closer examination of the signals two further aspects in the time domain may be
surmised:

• The integrated signal has far less edge steepness than the input signal. This would
imply the suppression of higher frequencies in the frequency domain, i.e. a lowpass
characteristic

• The curve of the integrated signal looks as if the mean value were formed successively
from the input signal. The curve decreases as soon as the input signal is negative and
increases once it is positive.

In order to test the first assumption we select a periodic rectangular signal. It has the
greatest conceivable edge steepness at the step points. The integrated signal ought not to
exhibit „vertical“ sides. The spectrum of the integrated signal ought to have a much
smaller proportion of higher frequencies. At the same time the curve of the rectangular
signal is constant - apart from the step points. This makes possible an answer to the
question as to what an integrator does with a constant function.