Digital Signal Processing Reference
In-Depth Information
Ti me domain
No is e
LP fil ter
Integral
De ri vati ve
1,25
1,00
0,75
0,50
0,25
0,00
-0,25
-0,50
-0,75
-1,00
-1,25
0,05
0,04
0,03
0,02
0,01
0,00
-0,01
-0,02
-0,03
1,25
1,00
0,75
0,50
0,25
0,00
-0,25
-0,50
-0,75
-1,00
-1,25
Input signal (LP filtered noise)
Integrated signal at the output of the integrator
Differentiation of the integrated signal
50
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
ms
Illustration 133:
Visualisation of the relationship between integration and differentiation
Closer examination leads one to suspect that the middle signal after differentiation produces the upper
one. The appropriate experiment shows the agreement: the differentiation of the integrated signal
produces the original signal.
As we are focussing on differentiation it ought to strike an attentive observer that the
middle (integrated) signal when differentiated ought to give the input signal. The local
maxima and minima of the input signal are again to be found at the steepest points of the
input signal.
Closer examination bears out this surmise. Thus, differentiation appears to be the reversal
of integration. Is integration the reversal of differentiation? An experiment presented in
Illustration 133 bears this out too. The fact that in both instances random signals were
selected practically ensures that this is generally valid
Differentiation can be seen as the reversal of integration and
integration as the reversal of differentiation.
