Digital Signal Processing Reference
In-Depth Information
T ime domai n
FFT
Frequency do
Sine
Derivat ive
1,00
1,00
Frequency 8 Hz
0,50
0,75
Sine 8 Hz
0,00
0,50
-0,50
0,25
-1,00
100,00
0,00
55
50
45
40
35
30
25
20
15
10
5
0
50,00
Frequency 8 Hz
0,00
-50,00
Cosine 8 Hz
-100,00
50
150
250
350
450
550
650
750
850
950
2,5
7,5
12,5
17,5
22,5
27,5
32,5
37,5
ms
Hz
Illustration 129:
Experimental evidence: Differentiation of a sine shows a cosine
At first sight the differentiated signal of a sine has a cosine- and/or sine wave oscillation. But is it really
precisely a cosine? If it was not the case the differentiation would be non-linear.
In the frequency domain we have a reliable procedure to prove this. As the differentiated signal in the
frequency domain - like the sine at the input - has a single frequency of 4 Hz it can only be a linear pro-
cess. It causes a phase shift of
/2 (Cosine!). Its amplitude is strictly proportionally the frequency ... and
the amplitude of the input signal (why ?).
π
These results are not surprising and result directly from the general property of differen-
tiation which can be seen in Illustration 127.
•
A differentiated sine produces a cosine, that is it displaces the phase by
π
/2 rad. This could not be otherwise because the gradient is greatest at
the zero crossing; hence the differentiated signal is also greatest at this
point. That this is really surprisingly a cosine is proved by the
frequency domain (Illustration 129): a line of the same frequency is still
present there.
•
The higher the frequency the faster the sinusoidal signal changes (also
at the zero crossing). The amplitude of the differentiated voltage is also
strictly proportional to the frequency.
