Digital Signal Processing Reference
In-Depth Information
Reminder: Positional notation of numerical systems
The Babylonians had a number system with 60 (!) different numerals . We need it still
today in the measurement of time : 1 h = 60 min and 1 min = 60 s
Our decimal system uses 10 different number from 0 to 9. For example the number
4096 is as agreed an abbreviation of
4 · 1000 + 0 · 100 + 9 · 10 + 6 · 1 = 4 · 10
3
+ 0 · 10
2
+ 9 · 10
1
+ 6 · 10
0
Thus the number 10 represents the
basis
of our numerical system . The
position
of the
number indicates its "value", hence "positional notation of numerical systems "!
The decimal system has a set of values which range from 0 to 9. This basic set is
repeated over and over again to create large numbers .
In digital signal processing the Binary Number System is used
. There are various
reasons for this :
•
Electronic components and circuits with two switching conditions ("on" and
"off") can be implemented most simply .
•
Signals with two states are the most noise -resistant , because it is only necessary to
distinguish between "on" and "off"
•
The mathematical basis is very straightforward . The basic table for binary numbers
is very simple :
0 · 0 = 0; 1 · 0 = 0; 1 · 1 = 1
How much school years were needed in order to learn our multiplication tables
?
The base of this number system is 2. The binary number 101101 written out in full is
equivalent
1 · 2
5
+ 0 · 2
4
+ 1 · 2
3
+ 1 · 2
2
+ 0 · 2
1
+ 1 · 2
0
=
1 · 32 + 0 · 16 + 1 · 8 + 1 · 4 + 0 · 2 + 1 · 1 = 45
Hence : 101101
2
= 45
10
In words: "101101 to the base of 2 corresponds to 45 to the basis 10"
A little test : what would the number 45
10
be in the "three number system ", i.e. on the
base of 3?
Illustration 186:
Significance number systems
The only thing required in this topic is in principle the four basic arithmetical operations. However, you
should also be proficient in them in the binary number system.
In Illustration 185 a periodic sequence of dual numbers beginning at 0 and ending at 31 is
connected with a - purely computer-based - D/A converter. Accordingly, this results in a
periodic sawtooth and step-like signal curve at the output. For the instantaneous value
01101
2
at the input the value 13
10
results at the output. See in this connection Illustration
159 (significance number systems)
