X X c .... The transition probabilities are found to be
↓
↓
↓
A A A A B . B . C . . D . .
0 0 0 0 1 0 1 0 1 1 0 1 1 1
E . E . G . G . H . F . H
↓
X
b
0
0.6
0.4
c
1
0
0
Thus demonstrating the possibility of the compression, a compres-
Sion that was predicted quantitatively by the entropy of the origi-
nal message!
Ex. 1: Show that the coding gives a one-one correspondence between message
Sent and message received (except for a possible ambiguity in the first letter).
X 0.70
B 0.18
C 0.12
(Thus c → X must be 1 because c always went to either a or d; the
Transitions from a and from d need weighting by the (equilibrial)
Probabilities of being at a or d.) The new states have equilibrial
185
184
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
Probabilities of X, 20/35; b, 9/35; c, 6/35 and entropies of Hx,
Hb, 0.971; Hc, 0. So the entropy of the new series is 0.92
bits per letter— exactly the same as before!
This fact says uncompromisingly that no information was lost
When the d’s and a’s were merged to X’s. It says, therefore, that
There must be some way of restoring the original four-letter mes-
Sage from the three, of telling which of the X’s were a’s and which
Were d’s. Closer examination shows that this can be done, strik-
Ingly verifying the rather surprising prediction.
Ex.: How is
BbbcXbcXbbcXXXcXXbcXcXXXXXXXbb
To be de-coded to its original form?
In other sciences need not follow suit. In biology especially
“noise” will seldom refer to this particular source; more com-
Monly, the “noise” in one system will be due to some other mac-
Roscopic system from which the system under study cannot be
Completely isolated.
Should the two (or more) messages be completely and simultane-
Ously recoverable, by de-coding of the output, the concept of noise
Is of little use. Chiefly it is wanted when the two messages (one
Wanted, one unwanted) interact with some mutual destruction,
Making the coding not fully reversible. To see this occur let us go
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Back to the fundamental processes. The irreversibility must mean
That the variety is not sustained (S.8/6), and that distinct elements at
The inputs are represented at the output by one element. Consider
The case in which the input is a vector with two components,
The first having possible values of A, B or C
Second,,,,,,,, E, F or G.
Suppose the output is a variable that can take values 1, 2, …, 9,
And that the coding was
AE AF AG BE BF BG CE CF CG
642291375
If now the input message were the sequence B A C B A CA A B B,
While the “noise” gave simultaneously the sequence G F FE E E
G F G E, then the output would be
1, 4, 7, 2, 6, 3, 2, 4, 1, 2
NO I SE
It may happen that the whole input to a transducer can be
Divided into two or more components, and we wish to consider the
Components individually. This happened in Ex. 8/17/3, where the
Two messages were sent simultaneously through the same trans-
Ducer and recovered separately at the output. Sometimes, how-
Ever, the two inputs are not both completely deducible from the
Output. If we are interested solely in one of the input components,
As a source of variety, regarding the other as merely an unavoida-
Ble nuisance, then the situation is commonly described as that of
A “message corrupted by noise”.
It must be noticed that noise is in no intrinsic way distinguish-
Able from any other form of variety. Only when some recipient is
Given, who will state which of the two is important to him, is a dis-
Tinction between message and noise possible. Thus suppose that
Over a wire is coming both some conversation and some effects
From a cathode that is emitting irregularly. To someone who
Wants to hear the conversation, the variations at the cathode are
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“noise”; but to the engineer who is trying to make accurate meas-
Urements of what is going on at the cathode, the conversation is
“noise”. “Noise” is thus purely relative to some given recipient,
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