Digital Signal Processing Reference
In-Depth Information
Sine/delta
sin(x)/x
T ime domain
Spectr um
Freq. domain
1,00
0,100
0,75
0,075
0,50
0,050
0,25
0,025
0,00
0,000
-0,25
4,0
3,5
3,0
2,5
2,0
1,5
1,0
0,5
0,0
4,0
3,5
3,0
2,5
2,0
1,5
1,0
0,5
0,0
-0,5
-1,0
0,30
0,25
0,20
0,15
0,10
0,05
0,00
-0,05
0,0225
0,0200
0,0175
0,0150
0,0125
0,0100
0,0075
0,0050
0,0025
0,0000
-0,0025
50
150 250 350 450 550 650 750 850 950
-300
-200
-100
0
50 100
200
300
ms
Hz
Illustration 146:
Convolution in the frequency domain as the result of a multiplication in the time
domain
A simple explanation for the strange behaviour of the frequency domain in the case of the multiplication of
two signals in the time domain was already given at the end of Chapter 5 „The Symmetry principle“ -
periodic signals in the time domain have line spectra of equidistant frequencies. Thus equidistant lines in
the time domain must result in periodic spectra for reasons of symmetry. The symmetrical mirroring of the
LF-spectrum at every frequency of the
-
pulse is called
convolution
.
δ
The first step in the conversion of an analog signal to a digital
signal is always multiplication of the signal by a periodic
sequence of
-
pulses, that is a non-linear process. Thus a digital
signal above all in the frequency domain has properties which the
original analog signal did not have. Most of the problems in the
later retrieval of the original information of the analog signal
arise from this fact.
δ
These frequency-related phenomena are to be demonstrated from a measurement-techno-
logy point of view using the example of two forms of signal. The first selected is a sine
wave, the simplest form of signal. After that an Si-shaped curve (see Illustration 119) as
an almost ideal band-limited signal.
