Digital Signal Processing Reference
In-Depth Information
Ti me domai n
Sp ect r um
Fr eq. domai n
Rectangular
Integr al
Time domain
Frequency domain
4
5,5
5,0
4,5
4,0
3,5
3,0
2,5
2,0
1,5
1,0
0,5
0,0
0,250
0,225
0,200
0,175
0,150
0,125
0,100
0,075
0,050
0,025
0,000
3
+
2
1
0
-1
-
-2
-3
-4
0,50
0,45
0,40
0,35
0,30
0,25
0,20
0,15
0,10
0,05
0,00
50
150
250
350
450
550
650
750
850
950
0
25
50
75
100
ms
Hz
Illustration 135:
Integration of a periodic rectangular signal
We select a very straightforward signal curve in the case of the periodic rectangular signal. This reveals
the property of integration particularly clearly.
Constant positive segments lead to a rising straight line and constant negative segments lead to a falling
straight line. Apparently, seen from a geometrical point of view, the integral successively measures the
area
between the curve of the signal and the horizontal time axis. The „area“ of the first rectangular
signal is 4 V
125 ms = 0.5 Vs and it is precisely this value that the integration curve gives after 125 ms.
We must continue to differentiate between positive and negative „areas“ as the next - in terms of the abso-
lute value equally large - area is deducted, so that after 250 ms the value is again zero.
The frequency domain (right) provides unambiguous qualitative and quantitative pointers. Initially, the
integration quite clearly has a lowpass characteristic. Both signals - the rectangular and the triangular
signals - only contain the odd multiples of the base frequency (see Chapter 2). The rule for the decrease in
amplitudes is quite straightforward and can be measured using the cursor. For the sawtooth Û
n
=Û
1
/n
(e.g. Û
3
= Û
1
/3 etc) holds. For the spectrum of the triangular curve Û
n
= Û
1
/ n
2
(e.g. Û
3
= Û
1
/9) applies.
If the same frequencies of the spectra are compared with each other the result is Û
triangle
= Û
sawtooth
/2
∗
π
f
= Û
sawtooth
/
ω
. Integration as the reversal of differentiation also in the frequency domain is thus fully
borne out here.
The first supposition is fully borne out by the measurement shown in Illustration 135:
Integration exhibits lowpass behaviour. To put it accurately, the
frequency domain of the input signal is divided by 2
in
integration. The amplitudes of the higher frequencies are thus
disproportionately reduced in size.
π
f =
ω
