Digital Signal Processing Reference
In-Depth Information
1,00
0,75
0,50
0,25
0,00
-0,25
-0,50
-0,75
-1,00
0,040
0,035
0,030
0,025
0,020
0,015
0,010
0,005
0,000
1,00
0,75
0,50
0,25
0,00
-0,25
-0,50
-0,75
-1,00
1,00
0,75
0,50
Sinusoidal input
0,25
0,00
0,02
0,01
Integrated sine : (-) cosine
0,01
0,01
0,00
1,00
0,75
Differentiation of integrated sine
0,50
0,25
0,00
25
50
75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475
2,5 10,0 20,0 30,0
ms
Hz
Illustration 140:
On the integration of the sine wave
The sine (top) which has been integrated results in a (-)cosine-shaped curve (centre). This can be repro-
duced via the differentiation of the integrated signal, i.e. the reversal of the integration. The „gradient
curve“ of the integrated signal corresponds to the original sine (see top and bottom).
A function is continuous if the function can be drawn in a
mathematically correct way without putting the „pencil“ down,
that means without „step” points.
Only continuous functions can be differentiated because the
steepness at „step“ points is (theoretically) infinite.
Filters
When linear processes are listed filters are often forgotten although they are indispensable
in information technology and signal processing. The linear processes mentioned so far
referred to the time domain as far as their names are concerned. Thus the process
differentiation of a signal means that the differentiation is carried out in the time domain.
This is equivalent, as already explained, to multiplication in the frequency domain (am-
plitude spectrum) by
/2 (the differentia-
tion of a sine results in a cosine; this is the equivalent of this phase shift!)
ω
= 2
π
f, and to a shift of the phase spectrum by
π
