Who must say which information he wants to ignore.
The point is worth emphasis because, as one of the commonest
Sources of uninteresting variety in electronic systems is the ther-
Mal dance (Brownian movement) of the molecules and electrons,
Electronic engineers tend to use the word “noise” without qualifi-
Cation to mean this particular source. Within their speciality they
Will probably continue to use the word in this sense, but workers
186
↓
And the de-coding could give, for the first component, only the
Approximation
B, A, C, A or B, A, C, A or B, A, B, A or B.
Thus the original message to this input has been “corrupted” by
“noise” at the other input.
In this example the channel is quite capable of carrying the mes-
Sage without ambiguity if the noise is suppressed by the second
Input being held constant, at E say. For then the coding is one-one:
↓
A
6
B
2
C
3
And reversible.
It will be noticed that the interaction occurred because only
Eight of the nine possible output states were used. By this perma-
Nent restriction, the capacity of the channel was reduced.
187
A N I N T R O D UC T I O N T O C Y B E R NE T I C S
I N CESSA N T TR AN SMI SSIO N
Ex. 1: What is the coding, of first input to output, if the second output is kept con-
Stant (i) at F; (ii) at G?
Ex. 2: A system of three states — P, Q, R— is to transmit changes at two inputs,
α and β, each of which can take two states. The states of the inputs and of
The system change in step. Is noise-free transmission possible ?
Distortion. It should be noticed that falsification of a mes-
Sage is not necessarily identical with the effect of noise. “If a par-
Ticular transmitted signal always produces the same received
Signal, i.e. the received signal is a definite function of the trans-
Mitted signal, then the effect may be called distortion. If this func-
Tion has an inverse— no two transmitted signals producing the
Same received signal— distortion may be corrected, at least in
Principle, by merely performing the inverse functional operation
On the received signal.” (Shannon.)
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Ex. 1: Is the change by which the erect object falls on to the retina inverted a dis-
Tortion or a corruption ?
Ex. 2: A tension applied to a muscle evokes a steady stream of impulses whose
Frequency is not proportional to the tension. Is the deviation from propor-
Tionality a distortion or a corruption?
Ex. 3: (Continued.) If the nerve carrying the impulses is subjected to alcohol
Vapour of sufficient strength it will cease to conduct for all tensions. Is this
A distortion or a corruption?
Measure for the amount of information lost, but this interpretation
Leads to nonsense. Thus if, in the same transmission, the line were
Actually cut and the recipient simply tossed a coin to get a “mes-
Sage” he would get about a half of the symbols right, yet no infor-
Mation whatever would have been transmitted. Shannon has
Shown conclusively that the natural measure is the equivocation,
Which is calculated as follows.
First find the entropy over all possible classes:
Log 0.495 – 0.005 log 0.005
Log 0.005 – 0.495 log 0.495
Call this H1 it is 1.081 bits per symbol. Next collect together the
Received signals, and their probabilities; this gives the table
Symbol received 0
Probability0.5
Find its entropy:
– 0.5 log 0.5 – 0.5 log 0.5
Call this H2. It is 1.000 bits per symbol. Then the equivocation is
H1 – H2 : 0.081 bits per symbol.
The actual rate at which information is being transmitted,
Allowance being made for the effect of noise, is the entropy of the
Source, less the equivocation. The source here has entropy 1.000
Bits per symbol, as follows from:
Symbol sent01
Probability0.50.5
So the original amount supplied is 1.000 bits per symbol. Of this
Gets through and 0.081 is destroyed by noise.
Ex. 1: What is the equivocation of the transmission of S.9/19, if all nine combi-
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Nations of letters occur, in the long run, with equal frequency?
Ex. 2: (Continued.) What happens to the equivocation if the first input uses only
The symbols B and C, so that the combinations BE, BF, BG, CE, CF, CG
Occur with equal frequencies? Is the answer reasonable?
*Ex. 3: Prove the following rules, which are useful when we want to find the
Value of the expression– p loga p, and p is either very small or very near to 1:
(i) If p = xy, – p log a p = – xy ( log ax + log ay );
z × 10–z
(ii) If p = 10 , – p log a p = ------------------ ;-
Log 10a
1q(iii) If p is very close to 1, put 1 – p = q , and – p log a p = ------------ q – ---- …
--log ea 2
2
–z
1
0.5
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