Digital Signal Processing Reference
In-Depth Information
Pulse
Delay 01
Delay 02
Delay 03
Delay 04
Delay 05
Delay 06
Time domai n
S pectrum
Freq. domai n
Addition
1,00
0,005
0,004
0,75
0,003
0,50
0,002
0,001
0,25
0,000
0,00
1,00
-0,001
0,030
0,025
0,75
0,020
0,50
0,015
0,010
0,25
0,005
0,00
0,000
50
100
150
200
250
300
350
400
450
500
550
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
ms
Hz
Illustration 210:
Generation of a longer lasting pulse response
By means of an extremely simple circuit with the two elementary signal processes delay and addition, a
digital signal of any length can be generated from a single
-pulse. The last pulse has passed through all 6
delay processes etc. The addition of the seven time-displaced pulses result in the bottom signal.
δ
In the case of a rectangular output signal in the time domain one is not surprised at the (periodic)
Si-shaped curve in the frequency domain. But in actual fact we wanted things the other way round: a rect-
angular curve in the frequency domain (filtered). What must the pulse response look like for reasons of
symmetry and how would we have to complete the circuit?
Below you see the sum as superposition of these three pulse responses. It gives the filtered
instantaneous curve of the input signal. The cutoff frequency of the lowpass can be
deduced from this “ripple effect”.
The addition (or sum) of the - time-displaced - Si-shaped pulse
responses of the particular type of filter result in the filtered
output signal.
By the way, you encountered a digital filter of a special type (“comb filter”) in Illustration
126, Illustration 88 and Illustration 89. There you see what role the delaying process plays
in the case of digital filters. Instead of the constant amplitude spectrum of a single
δ
-pulse
the spectrum in the case of two
-pulses are cosine-shaped. You will find the explanation
for this in the text of Illustration 88. Certain frequencies at regular intervals (“comb-like
structures”) are not allowed to pass.
δ
