Reprinted with corrections
from
The Bell System Technical Journal,
Vol. 27, pp. 379–423, 623–656, July, October, 1948.
A Mathematical Theory of Communication
By C. E. SHANNON
I
NTRODUCTION
T
HE recent
development
of
various
methods
of
modulation
such
as
PCM
and
PPM
which
exchange
bandwidth for signal-to-noise ratio has intensified the interest in a general theory of communication. A
basis for such a theory is contained in the important papers of Nyquist
1
and Hartley
2
on this subject.
In the
present paper we will extend the theory to include a number of new factors, in particular the effect of noise
in the channel, and the savings possible due to the statistical structure of the original message and due to the
nature of the final destination of the information.
The
fundamental
problem
of
communication
is
that
of
reproducing
at
one
point
either
exactly
or
ap-
proximately a message selected at another point.
Frequently the messages have
meaning
; that is they refer
to or are correlated according to some system with certain physical or conceptual entities.
These semantic
aspects of communication are irrelevant to the engineering problem. The significant aspect is that the actual
message is one
selected from a set
of possible messages.
The system must be designed to operate for each
possible selection, not just the one which will actually be chosen since this is unknown at the time of design.
If the number of messages in the set is finite then this number or any monotonic function of this number
can
be
regarded as
a
measure
of
the
information produced
when
one
message
is
chosen
from
the
set,
all
choices
being
equally
likely.
As
was
pointed
out
by
Hartley
the
most
natural
choice
is
the
logarithmic
function.
Although this definition must be generalized considerably when we consider the influence of the
statistics
of
the
message
and
when
we
have
a
continuous
range
of
messages,
we
will
in
all
cases
use
an
essentially logarithmic measure.
The logarithmic measure is more convenient for various reasons:
1.
It is practically more useful.
Parameters of engineering importance such as time, bandwidth, number
of relays,
etc.,
tend to
vary linearly with
the logarithm of
the number of possibilities.
For
example,
adding one relay to a group doubles the number of possible states of the relays. It adds 1 to the base 2
logarithm of
this
number.
Doubling
the
time
roughly
squares
the
number
of
possible
messages,
or
doubles the logarithm, etc.
2.
It is nearer to our intuitive feeling as to the proper measure.
This is closely related to (1) since we in-
tuitively measures entities by linear comparison with common standards. One feels, for example, that
two punched cards should have twice the capacity of one for information storage,
and two
identical
channels twice the capacity of one for transmitting information.
3.
It is mathematically more suitable.
Many of the limiting operations are simple in terms of the loga-
rithm but would require clumsy restatement in terms of the number of possibilities.
The choice of a logarithmic base corresponds to the choice of a unit for measuring information.
If the
base
2
is
used
the
resulting
units
may
be
called
binary
digits,
or
more
briefly
bits,
a
word
suggested
by
J.
W.
Tukey.
A device with
two
stable positions,
such
as a
relay or
a flip-flop circuit,
can store one bit
of
information.
N
such devices can store
N
bits, since the total number of possible states is 2
N
and log
2
2
N
=
N
.
If the base 10 is used the units may be called decimal digits.
Since
log
2
M
=
log
10
M
=
log
10
2
=
3
:
32log
10
M
;
1
Nyquist, H., “Certain
Factors
Affecting
Telegraph
Speed,”
Bell System Technical
Journal,
April 1924, p. 324;
“Certain
Topics in
Telegraph Transmission Theory,”
A.I.E.E. Trans.,
v. 47, April 1928, p. 617.
2
Hartley, R. V. L., “Transmission of Information,”
Bell System Technical Journal,
July 1928, p. 535.
1
INFORMATION
SOURCE
MESSAGE
TRANSMITTER
SIGNAL
RECEIVED
SIGNAL
RECEIVER
MESSAGE
DESTINATION
NOISE
SOURCE
Fig. 1—Schematic diagram of a general communication system.
a decimal digit is about 3
1
3
bits.
A digit wheel on a desk
computing machine has ten
stable positions and
therefore has a storage capacity of one decimal digit. In analytical work where integration and differentiation
are involved the base
e
is sometimes useful.
The resulting units of information will be called natural units.
Change from the base
a
to base
b
merely requires multiplication by log
b
a
.
By
a
communication system
we
will
mean
a
system
of
the
type
indicated
schematically
in
Fig.
1.
It
consists of essentially five parts:
1.
An
information source
which produces a message or sequence of messages to be communicated to the
receiving terminal.
The message may be of various types:
(a) A sequence of letters as in a telegraph
of
teletype
system;
(b)
A
single
function
of
time
f
(
t
)
as
in
radio
or
telephony;
(c)
A
function
of
time and other variables as in black and white television — here the message may be thought of as a
function
f
(
x
;
y
;
t
)
of two space coordinates and time, the light intensity at point
(
x
;
y
)
and time
t
on a
pickup tube plate; (d) Two or more functions of time, say
f
(
t
)
,
g
(
t
)
,
h
(
t
)
— this is the case in “three-
dimensional” sound transmission or if the system is intended to service several individual channels in
multiplex; (e) Several functions of several variables — in color television the message consists of three
functions
f
(
x
;
y
;
t
)
,
g
(
x
;
y
;
t
)
,
h
(
x
;
y
;
t
)
defined in a three-dimensional continuum — we may also think
of
these
three
functions
as
components
of
a
vector
field
defined
in
the
region
—
similarly,
several
black
and
white
television
sources
would
produce
“messages”
consisting
of
a
number
of
functions
of three variables;
(f) Various combinations also occur, for example in television with an associated
audio channel.
2.
A
transmitter
which
operates
on
the
message
in
some
way
to
produce
a
signal
suitable
for
trans-
mission
over
the
channel.
In
telephony
this
operation
consists
merely
of
changing
sound
pressure
into
a
proportional electrical current.
In
telegraphy we
have
an
encoding operation which
produces
a
sequence of dots,
dashes and spaces on
the channel corresponding to
the message.
In
a
multiplex
PCM
system
the
different speech
functions
must
be
sampled,
compressed,
quantized
and
encoded,
and
finally
interleaved
properly
to
construct
the
signal.
Vocoder
systems,
television
and
frequency
modulation are other examples of complex operations applied to the message to obtain the signal.
3.
The
channel
is merely the medium used to transmit the signal from transmitter to receiver.
It may be
a pair of wires, a coaxial cable, a band of radio frequencies, a beam of light, etc.
4.
The
receiver
ordinarily performs the inverse operation of that done by the transmitter, reconstructing
the message from the signal.
5.
The
destination
is the person (or thing) for whom the message is intended.
We
wish
to
consider
certain
general problems involving communication systems.
To
do
this
it
is
first
necessary to represent the various elements involved as mathematical entities, suitably idealized from their
2
physical counterparts. We may roughly classify communication systems into three main categories: discrete,
continuous and
mixed.
By
a
discrete system
we
will
mean
one in
which
both
the
message and
the signal
are a sequence of discrete symbols.
A typical case is telegraphy where the message is a sequence of letters
and the signal a sequence of dots, dashes and spaces.
A continuous system is one in which the message and
signal
are both
treated
as
continuous functions,
e.g.,
radio
or
television.
A
mixed
system
is
one in
which
both discrete and continuous variables appear, e.g., PCM transmission of speech.
We
first
consider
the
discrete
case.
This
case
has
applications not
only
in
communication theory,
but
also in the theory of computing machines, the design of telephone exchanges and other fields.
In addition
the discrete case forms a foundation for the continuous and mixed cases which will be treated in the second
half of the paper.
PART I: DISCRETE NOISELESS SYSTEMS
1.
T
HE
D
ISCRETE
N
OISELESS
C
HANNEL
Teletype and telegraphy are two simple examples of a discrete channel for transmitting information.
Gen-
erally, a discrete channel will mean a system whereby a sequence of choices from a finite set of elementary
symbols
S
1
;
:
:
:
;
S
n
can be transmitted from one point to another.
Each of the symbols
S
i
is assumed to have
a
certain
duration
in
time
t
i
seconds
(not
necessarily
the
same
for
different
S
i
,
for
example
the
dots
and
dashes in telegraphy).
It is not required that all possible sequences of the
S
i
be capable of transmission on
the system;
certain sequences only may be allowed.
These will
be possible signals for the channel.
Thus
in telegraphy suppose the symbols are:
(1) A dot, consisting of line closure for a unit of time and then line
open for a unit of time; (2) A dash, consisting of three time units of closure and one unit open; (3) A letter
space consisting of, say, three units of line open; (4) A word space of six units of line open. We might place
the restriction on allowable sequences that no spaces follow each other (for if two letter spaces are adjacent,
it
is
identical with
a word
space).
The question we now consider is
how one can
measure the capacity
of
such a channel to transmit information.
In
the
teletype
case
where
all
symbols
are
of
the
same
duration,
and
any
sequence of
the
32
symbols
is
allowed
the
answer
is
easy.
Each
symbol represents five
bits
of
information.
If
the
system
transmits
n
symbols per second it is natural to say that the channel has a capacity of 5
n
bits per second.
This does not
mean that the teletype channel will always be transmitting information at this rate — this is the maximum
possible rate and whether or not the actual rate reaches this maximum depends on the source of information
which feeds the channel, as will appear later.
In the more general case with different lengths of symbols and constraints on the allowed sequences, we
make the following definition:
Definition: The capacity
C
of a discrete channel is given by
C
=
Lim
T
!
∞
log
N
(
T
)
T
where
N
(
T
)
is the number of allowed signals of duration
T
.
It is easily seen that in the teletype case this reduces to the previous result.
It can be shown that the limit
in
question
will
exist
as
a
finite
number
in
most
cases
of
interest.
Suppose
all
sequences
of
the
symbols
S
1
;
:
:
:
;
S
n
are
allowed
and
these
symbols
have
durations
t
1
;
:
:
:
;
t
n
.
What
is
the
channel
capacity?
If
N
(
t
)
represents the number of sequences of duration
t
we have
N
(
t
)
=
N
(
t
t
1
)
+
N
(
t
t
2
)
+
+
N
(
t
t
n
)
:
The
total
number
is
equal
to
the
sum
of
the
numbers
of
sequences
ending
in
S
1
;
S
2
;
:
:
:
;
S
n
and
these
are
N
(
t
t
1
)
;
N
(
t
t
2
)
;
:
:
:
;
N
(
t
t
n
)
, respectively.
According to a well-known result in finite differences,
N
(
t
)
is then asymptotic for large
t
to
X
t
0
where
X
0
is the largest real solution of the characteristic equation:
X
t
1
+
X
t
2
+
+
X
t
n
=
1
3
and therefore
C
=
log
X
0
:
In case there are restrictions on allowed sequences we may still often obtain a difference equation of this
type and find
C
from the characteristic equation. In the telegraphy case mentioned above
N
(
t
)
=
N
(
t
2
)
+
N
(
t
4
)
+
N
(
t
5
)
+
N
(
t
7
)
+
N
(
t
8
)
+
N
(
t
10
)
as
we
see
by
counting
sequences
of
symbols
according
to
the
last
or
next
to
the
last
symbol
occurring.
Hence
C
is
log
0
where
0
is the positive root of 1
=
2
+
4
+
5
+
7
+
8
+
10
.
Solving this we find
C
=
0
:
539.
A
very
general
type
of
restriction
which
may
be
placed
on
allowed
sequences
is
the
following:
We
imagine a number of possible states
a
1
;
a
2
;
:
:
:
;
a
m
.
For each state only certain symbols from the set
S
1
;
:
:
:
;
S
n
can
be transmitted
(different subsets
for the different states).
When
one of
these
has been
transmitted the
state
changes
to
a
new
state
depending
both
on
the
old
state
and
the
particular
symbol
transmitted.
The
telegraph case is a simple example of this.
There are two states depending on whether or not a space was
the last symbol transmitted.
If so, then only a dot or a dash can be sent next and the state always changes.
If not, any symbol can be transmitted and the state changes if a space is sent, otherwise it remains the same.
The conditions can be indicated in a linear graph as shown in Fig. 2.
The junction points correspond to the
DASH
DOT
DASH
DOT
LETTER
SPACE
WORD
SPACE
Fig. 2—Graphical representation of the constraints on telegraph symbols.
states and the lines indicate the symbols possible in a state and the resulting state.
In Appendix 1 it is shown
that if the conditions on allowed sequences can be described in this form
C
will exist and can be calculated
in accordance with the following result:
Theorem 1:
Let
b
(
s
)
ij
be the duration of the
s
th
symbol which is allowable in state
i
and leads to state
j
.
Then the channel capacity
C
is equal to log
W
where
W
is the largest real root of the determinant equation:
∑
s
W
b
(
s
)
ij
ij
=
0
where
ij
=
1 if
i
=
j
and is zero otherwise.
For example, in the telegraph case (Fig. 2) the determinant is:
1
(
W
2
+
W
4
)
(
W
3
+
W
6
)
(
W
2
+
W
4
1
)
=
0
:
On expansion this leads to the equation given above for this case.
2.
T
HE
D
ISCRETE
S
OURCE
OF
I
NFORMATION
We have seen that under very general conditions the logarithm of the number of possible signals in a discrete
channel increases linearly with
time.
The capacity to
transmit information can
be specified by giving this
rate of increase, the number of bits per second required to specify the particular signal used.
We now consider the information source. How is an information source to be described mathematically,
and how much information in bits per second is produced in a given source?
The main point at issue is the
effect of statistical knowledge about the source in reducing the required capacity of the channel, by the use
4
of proper encoding of the information. In telegraphy, for example, the messages to be transmitted consist of
sequences of letters. These sequences, however, are not completely random. In general, they form sentences
and have the statistical structure of, say, English.
The letter E occurs more frequently than Q, the sequence
TH
more frequently than
XP,
etc.
The existence of
this
structure allows
one to
make a
saving
in
time
(or
channel capacity) by properly encoding the message sequences into signal sequences.
This is already done
to a limited extent in telegraphy by using the shortest channel symbol, a dot, for the most common English
letter E; while the infrequent letters, Q, X, Z are represented by longer sequences of dots and dashes.
This
idea is carried still further in certain commercial codes where common words and phrases are represented
by
four- or
five-letter
code
groups with
a
considerable saving
in
average time.
The standardized
greeting
and anniversary telegrams now in use extend this to the point of encoding a sentence or two into a relatively
short sequence of numbers.
We can think of a discrete source as generating the message, symbol by symbol.
It will choose succes-
sive symbols according to
certain probabilities depending, in
general, on
preceding choices as well
as the
particular symbols
in
question.
A
physical system,
or
a
mathematical model of
a
system
which
produces
such a sequence of symbols governed by a set of probabilities, is known as a stochastic process.
3
We may
consider a discrete source, therefore, to be represented by a stochastic process.
Conversely, any stochastic
process which produces a discrete sequence of symbols chosen from a finite set may be considered a discrete
source. This will include such cases as:
1.
Natural written languages such as English, German, Chinese.
2.
Continuous information
sources
that
have
been
rendered
discrete
by
some
quantizing
process.
For
example, the quantized speech from a PCM transmitter, or a quantized television signal.
3.
Mathematical
cases
where
we
merely
define
abstractly
a
stochastic
process
which
generates
a
se-
quence of symbols.
The following are examples of this last type of source.
(A)
Suppose we have five letters A, B, C, D, E which are chosen each with probability .2, successive
choices being independent.
This
would lead
to
a
sequence of
which the
following is
a
typical
example.
B D C B C E C C C A D C B D D A A E C E E A
A B B D A E E C A C E E B A E E C B C E A D.
This was constructed with the use of a table of random numbers.
4
(B)
Using the same five letters let the probabilities be .4, .1, .2, .2, .1, respectively, with successive
choices independent. A typical message from this source is then:
A A A C D C B D C E A A D A D A C E D A
E A D C A B E D A D D C E C A A A A A D.
(C)
A
more complicated
structure is
obtained
if
successive
symbols are
not
chosen
independently
but
their
probabilities
depend
on
preceding
letters.
In
the
simplest
case
of
this
type
a
choice
depends only
on the
preceding letter
and
not on
ones before that.
The statistical
structure can
then be described by a set of transition probabilities
p
i
(
j
)
, the probability that letter
i
is followed
by letter
j
.
The indices
i
and
j
range over all the possible symbols.
A second equivalent way of
specifying the structure is to give the “digram” probabilities
p
(
i
;
j
)
, i.e., the relative frequency of
the digram
i
j
.
The letter frequencies
p
(
i
)
, (the probability of letter
i
), the transition probabilities
3
See, for example,
S. Chandrasekhar,
“Stochastic
Problems in Physics and Astronomy,”
Reviews
of Modern Physics
, v. 15, No. 1,
January 1943, p. 1.
4
Kendall and Smith,
Tables of Random Sampling Numbers,
Cambridge, 1939.
5
p
i
(
j
)
and the digram probabilities
p
(
i
;
j
)
are related by the following formulas:
p
(
i
)
=
∑
j
p
(
i
;
j
)
=
∑
j
p
(
j
;
i
)
=
∑
j
p
(
j
)
p
j
(
i
)
p
(
i
;
j
)
=
p
(
i
)
p
i
(
j
)
∑
j
p
i
(
j
)
=
∑
i
p
(
i
)
=
∑
i
;
j
p
(
i
;
j
)
=
1
:
As a specific example suppose there are three letters A, B, C with the probability tables:
p
i
(
j
)
j
A
B
C
A
0
4
5
1
5
i
B
1
2
1
2
0
C
1
2
2
5
1
10
i
p
(
i
)
A
9
27
B
16
27
C
2
27
p
(
i
;
j
)
j
A
B
C
A
0
4
15
1
15
i
B
8
27
8
27
0
C
1
27
4
135
1
135
A typical message from this source is the following:
A B B A B A B A B A B A B A B B B A B B B B B A B A B A B A B A B B B A C A C A B
B A B B B B A B B A B A C B B B A B A.
The next increase in complexity would involve trigram frequencies but no more.
The choice of
a letter would depend on the preceding two letters but not on the message before that point.
A
set of trigram frequencies
p
(
i
;
j
;
k
)
or equivalently a set of transition probabilities
p
ij
(
k
)
would
be required.
Continuing in this way one obtains successively more complicated stochastic pro-
cesses.
In the general
n
-gram case a set of
n
-gram probabilities
p
(
i
1
;
i
2
;
:
:
:
;
i
n
)
or of transition
probabilities
p
i
1
;
i
2
;:::;
i
n
1
(
i
n
)
is required to specify the statistical structure.
(D)
Stochastic
processes
can
also
be
defined
which
produce
a
text
consisting
of
a
sequence
of
“words.”
Suppose
there
are
five
letters
A,
B,
C,
D,
E
and
16
“words”
in
the
language
with
associated probabilities:
.10 A
.16 BEBE
.11 CABED
.04 DEB
.04 ADEB
.04 BED
.05 CEED
.15 DEED
.05 ADEE
.02 BEED
.08 DAB
.01 EAB
.01 BADD
.05 CA
.04 DAD
.05 EE
Suppose successive “words” are chosen independently and are separated by a space.
A typical
message might be:
DAB EE A BEBE DEED DEB ADEE ADEE EE DEB BEBE BEBE BEBE ADEE BED DEED
DEED CEED ADEE A DEED DEED BEBE CABED BEBE BED DAB DEED ADEB.
If
all
the words are of
finite length
this process is
equivalent to
one of
the preceding type,
but
the
description may
be
simpler
in
terms
of
the
word
structure and
probabilities.
We
may
also
generalize here and introduce transition probabilities between words, etc.
These
artificial
languages
are
useful
in
constructing
simple
problems
and
examples
to
illustrate
vari-
ous
possibilities.
We
can
also
approximate to
a
natural language by
means
of
a
series
of
simple
artificial
languages.
The zero-order approximation is obtained by choosing all letters with the same probability and
independently.
The first-order approximation is obtained by choosing successive letters independently but
each
letter
having
the
same
probability
that
it
has
in
the
natural
language.
5
Thus,
in
the
first-order
ap-
proximation
to
English,
E
is
chosen
with
probability
.12
(its
frequency
in
normal
English)
and
W
with
probability
.02,
but
there
is
no
influence
between
adjacent
letters
and
no
tendency
to
form
the
preferred
5
Letter, digram and trigram frequencies are given in
Secret and Urgent
by Fletcher Pratt, Blue Ribbon Books, 1939.
Word frequen-
cies are tabulated in
Relative Frequency of English Speech Sounds,
G. Dewey, Harvard University Press, 1923.
6
digrams such
as
TH,
ED,
etc.
In the second-order approximation, digram structure is
introduced.
After
a
letter
is
chosen,
the
next
one
is
chosen
in
accordance with
the
frequencies with
which
the
various
letters
follow
the
first
one.
This
requires
a
table
of
digram
frequencies
p
i
(
j
)
.
In
the
third-order approximation,
trigram structure is introduced.
Each letter is chosen with probabilities which depend on the preceding two
letters.
3.
T
HE
S
ERIES
OF
A
PPROXIMATIONS
TO
E
NGLISH
To give a visual idea of how this series of processes approaches a language, typical sequences in the approx-
imations to English have been constructed and are given below.
In all cases we have assumed a 27-symbol
“alphabet,” the 26 letters and a space.
1.
Zero-order approximation (symbols independent and equiprobable).
XFOML
RXKHRJFFJUJ
ZLPWCFWKCYJ
FFJEYVKCQSGHYD
QPAAMKBZAACIBZL-
HJQD.
2.
First-order approximation (symbols independent but with frequencies of English text).
OCRO
HLI
RGWR
NMIELWIS
EU
LL
NBNESEBYA
TH
EEI
ALHENHTTPA
OOBTTVA
NAH BRL.
3.
Second-order approximation (digram structure as in English).
ON
IE
ANTSOUTINYS
ARE
T
INCTORE
ST
BE
S
DEAMY
ACHIN
D
ILONASIVE
TU-
COOWE AT TEASONARE FUSO TIZIN ANDY TOBE SEACE CTISBE.
4.
Third-order approximation (trigram structure as in English).
IN NO IST LAT WHEY CRATICT FROURE BIRS GROCID PONDENOME OF DEMONS-
TURES OF THE REPTAGIN IS REGOACTIONA OF CRE.
5.
First-order word approximation. Rather than continue with tetragram,
:
:
:
,
n
-gram structure it is easier
and
better
to
jump
at
this
point to
word
units.
Here
words
are
chosen
independently but
with
their
appropriate frequencies.
REPRESENTING AND SPEEDILY IS AN GOOD APT OR COME CAN DIFFERENT NAT-
URAL HERE HE THE A IN CAME THE TO OF TO EXPERT GRAY COME TO FURNISHES
THE LINE MESSAGE HAD BE THESE.
6.
Second-order word approximation.
The word transition probabilities are correct but no further struc-
ture is included.
THE HEAD AND IN FRONTAL ATTACK ON AN ENGLISH WRITER THAT THE CHAR-
ACTER OF THIS POINT IS THEREFORE ANOTHER METHOD FOR THE LETTERS THAT
THE TIME OF WHO EVER TOLD THE PROBLEM FOR AN UNEXPECTED.
The resemblance to ordinary English text increases quite noticeably at each of the above steps.
Note that
these samples have reasonably good structure out to about twice the range that is taken into account in their
construction.
Thus in
(3)
the
statistical
process insures reasonable text
for two-letter
sequences,
but four-
letter sequences from the sample can usually be fitted into good sentences.
In (6) sequences of four or more
words can easily be placed in sentences without unusual or strained constructions.
The particular sequence
of ten words “attack on an English writer that the character of this” is not at all unreasonable. It appears then
that a sufficiently complex stochastic process will give a satisfactory representation of a discrete source.
The
first
two
samples
were
constructed
by
the
use
of
a
book
of
random
numbers in
conjunction with
(for example 2) a
table of
letter frequencies.
This method might have been continued for (3),
(4) and (5),
since digram, trigram and word frequency tables are available,
but a simpler equivalent method was used.
7
To construct (3) for example, one opens a book at random and selects a letter at random on the page.
This
letter
is
recorded.
The book is
then
opened to
another page and
one reads until
this letter
is
encountered.
The
succeeding letter
is
then
recorded.
Turning to
another page this
second
letter
is
searched
for
and
the
succeeding letter
recorded, etc.
A
similar
process was
used for (4),
(5)
and (6).
It would be interesting if
further approximations could be constructed, but the labor involved becomes enormous at the next stage.
4.
G
RAPHICAL
R
EPRESENTATION
OF
A
M
ARKOFF
P
ROCESS
Stochastic processes of the type described above are known mathematically as discrete Markoff processes
and
have been
extensively studied in
the
literature.
6
The general case
can
be described as follows:
There
exist a finite number of
possible “states”
of a system;
S
1
;
S
2
;
:
:
:
;
S
n
.
In
addition there is a set
of transition
probabilities;
p
i
(
j
)
the probability that if
the system
is
in
state
S
i
it will
next go to
state
S
j
.
To
make this
Markoff process into an information source we need only assume that a letter is produced for each transition
from one state to another. The states will correspond to the “residue of influence” from preceding letters.
The situation can be represented graphically as shown in Figs. 3, 4 and 5.
The “states” are the junction
A
B
C
D
E
.
.
.
.
.
Fig. 3—A graph corresponding to the source in example B.
points in the graph and the probabilities and letters produced for a transition are given beside the correspond-
ing line.
Figure 3 is for the example B in Section 2,
while Fig. 4 corresponds to the example C.
In Fig. 3
A
A
B
B
B
C
C
.
.
.
.
.
.
.
Fig. 4—A graph corresponding to the source in example C.
there is only one state since successive letters are independent.
In Fig. 4 there are as many states as letters.
If a trigram example were constructed there would be at most
n
2
states corresponding to the possible pairs
of letters preceding the one being chosen.
Figure 5 is a graph for the case of word structure in example D.
Here S corresponds to the “space” symbol.
5.
E
RGODIC
AND
M
IXED
S
OURCES
As
we
have
indicated
above a
discrete
source
for
our
purposes
can
be
considered
to
be
represented
by
a
Markoff process.
Among the
possible discrete
Markoff processes there is
a
group with
special
properties
of
significance
in
communication
theory.
This
special
class
consists
of
the
“ergodic”
processes
and
we
shall call the corresponding sources ergodic sources. Although a rigorous definition of an ergodic process is
somewhat involved, the general idea is simple. In an ergodic process every sequence produced by the process
6
For a
detailed
treatment
see
M.
Fr´echet,
M´ethode
des
fonctions
arbitraires.
Th´eorie
des
´ev´enements
en
chaˆıne
dans
le
cas
d’un
nombre fini d’´etats possibles
.
Paris, Gauthier-Villars,
1938.
8
is
the
same
in
statistical
properties.
Thus
the
letter
frequencies,
digram
frequencies,
etc.,
obtained
from
particular
sequences,
will,
as
the
lengths
of
the
sequences
increase,
approach
definite
limits
independent
of
the particular sequence.
Actually
this
is
not true of
every sequence but
the set
for which
it
is
false
has
probability zero.
Roughly the ergodic property means statistical homogeneity.
All the examples of artificial languages given above are ergodic. This property is related to the structure
of the corresponding graph.
If the graph has the following two
properties
7
the corresponding process will
be ergodic:
1.
The graph does not consist of two isolated parts A and B such that it is impossible to go from junction
points in part A to junction points in part B along lines of the graph in the direction of arrows and also
impossible to go from junctions in part B to junctions in part A.
2.
A closed series of lines in the graph with all arrows on the lines pointing in the same orientation will
be called a “circuit.” The “length” of a circuit is the number of lines in it. Thus in Fig. 5 series BEBES
is a circuit of length 5. The second property required is that the greatest common divisor of the lengths
of all circuits in the graph be one.
S
S
S
A
A
A
A
A
B
B
B
B
B
B
B
C
D
D
D
D
D
D
E
E
E
E
E
E
E
E
E
E
E
Fig. 5—A graph corresponding to the source in example D.
If the first condition is satisfied but the second one violated by having the greatest common divisor equal
to
d
>
1, the sequences have a certain type of periodic structure.
The various sequences fall into
d
different
classes which are statistically the same apart from a shift of the origin (i.e., which letter in the sequence is
called
letter
1).
By
a shift of
from 0
up to
d
1 any sequence can
be made statistically
equivalent to
any
other.
A
simple
example
with
d
=
2
is
the
following:
There
are
three
possible
letters
a
;
b
;
c
.
Letter
a
is
followed with either
b
or
c
with probabilities
1
3
and
2
3
respectively. Either
b
or
c
is always followed by letter
a
.
Thus a typical sequence is
a b a c a c a c a b a c a b a b a c a c
:
This type of situation is not of much importance for our work.
If the first condition is violated the graph may be separated into a set of subgraphs each of which satisfies
the first condition. We will assume that the second condition is also satisfied for each subgraph. We have in
this case what may be called a “mixed” source made up of a number of pure components.
The components
correspond to the various subgraphs. If
L
1
,
L
2
,
L
3
;
:
:
:
are the component sources we may write
L
=
p
1
L
1
+
p
2
L
2
+
p
3
L
3
+
7
These are restatements in terms of the graph of conditions given in Fr´echet.
9
where
p
i
is the probability of the component source
L
i
.
Physically the situation represented is this:
There are several different sources
L
1
,
L
2
,
L
3
;
:
:
:
which are
each of
homogeneous statistical
structure (i.e.,
they are ergodic).
We
do not know
a
priori
which is to
be
used,
but once the sequence starts in a given pure component
L
i
,
it continues indefinitely according to the
statistical structure of that component.
As
an
example one may
take two
of
the
processes defined
above and
assume
p
1
=
:
2
and
p
2
=
:
8.
A
sequence from the mixed source
L
=
:
2
L
1
+
:
8
L
2
would be obtained by choosing first
L
1
or
L
2
with probabilities .2 and .8 and after this choice generating a
sequence from whichever was chosen.
Except when the contrary is stated we shall assume a source to be ergodic. This assumption enables one
to identify averages along a sequence with averages over the ensemble of possible sequences (the probability
of
a
discrepancy
being
zero).
For
example
the
relative
frequency
of
the
letter
A
in
a
particular
infinite
sequence will be, with probability one, equal to its relative frequency in the ensemble of sequences.
If
P
i
is the probability of state
i
and
p
i
(
j
)
the transition probability to state
j
, then for the process to be
stationary it is clear that the
P
i
must satisfy equilibrium conditions:
P
j
=
∑
i
P
i
p
i
(
j
)
:
In the ergodic case it can be shown that with any starting conditions the probabilities
P
j
(
N
)
of being in state
j
after
N
symbols, approach the equilibrium values as
N
!
∞
.
6.
C
HOICE
,
U
NCERTAINTY
AND
E
NTROPY
We
have represented a discrete information source as a Markoff process.
Can
we define a quantity which
will measure, in some sense, how much information is “produced” by such a process, or better, at what rate
information is produced?
Suppose we
have
a
set
of
possible
events
whose
probabilities of
occurrence are
p
1
;
p
2
;
:
:
:
;
p
n
.
These
probabilities are known but that is all we know concerning which event will occur.
Can we find a measure
of how much “choice” is involved in the selection of the event or of how uncertain we are of the outcome?
If there is such a measure, say
H
(
p
1
;
p
2
;
:
:
:
;
p
n
)
, it is reasonable to require of it the following properties:
1.
H
should be continuous in the
p
i
.
2.
If all the
p
i
are equal,
p
i
=
1
n
,
then
H
should be a monotonic increasing function of
n
.
With equally
likely events there is more choice, or uncertainty, when there are more possible events.
3.
If a
choice be broken down into two successive choices, the original
H
should be the weighted sum
of the individual values of
H
.
The meaning of this is
illustrated in
Fig. 6.
At the left we have three
/
/
/
/
/
/
/
/
/
/
Fig. 6—Decomposition of a choice from three possibilities.
possibilities
p
1
=
1
2
,
p
2
=
1
3
,
p
3
=
1
6
.
On the right we first choose between two possibilities each with
probability
1
2
, and if the second occurs make another choice with probabilities
2
3
,
1
3
.
The final results
have the same probabilities as before. We require, in this special case, that
H
(
1
2
;
1
3
;
1
6
)
=
H
(
1
2
;
1
2
)
+
1
2
H
(
2
3
;
1
3
)
:
The coefficient
1
2
is because this second choice only occurs half the time.
10
In Appendix 2, the following result is established:
Theorem 2:
The only
H
satisfying the three above assumptions is of the form:
H
=
K
n
∑
i
=
1
p
i
log
p
i
where
K
is a positive constant.
This theorem, and the assumptions required for its proof, are in no way necessary for the present theory.
It is given chiefly to lend a certain plausibility to some of our later definitions. The real justification of these
definitions, however, will reside in their implications.
Quantities of the form
H
=
∑
p
i
log
p
i
(the constant
K
merely amounts to a choice of a unit of measure)
play a central role in information theory as measures of information, choice and uncertainty. The form of
H
will be recognized as that of entropy as defined in certain formulations of statistical mechanics
8
where
p
i
is
the probability of a system being in cell
i
of its phase space.
H
is then, for example, the
H
in Boltzmann’s
famous
H
theorem. We shall call
H
=
∑
p
i
log
p
i
the entropy of the set of probabilities
p
1
;
:
:
:
;
p
n
.
If
x
is a
chance variable we will write
H
(
x
)
for its entropy; thus
x
is not an argument of a function but a label for a
number, to differentiate it from
H
(
y
)
say, the entropy of the chance variable
y
.
The entropy in the case of two possibilities with probabilities
p
and
q
=
1
p
, namely
H
=
(
p
log
p
+
q
log
q
)
is plotted in Fig. 7 as a function of
p
.
H
BITS
p
0
.
.
.
.
.
.
.
.
.
.0
0
.
.
.
.
.
.
.
.
.
.0
Fig. 7—Entropy in the case of two possibilities with probabilities
p
and
(
1
p
)
.
The
quantity
H
has
a
number
of
interesting
properties
which
further
substantiate
it
as
a
reasonable
measure of choice or information.
1.
H
=
0 if and only if all the
p
i
but one are zero, this one having the value unity.
Thus only when we
are certain of the outcome does
H
vanish. Otherwise
H
is positive.
2.
For
a
given
n
,
H
is
a
maximum
and
equal
to
log
n
when
all
the
p
i
are
equal
(i.e.,
1
n
).
This
is
also
intuitively the most uncertain situation.
8
See, for example, R. C. Tolman,
Principles of Statistical
Mechanics,
Oxford, Clarendon, 1938.
11
3. Suppose there are two events,
x
and
y
, in question with
m
possibilities for the first and
n
for the second.
Let
p
(
i
;
j
)
be the probability of the joint occurrence of
i
for the first and
j
for the second. The entropy of the
joint event is
H
(
x
;
y
)
=
∑
i
;
j
p
(
i
;
j
)
log
p
(
i
;
j
)
while
H
(
x
)
=
∑
i
;
j
p
(
i
;
j
)
log
∑
j
p
(
i
;
j
)
H
(
y
)
=
∑
i
;
j
p
(
i
;
j
)
log
∑
i
p
(
i
;
j
)
:
It is easily shown that
H
(
x
;
y
)
H
(
x
)
+
H
(
y
)
with equality only if the events are independent (i.e.,
p
(
i
;
j
)
=
p
(
i
)
p
(
j
)
).
The uncertainty of a joint event is
less than or equal to the sum of the individual uncertainties.
4.
Any change toward equalization of the probabilities
p
1
;
p
2
;
:
:
:
;
p
n
increases
H
.
Thus if
p
1
<
p
2
and
we increase
p
1
, decreasing
p
2
an equal amount so that
p
1
and
p
2
are more nearly equal, then
H
increases.
More generally, if we perform any “averaging” operation on the
p
i
of the form
p
0
i
=
∑
j
a
ij
p
j
where
∑
i
a
ij
=
∑
j
a
ij
=
1, and all
a
ij
0, then
H
increases (except in the special case where this transfor-
mation amounts to no more than a permutation of the
p
j
with
H
of course remaining the same).
5.
Suppose there are two chance events
x
and
y
as in 3, not necessarily independent.
For any particular
value
i
that
x
can assume there is a conditional probability
p
i
(
j
)
that
y
has the value
j
.
This is given by
p
i
(
j
)
=
p
(
i
;
j
)
∑
j
p
(
i
;
j
)
:
We define the
conditional entropy
of
y
,
H
x
(
y
)
as the average of the entropy of
y
for each value of
x
, weighted
according to the probability of getting that particular
x
.
That is
H
x
(
y
)
=
∑
i
;
j
p
(
i
;
j
)
log
p
i
(
j
)
:
This quantity measures how uncertain we are of
y
on the average when we know
x
.
Substituting the value of
p
i
(
j
)
we obtain
H
x
(
y
)
=
∑
i
;
j
p
(
i
;
j
)
log
p
(
i
;
j
)
+
∑
i
;
j
p
(
i
;
j
)
log
∑
j
p
(
i
;
j
)
=
H
(
x
;
y
)
H
(
x
)
or
H
(
x
;
y
)
=
H
(
x
)
+
H
x
(
y
)
:
The uncertainty (or entropy) of the joint event
x
;
y
is the uncertainty of
x
plus the uncertainty of
y
when
x
is
known.
6.
From 3 and 5 we have
H
(
x
)
+
H
(
y
)
H
(
x
;
y
)
=
H
(
x
)
+
H
x
(
y
)
:
Hence
H
(
y
)
H
x
(
y
)
:
The uncertainty of
y
is never increased by knowledge of
x
.
It will be decreased unless
x
and
y
are independent
events, in which case it is not changed.
12
7.
T
HE
E
NTROPY
OF
AN
I
NFORMATION
S
OURCE
Consider a discrete source of the finite state type considered above.
For each possible state
i
there will be a
set of probabilities
p
i
(
j
)
of producing the various possible symbols
j
.
Thus there is an entropy
H
i
for each
state.
The entropy of the source will be defined as the average of these
H
i
weighted in accordance with the
probability of occurrence of the states in question:
H
=
∑
i
P
i
H
i
=
∑
i
;
j
P
i
p
i
(
j
)
log
p
i
(
j
)
:
This is the entropy of the source per symbol of text.
If the Markoff process is proceeding at a definite time
rate there is also an entropy per second
H
0
=
∑
i
f
i
H
i
where
f
i
is the average frequency (occurrences per second) of state
i
.
Clearly
H
0
=
mH
where
m
is the average number of symbols produced per second.
H
or
H
0
measures the amount of informa-
tion generated by the source per symbol or per second.
If the logarithmic base is 2, they will represent bits
per symbol or per second.
If successive symbols are independent then
H
is simply
∑
p
i
log
p
i
where
p
i
is the probability of sym-
bol
i
.
Suppose in this case we consider a long message of
N
symbols.
It will contain with high probability
about
p
1
N
occurrences of the first symbol,
p
2
N
occurrences of the second, etc. Hence the probability of this
particular message will be roughly
p
=
p
p
1
N
1
p
p
2
N
2
p
p
n
N
n
or
log
p
:
=
N
∑
i
p
i
log
p
i
log
p
:
=
NH
H
:
=
log1
=
p
N
:
H
is thus approximately the logarithm of the reciprocal probability of a typical long sequence divided by the
number of symbols in the sequence.
The same result holds for any source.
Stated more precisely we have
(see Appendix 3):
Theorem 3:
Given any
>
0 and
>
0, we can find an
N
0
such that the sequences of any length
N
N
0
fall into two classes:
1.
A set whose total probability is less than
.
2.
The remainder, all of whose members have probabilities satisfying the inequality
log
p
1
N
H
<
:
In other words we are almost certain to have
log
p
1
N
very close to
H
when
N
is large.
A closely related result deals with the number of sequences of various probabilities.
Consider again the
sequences
of
length
N
and
let
them
be
arranged
in
order
of
decreasing
probability.
We
define
n
(
q
)
to
be
the
number we
must
take
from this
set
starting
with
the
most
probable one in
order
to
accumulate a
total
probability
q
for those taken.
13
Theorem 4:
Lim
N
!
∞
log
n
(
q
)
N
=
H
when
q
does not equal 0 or 1.
We may interpret log
n
(
q
)
as the number of bits required to specify the sequence when we consider only
the most probable sequences with a total probability
q
.
Then
log
n
(
q
)
N
is the number of bits per symbol for
the specification.
The theorem says that for large
N
this will be independent of
q
and equal to
H
.
The rate
of growth of the logarithm of the number of reasonably probable sequences is given by
H
, regardless of our
interpretation of “reasonably probable.” Due to these results, which are proved in Appendix 3, it is possible
for most purposes to treat the long sequences as though there were just 2
HN
of them, each with a probability
2
HN
.
The
next
two
theorems
show
that
H
and
H
0
can
be
determined
by
limiting
operations
directly
from
the statistics of the message sequences, without reference to the states and transition probabilities between
states.
Theorem 5:
Let
p
(
B
i
)
be the probability of a sequence
B
i
of symbols from the source. Let
G
N
=
1
N
∑
i
p
(
B
i
)
log
p
(
B
i
)
where the sum is over all sequences
B
i
containing
N
symbols.
Then
G
N
is a monotonic decreasing function
of
N
and
Lim
N
!
∞
G
N
=
H
:
Theorem 6:
Let
p
(
B
i
;
S
j
)
be
the
probability
of
sequence
B
i
followed
by
symbol
S
j
and
p
B
i
(
S
j
)
=
p
(
B
i
;
S
j
)
=
p
(
B
i
)
be the conditional probability of
S
j
after
B
i
.
Let
F
N
=
∑
i
;
j
p
(
B
i
;
S
j
)
log
p
B
i
(
S
j
)
where
the
sum
is
over
all
blocks
B
i
of
N
1
symbols
and
over
all
symbols
S
j
.
Then
F
N
is
a
monotonic
decreasing function of
N
,
F
N
=
NG
N
(
N
1
)
G
N
1
;
G
N
=
1
N
N
∑
n
=
1
F
n
;
F
N
G
N
;
and Lim
N
!
∞
F
N
=
H
.
These results are derived in Appendix 3. They show that a series of approximations to
H
can be obtained
by considering only the statistical structure of the sequences extending over 1
;
2
;
:
:
:
;
N
symbols.
F
N
is the
better
approximation.
In
fact
F
N
is
the
entropy
of
the
N
th
order
approximation to
the
source
of
the
type
discussed
above.
If
there
are
no
statistical
influences
extending
over
more
than
N
symbols,
that
is
if
the
conditional probability of the next symbol knowing the preceding
(
N
1
)
is not changed by a knowledge of
any before that, then
F
N
=
H
.
F
N
of course is the conditional entropy of the next symbol when the
(
N
1
)
preceding ones are known, while
G
N
is the entropy per symbol of blocks of
N
symbols.
The ratio of the entropy of a source to the maximum value it could have while still restricted to the same
symbols will be called its
relative entropy
. This is the maximum compression possible when we encode into
the same alphabet.
One minus the relative entropy is the
redundancy
.
The redundancy of ordinary English,
not
considering
statistical
structure
over
greater
distances
than
about
eight
letters,
is
roughly
50%.
This
means that when we write English half of what we write is determined by the structure of the language and
half is chosen freely.
The figure 50% was found by several independent methods which all gave results in
14
this neighborhood. One is by calculation of the entropy of the approximations to English. A second method
is
to
delete a
certain
fraction of
the letters from a
sample of
English text and
then
let someone attempt to
restore them.
If they can
be restored when 50% are deleted the redundancy must be greater than 50%.
A
third method depends on certain known results in cryptography.
Two
extremes of
redundancy in
English
prose are represented by Basic
English and
by James
Joyce’s
book “Finnegans Wake”.
The Basic English vocabulary is limited to 850 words and the redundancy is very
high.
This is reflected in the expansion that occurs when a passage is translated into Basic English.
Joyce
on the other hand enlarges the vocabulary and is alleged to achieve a compression of semantic content.
The
redundancy
of
a
language
is
related
to
the
existence
of
crossword
puzzles.
If
the
redundancy
is
zero
any
sequence of
letters
is
a
reasonable text in
the
language and
any
two-dimensional array
of
letters
forms a crossword puzzle. If the redundancy is too high the language imposes too many constraints for large
crossword puzzles to be possible. A more detailed analysis shows that if we assume the constraints imposed
by the language are of a rather chaotic and random nature, large crossword puzzles are just possible when
the redundancy is 50%. If the redundancy is 33%, three-dimensional crossword puzzles should be possible,
etc.
8.
R
EPRESENTATION
OF
THE
E
NCODING
AND
D
ECODING
O
PERATIONS
We
have
yet
to
represent
mathematically
the
operations
performed
by
the
transmitter
and
receiver
in
en-
coding and decoding the information.
Either of these will be called a discrete transducer.
The input to the
transducer is a sequence of input symbols and its output a sequence of output symbols. The transducer may
have an internal memory so that its output depends not only on the present input symbol but also on the past
history. We assume that the internal memory is finite, i.e., there exist a finite number
m
of possible states of
the transducer and that its output is a function of the present state and the present input symbol.
The next
state will be a second function of these two quantities. Thus a transducer can be described by two functions:
y
n
=
f
(
x
n
;
n
)
n
+
1
=
g
(
x
n
;
n
)
where
x
n
is the
n
th
input symbol,
n
is the state of the transducer when the
n
th
input symbol is introduced,
y
n
is the output symbol (or sequence of output symbols) produced when
x
n
is introduced if the state is
n
.
If the output symbols of one transducer can be identified with the input symbols of a second, they can be
connected in tandem and the result is also a transducer.
If there exists a second transducer which operates
on the output of the first and recovers the original input, the first transducer will be called non-singular and
the second will be called its inverse.
Theorem 7:
The
output
of
a
finite
state
transducer
driven
by
a
finite
state
statistical
source
is
a
finite
state statistical source, with entropy (per unit time) less than or equal to that of the input.
If the transducer
is non-singular they are equal.
Let
represent the state of the source, which produces a sequence of symbols
x
i
; and let
be the state of
the transducer, which produces, in its output, blocks of symbols
y
j
. The combined system can be represented
by the “product state space” of pairs
(
;
)
.
Two points in the space
(
1
;
1
)
and
(
2
;
2
)
, are connected by
a line if
1
can produce an
x
which changes
1
to
2
, and this line is given the probability of that
x
in this
case.
The line is labeled with the block of
y
j
symbols produced by the transducer. The entropy of the output
can be calculated as the weighted sum over the states.
If we sum first on
each resulting term is less than or
equal to the corresponding term for
, hence the entropy is not increased.
If the transducer is non-singular
let its output be connected to the inverse transducer. If
H
0
1
,
H
0
2
and
H
0
3
are the output entropies of the source,
the first and second transducers respectively, then
H
0
1
H
0
2
H
0
3
=
H
0
1
and therefore
H
0
1
=
H
0
2
.
15
Suppose we have a system of constraints on possible sequences of the type which can be represented by
a linear graph as in Fig. 2.
If probabilities
p
(
s
)
ij
were assigned to the various lines connecting state
i
to state
j
this would become a source. There is one particular assignment which maximizes the resulting entropy (see
Appendix 4).
Theorem 8:
Let
the
system
of
constraints
considered
as
a
channel
have
a
capacity
C
=
log
W
.
If
we
assign
p
(
s
)
ij
=
B
j
B
i
W
`
(
s
)
ij
where
`
(
s
)
ij
is the duration of the
s
th
symbol leading from state
i
to state
j
and the
B
i
satisfy
B
i
=
∑
s
;
j
B
j
W
`
(
s
)
ij
then
H
is maximized and equal to
C
.
By proper assignment of the transition probabilities the entropy of symbols on a channel can be maxi-
mized at the channel capacity.
9.
T
HE
F
UNDAMENTAL
T
HEOREM
FOR
A
N
OISELESS
C
HANNEL
We
will now justify our interpretation of
H
as the rate of generating information by proving that
H
deter-
mines the channel capacity required with most efficient coding.
Theorem 9:
Let
a
source have
entropy
H
(
bits
per
symbol
)
and
a
channel have a
capacity
C
(
bits
per
second
)
.
Then it is possible to encode the output of the source in such a way as to transmit at the average
rate
C
H
symbols per second over the channel where
is arbitrarily small.
It is not possible to transmit at
an average rate greater than
C
H
.
The converse part of the theorem, that
C
H
cannot be exceeded, may be proved by noting that the entropy
of the channel input per second is equal to that of the source, since the transmitter must be non-singular, and
also this entropy cannot exceed the channel capacity. Hence
H
0
C
and the number of symbols per second
=
H
0
=
H
C
=
H
.
The first
part of
the theorem will be proved in
two
different ways.
The first
method is to
consider the
set of all sequences of
N
symbols produced by the source. For
N
large we can divide these into two groups,
one containing less
than 2
(
H
+
)
N
members and the second
containing less
than 2
RN
members (where
R
is
the logarithm of the number of different symbols) and having a total probability less than
.
As
N
increases
and
approach zero.
The number of signals of duration
T
in the channel is greater than 2
(
C
)
T
with
small when
T
is large. if we choose
T
=
H
C
+
N
then there will be a sufficient number of sequences of channel symbols for the high probability group when
N
and
T
are sufficiently large (however small
) and also some additional ones. The high probability group
is
coded
in
an
arbitrary
one-to-one way
into
this
set.
The
remaining
sequences
are
represented
by
larger
sequences,
starting
and
ending
with
one
of
the
sequences
not
used
for
the
high
probability
group.
This
special sequence acts as a start and stop signal for a different code.
In between a sufficient time is allowed
to give enough different sequences for all the low probability messages.
This will require
T
1
=
R
C
+
'
N
where
'
is small.
The mean rate of transmission in message symbols per second will then be greater than
(
1
)
T
N
+
T
1
N
#
1
=
(
1
)
H
C
+
+
R
C
+
'
1
:
16
As
N
increases
,
and
'
approach zero and the rate approaches
C
H
.
Another method of performing this coding and thereby proving the theorem can be described as follows:
Arrange
the
messages
of
length
N
in
order
of
decreasing
probability
and
suppose
their
probabilities
are
p
1
p
2
p
3
p
n
.
Let
P
s
=
∑
s
1
1
p
i
; that is
P
s
is the cumulative probability up to, but not including,
p
s
.
We first encode into a binary system.
The binary code for message
s
is obtained by expanding
P
s
as a binary
number. The expansion is carried out to
m
s
places, where
m
s
is the integer satisfying:
log
2
1
p
s
m
s
<
1
+
log
2
1
p
s
:
Thus the messages of high probability are represented by short codes and those of low probability by long
codes.
From these inequalities we have
1
2
m
s
p
s
<
1
2
m
s
1
:
The code for
P
s
will differ from all succeeding ones in one or more of its
m
s
places, since all the remaining
P
i
are at least
1
2
ms
larger and their binary expansions therefore differ in the first
m
s
places.
Consequently all
the codes are different and it is possible to recover the message from its code.
If the channel sequences are
not already sequences of binary digits, they can be ascribed binary numbers in an arbitrary fashion and the
binary code thus translated into signals suitable for the channel.
The average number
H
0
of
binary
digits
used
per
symbol of
original message
is
easily
estimated.
We
have
H
0
=
1
N
∑
m
s
p
s
:
But,
1
N
∑
log
2
1
p
s
p
s
1
N
∑
m
s
p
s
<
1
N
∑
1
+
log
2
1
p
s
p
s
and therefore,
G
N
H
0
<
G
N
+
1
N
As
N
increases
G
N
approaches
H
, the entropy of the source and
H
0
approaches
H
.
We
see
from
this
that
the
inefficiency
in
coding,
when
only
a
finite
delay
of
N
symbols
is
used,
need
not
be
greater
than
1
N
plus
the
difference
between
the
true
entropy
H
and
the
entropy
G
N
calculated
for
sequences of length
N
.
The per cent excess time needed over the ideal is therefore less than
G
N
H
+
1
HN
1
:
This
method
of
encoding
is
substantially
the
same
as
one
found
independently
by
R.
M.
Fano.
9
His
method is to arrange the messages of length
N
in order of decreasing probability. Divide this series into two
groups of
as
nearly
equal probability
as
possible.
If
the
message
is
in
the
first
group
its
first
binary
digit
will
be
0,
otherwise
1.
The
groups
are
similarly
divided
into
subsets
of
nearly
equal
probability
and
the
particular
subset
determines the
second
binary
digit.
This
process is
continued until each
subset
contains
only one message. It is easily seen that apart from minor differences (generally in the last digit) this amounts
to the same thing as the arithmetic process described above.
10.
D
ISCUSSION
AND
E
XAMPLES
In order to obtain the maximum power transfer from a generator to a load, a transformer must in general be
introduced so that the generator as seen from the load has the load resistance.
The situation here is roughly
analogous.
The transducer which does the encoding should match the source to the channel in a statistical
sense.
The source as seen from the channel through the transducer should have the same statistical structure
9
Technical Report No. 65, The Research Laboratory of Electronics, M.I.T., March 17, 1949.
17
as the source which maximizes the entropy in
the channel.
The content of Theorem 9 is that,
although an
exact match is not in general possible, we can approximate it as closely as desired.
The ratio of the actual
rate of transmission to the capacity
C
may be called the efficiency of the coding system.
This is of course
equal to the ratio of the actual entropy of the channel symbols to the maximum possible entropy.
In
general,
ideal
or
nearly
ideal
encoding requires
a
long
delay
in
the
transmitter
and
receiver.
In
the
noiseless case which we have been considering, the main function of this delay is to allow reasonably good
matching
of
probabilities
to
corresponding
lengths
of
sequences.
With
a
good
code
the
logarithm
of
the
reciprocal probability of a long message must be proportional to the duration of the corresponding signal, in
fact
log
p
1
T
C
must be small for all but a small fraction of the long messages.
If a source can produce only one particular message its entropy is zero, and no channel is required.
For
example,
a computing machine set up
to
calculate the successive digits
of
produces a
definite sequence
with no chance element.
No channel is required to “transmit” this to another point.
One could construct a
second machine to compute the same sequence at the point. However, this may be impractical. In such a case
we can choose to ignore some or all of the statistical knowledge we have of the source.
We might consider
the digits of
to be a random sequence in that we construct a system capable of sending any sequence of
digits.
In a similar way we may choose to use some of our statistical knowledge of English in constructing
a
code,
but
not
all
of
it.
In
such
a
case
we
consider
the
source with
the
maximum entropy
subject
to
the
statistical conditions we wish
to retain.
The entropy of this source determines the channel capacity which
is necessary and sufficient.
In the
example the only information retained is that all the digits are chosen
from the set
0
;
1
;
:
:
:
;
9.
In
the case of
English one might wish
to
use the statistical
saving
possible due to
letter frequencies, but nothing else.
The maximum entropy source is then the first approximation to English
and its entropy determines the required channel capacity.
As
a
simple
example
of
some
of
these
results
consider
a
source
which
produces a
sequence
of
letters
chosen from among
A
,
B
,
C
,
D
with probabilities
1
2
,
1
4
,
1
8
,
1
8
, successive symbols being chosen independently.
We have
H
=
1
2
log
1
2
+
1
4
log
1
4
+
2
8
log
1
8
=
7
4
bits per symbol
:
Thus we can approximate a coding system to encode messages from this source into binary digits with an
average of
7
4
binary digit per symbol. In this case we can actually achieve the limiting value by the following
code (obtained by the method of the second proof of Theorem 9):
A
0
B
10
C
110
D
111
The average number of binary digits used in encoding a sequence of
N
symbols will be
N
1
2
1
+
1
4
2
+
2
8
3
=
7
4
N
:
It is easily seen
that the binary digits 0,
1
have probabilities
1
2
,
1
2
so
the
H
for the coded sequences is one
bit per symbol.
Since, on the average, we have
7
4
binary symbols per original letter, the entropies on a time
basis are the same.
The maximum possible entropy for the original set is log4
=
2, occurring when
A
,
B
,
C
,
D
have probabilities
1
4
,
1
4
,
1
4
,
1
4
.
Hence the relative entropy is
7
8
.
We can translate the binary sequences into
the original set of symbols on a two-to-one basis by the following table:
00
A
0
01
B
0
10
C
0
11
D
0
18
This double process then encodes the original message into the same symbols but with an average compres-
sion ratio
7
8
.
As a second example consider a source which produces a sequence of
A
’s and
B
’s with probability
p
for
A
and
q
for
B
.
If
p
q
we have
H
=
log
p
p
(
1
p
)
1
p
=
p
log
p
(
1
p
)
(
1
p
)
=
p
:
=
p
log
e
p
:
In such a case one can construct a fairly good coding of the message on a 0, 1 channel by sending a special
sequence, say 0000, for the infrequent symbol
A
and then a sequence indicating the
number
of
B
’s following
it.
This
could
be indicated
by
the
binary representation with
all
numbers containing the
special sequence
deleted. All numbers up to 16 are represented as usual; 16 is represented by the next binary number after 16
which does not contain four zeros, namely 17
=
10001, etc.
It can be shown that as
p
!
0 the coding approaches ideal provided the length of the special sequence is
properly adjusted.
PART II: THE DISCRETE CHANNEL WITH NOISE
11.
R
EPRESENTATION
OF
A
N
OISY
D
ISCRETE
C
HANNEL
We now consider the case where the signal is perturbed by noise during transmission or at one or the other
of
the
terminals.
This
means
that
the
received
signal
is
not
necessarily
the
same
as
that
sent
out
by
the
transmitter.
Two
cases
may
be
distinguished.
If
a
particular transmitted
signal
always
produces the
same
received signal, i.e., the received signal is a definite function of the transmitted signal, then the effect may be
called distortion.
If this function 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.
The case of interest here is that in which the signal does not always undergo the same change in trans-
mission.
In this case we may assume the received signal
E
to be a function of the transmitted signal
S
and a
second variable, the noise
N
.
E
=
f
(
S
;
N
)
The noise is considered to be a chance variable just as the message was above.
In general it may be repre-
sented by a suitable stochastic process.
The most general type of noisy discrete channel we shall consider
is a generalization of the finite state noise-free channel described previously.
We assume a finite number of
states and a set of probabilities
p
;
i
(
;
j
)
:
This is the probability, if the channel is in state
and symbol
i
is transmitted, that symbol
j
will be received
and the channel left in state
.
Thus
and
range over the possible states,
i
over the possible transmitted
signals and
j
over the possible received signals. In the case where successive symbols are independently per-
turbed by the noise there is only one state, and the channel is described by the set of transition probabilities
p
i
(
j
)
, the probability of transmitted symbol
i
being received as
j
.
If a noisy channel is fed by a source there are two statistical processes at work: the source and the noise.
Thus there are a
number of
entropies that
can
be calculated.
First
there is
the entropy
H
(
x
)
of
the source
or
of
the
input
to
the
channel
(these
will
be
equal
if
the
transmitter
is
non-singular).
The
entropy
of
the
output of the channel, i.e., the received signal, will be denoted by
H
(
y
)
.
In the noiseless case
H
(
y
)
=
H
(
x
)
.
The joint entropy of input and output will be
H
(
xy
)
.
Finally there are two conditional entropies
H
x
(
y
)
and
H
y
(
x
)
, the entropy of the output when the input is known and conversely.
Among these quantities we have
the relations
H
(
x
;
y
)
=
H
(
x
)
+
H
x
(
y
)
=
H
(
y
)
+
H
y
(
x
)
:
All of these entropies can be measured on a per-second or a per-symbol basis.
19
12.
E
QUIVOCATION
AND
C
HANNEL
C
APACITY
If
the
channel
is
noisy
it
is
not
in
general
possible
to
reconstruct the
original
message
or
the
transmitted
signal with
certainty
by any operation on the received signal
E
.
There are,
however, ways of transmitting
the information which are optimal in combating noise.
This is the problem which we now consider.
Suppose there are two possible symbols 0 and 1, and we are transmitting at a rate of 1000 symbols per
second with
probabilities
p
0
=
p
1
=
1
2
.
Thus our source is
producing information at
the rate of
1000 bits
per
second.
During
transmission
the
noise
introduces errors
so
that,
on
the
average,
1
in
100
is
received
incorrectly (a 0 as 1, or 1 as 0).
What is the rate of transmission of information?
Certainly less than 1000
bits
per
second
since
about
1%
of
the
received
symbols
are
incorrect.
Our
first
impulse
might
be
to
say
the
rate
is
990 bits
per
second,
merely
subtracting the
expected number of
errors.
This
is
not satisfactory
since it fails to take into account the recipient’s lack of knowledge of where the errors occur.
We may carry
it to
an extreme case
and suppose the
noise so
great that the received symbols are entirely independent of
the transmitted symbols.
The probability of receiving 1 is
1
2
whatever was transmitted and similarly for 0.
Then about half of the received symbols are correct due to chance alone, and we would be giving the system
credit for transmitting 500 bits per second while actually no information is being transmitted at all.
Equally
“good” transmission would
be obtained by dispensing with
the channel entirely and
flipping a
coin
at the
receiving point.
Evidently the proper correction to apply to the amount of information transmitted is the amount of this
information which is missing in the received signal, or alternatively the uncertainty when we have received
a signal of what was actually sent.
From our previous discussion of entropy as a measure of uncertainty it
seems reasonable to use the conditional entropy of the message, knowing the received signal, as a measure
of this missing information.
This is indeed the proper definition, as we shall see later.
Following this idea
the
rate
of
actual
transmission,
R
,
would
be
obtained
by
subtracting
from
the
rate
of
production (i.e.,
the
entropy of the source) the average rate of conditional entropy.
R
=
H
(
x
)
H
y
(
x
)
The conditional entropy
H
y
(
x
)
will, for convenience, be called the equivocation. It measures the average
ambiguity of the received signal.
In the example considered above, if a 0 is received the
a posteriori
probability that a 0 was transmitted
is .99, and that a 1 was transmitted is .01. These figures are reversed if a 1 is received. Hence
H
y
(
x
)
=
[
:
99log
:
99
+
0
:
01log0
:
01
]
=
:
081 bits/symbol
or 81 bits per second. We may say that the system is transmitting at a rate 1000
81
=
919 bits per second.
In the extreme case where a 0 is equally likely to be received as a 0 or 1 and similarly for 1, the
a posteriori
probabilities are
1
2
,
1
2
and
H
y
(
x
)
=
1
2
log
1
2
+
1
2
log
1
2
=
1 bit per symbol
or 1000 bits per second. The rate of transmission is then 0 as it should be.
The following theorem gives a direct intuitive interpretation of the equivocation and also serves to justify
it
as the unique appropriate measure.
We
consider a communication system
and
an observer (or
auxiliary
device) who can see both what is sent and what is recovered (with errors due to noise).
This observer notes
the errors in the recovered message and transmits data to the receiving point over a “correction channel” to
enable the receiver to correct the errors. The situation is indicated schematically in Fig. 8.
Theorem 10:
If
the
correction
channel
has
a
capacity
equal
to
H
y
(
x
)
it
is
possible
to
so
encode
the
correction data as to send it over this channel and correct all but an arbitrarily small fraction
of the errors.
This is not possible if the channel capacity is less than
H
y
(
x
)
.
20
SOURCE
M
TRANSMITTER
RECEIVER
CORRECTING
DEVICE
OBSERVER
M
0
M
CORRECTION
DATA
Fig. 8—Schematic diagram of a correction system.
Roughly
then,
H
y
(
x
)
is
the
amount
of
additional
information
that
must
be
supplied
per
second
at
the
receiving point to correct the received message.
To
prove
the
first
part,
consider
long
sequences
of
received
message
M
0
and
corresponding
original
message
M
.
There will be logarithmically
TH
y
(
x
)
of the
M
’s which could reasonably have produced each
M
0
.
Thus we have
TH
y
(
x
)
binary digits to send each
T
seconds. This can be done with
frequency of errors
on a channel of capacity
H
y
(
x
)
.
The second part can be proved by noting, first, that for any discrete chance variables
x
,
y
,
z
H
y
(
x
;
z
)
H
y
(
x
)
:
The left-hand side can be expanded to give
H
y
(
z
)
+
H
yz
(
x
)
H
y
(
x
)
H
yz
(
x
)
H
y
(
x
)
H
y
(
z
)
H
y
(
x
)
H
(
z
)
:
If we identify
x
as the output of the source,
y
as the received signal and
z
as the signal sent over the correction
channel, then the right-hand side is the equivocation less the rate of transmission over the correction channel.
If the capacity of this channel is less than the equivocation the right-hand side will be greater than zero and
H
yz
(
x
)
>
0.
But this is the uncertainty of what was sent, knowing both the received signal and the correction
signal.
If this is greater than zero the frequency of errors cannot be arbitrarily small.
Example:
Suppose the errors occur at random in a sequence of binary digits:
probability
p
that a digit is wrong
and
q
=
1
p
that
it
is
right.
These
errors
can
be
corrected
if
their
position
is
known.
Thus
the
correction
channel
need
only
send
information
as
to
these
positions.
This
amounts
to
transmitting
from a source which produces binary digits with probability
p
for 1 (incorrect) and
q
for 0 (correct).
This requires a channel of capacity
[
p
log
p
+
q
log
q
]
which is the equivocation of the original system.
The rate of transmission
R
can be written in two other forms due to the identities noted above. We have
R
=
H
(
x
)
H
y
(
x
)
=
H
(
y
)
H
x
(
y
)
=
H
(
x
)
+
H
(
y
)
H
(
x
;
y
)
:
21
The first defining expression has already been interpreted as the amount of information sent less the uncer-
tainty of what was sent. The second measures the amount received less the part of this which is due to noise.
The third is the sum of the two amounts less the joint entropy and therefore in a sense is the number of bits
per second common to the two.
Thus all three expressions have a certain intuitive significance.
The capacity
C
of
a
noisy channel should
be the maximum possible rate of
transmission,
i.e.,
the rate
when the source is properly matched to the channel. We therefore define the channel capacity by
C
=
Max
H
(
x
)
H
y
(
x
)
where the maximum is with respect to all possible information sources used as input to the channel.
If the
channel is noiseless,
H
y
(
x
)
=
0. The definition is then equivalent to that already given for a noiseless channel
since the maximum entropy for the channel is its capacity.
13.
T
HE
F
UNDAMENTAL
T
HEOREM
FOR
A
D
ISCRETE
C
HANNEL
WITH
N
OISE
It may seem surprising that we should define a definite capacity
C
for a noisy channel since we can never
send certain information in such a case.
It is clear, however, that by sending the information in a redundant
form the probability of errors can be reduced.
For example, by repeating the message many times and by a
statistical study of the different received versions of the message the probability of errors could be made very
small.
One
would
expect,
however,
that to
make this
probability of
errors approach zero,
the
redundancy
of the encoding must increase indefinitely, and the rate of transmission therefore approach zero.
This is by
no means true.
If it were, there would not be a very well defined capacity, but only a capacity for a given
frequency of
errors,
or
a given equivocation;
the capacity going down as
the error
requirements are made
more stringent. Actually the capacity
C
defined above has a very definite significance.
It is possible to send
information at the rate
C
through the channel
with as small a frequency of errors or equivocation as desired
by proper encoding.
This statement is not true for any rate greater than
C
.
If an attempt is made to transmit
at a higher rate than
C
, say
C
+
R
1
, then there will necessarily be an equivocation equal to or greater than the
excess
R
1
.
Nature takes payment by requiring just that much uncertainty, so that we are not actually getting
any more than
C
through correctly.
The situation is indicated in Fig. 9.
The rate of information into the channel is plotted horizontally and
the equivocation vertically.
Any point above the heavy line in the shaded region can be attained and those
below cannot.
The points on the line cannot in general be attained, but there will usually be two points on
the line that can.
These results are the main justification for the definition of
C
and will now be proved.
Theorem 11:
Let a discrete channel have the capacity
C
and a discrete source the entropy per second
H
.
If
H
C
there exists a coding system such that the output of the source can be transmitted over the channel
with
an arbitrarily small frequency of errors (or an
arbitrarily small equivocation).
If
H
>
C
it is
possible
to encode the source so that the equivocation is less than
H
C
+
where
is arbitrarily small.
There is no
method of encoding which gives an equivocation less than
H
C
.
The
method
of
proving
the
first
part
of
this
theorem
is
not
by
exhibiting
a
coding
method
having
the
desired properties, but by showing that such a code must exist in a certain group of codes.
In fact we will
ATTAINABLE
REGION
C
H
(
x
)
H
y
(
x
)
SLOPE
=
1.0
Fig. 9—The equivocation possible for a given input entropy to a channel.
22
average the frequency of errors over this group and show that this average can be made less than
.
If the
average of a set of numbers is less than
there must exist at least one in the set which is less than
.
This
will establish the desired result.
The capacity
C
of a noisy channel has been defined as
C
=
Max
H
(
x
)
H
y
(
x
)
where
x
is the input and
y
the output.
The maximization is over all sources which might be used as input to
the channel.
Let
S
0
be a source which achieves the maximum capacity
C
.
If this maximum is not actually achieved
by
any
source let
S
0
be a
source which approximates to
giving the
maximum rate.
Suppose
S
0
is
used
as
input to the channel. We consider the possible transmitted and received sequences of a long duration
T
.
The
following will be true:
1.
The transmitted sequences fall into two classes, a high probability group with about 2
TH
(
x
)
members
and the remaining sequences of small total probability.
2.
Similarly
the
received
sequences
have
a
high
probability
set
of
about
2
TH
(
y
)
members
and
a
low
probability set of remaining sequences.
3.
Each high probability output could be produced by about 2
TH
y
(
x
)
inputs.
The probability of all other
cases has a small total probability.
All the
’s and
’s
implied by the words “small” and “about” in
these statements approach zero as
we
allow
T
to increase and
S
0
to approach the maximizing source.
The
situation
is
summarized
in
Fig.
10
where
the
input
sequences
are
points
on
the
left
and
output
sequences points on
the
right.
The fan
of
cross
lines
represents the
range of
possible
causes for
a
typical
output.
M
E
2
H
(
x
)
T
HIGH
PROBABILITY
MESSAGES
2
H
(
y
)
T
HIGH
PROBABILITY
RECEIVED
SIGNALS
2
H
y
(
x
)
T
REASONABLE
CAUSES
FOR
EACH
E
2
H
x
(
y
)
T
REASONABLE
EFFECTS
FOR
EACH
M
Fig. 10—Schematic representation of the relations between inputs and outputs in a channel.
Now suppose we have another source producing information at rate
R
with
R
<
C
.
In the period
T
this
source will have 2
TR
high probability messages.
We wish to associate these with a selection of the possible
channel
inputs
in
such
a
way
as
to
get
a
small
frequency of
errors.
We
will
set
up
this
association
in
all
23
possible ways
(using,
however, only the
high
probability group of
inputs as
determined by
the source
S
0
)
and
average
the
frequency
of
errors
for
this
large
class
of
possible
coding
systems.
This
is
the
same
as
calculating the frequency of errors for a random association of the messages and channel inputs of duration
T
.
Suppose a particular output
y
1
is observed.
What is the probability of more than one message in the set
of possible causes of
y
1
?
There are 2
TR
messages distributed at random in 2
TH
(
x
)
points.
The probability of
a particular point being a message is thus
2
T
(
R
H
(
x
))
:
The probability that none of the points in the fan is a message (apart from the actual originating message) is
P
=
1
2
T
(
R
H
(
x
))
2
THy
(
x
)
:
Now
R
<
H
(
x
)
H
y
(
x
)
so
R
H
(
x
)
=
H
y
(
x
)
with
positive.
Consequently
P
=
1
2
TH
y
(
x
)
T
2
THy
(
x
)
approaches (as
T
!
∞
)
1
2
T
:
Hence the probability of an error approaches zero and the first part of the theorem is proved.
The second part of the theorem is easily shown by noting that we could merely send
C
bits per second
from
the
source,
completely
neglecting
the
remainder
of
the
information
generated.
At
the
receiver
the
neglected part gives an equivocation
H
(
x
)
C
and the part transmitted need only add
.
This limit can also
be attained in many other ways, as will be shown when we consider the continuous case.
The last statement of the theorem is a simple consequence of our definition of
C
. Suppose we can encode
a source with
H
(
x
)
=
C
+
a
in such a way as to obtain an equivocation
H
y
(
x
)
=
a
with
positive.
Then
R
=
H
(
x
)
=
C
+
a
and
H
(
x
)
H
y
(
x
)
=
C
+
with
positive. This contradicts the definition of
C
as the maximum of
H
(
x
)
H
y
(
x
)
.
Actually
more
has
been
proved
than
was
stated
in
the
theorem.
If
the
average
of
a
set
of
numbers is
within
of of their maximum, a fraction of at most
p
can be more than
p
below the maximum. Since
is
arbitrarily small we can say that almost all the systems are arbitrarily close to the ideal.
14.
D
ISCUSSION
The demonstration of Theorem 11,
while
not a
pure existence proof, has some of the deficiencies of
such
proofs.
An attempt to obtain a good approximation to ideal coding by following the method of the proof is
generally impractical. In fact, apart from some rather trivial cases and certain limiting situations, no explicit
description
of
a
series
of
approximation
to
the
ideal
has
been
found.
Probably
this
is
no
accident
but
is
related to the difficulty of giving an explicit construction for a good approximation to a random sequence.
An approximation to the ideal would have the property that if the signal is altered in a reasonable way
by
the
noise,
the
original
can
still
be
recovered.
In
other
words
the
alteration
will
not
in
general bring
it
closer to another reasonable signal than the original. This is accomplished at the cost of a certain amount of
redundancy in the coding.
The redundancy must be introduced in the proper way to combat the particular
noise
structure
involved.
However,
any
redundancy
in
the
source
will
usually
help
if
it
is
utilized
at
the
receiving
point.
In
particular,
if
the
source
already
has
a
certain
redundancy
and
no
attempt
is
made
to
eliminate it in matching to the channel, this redundancy will help combat noise.
For example, in a noiseless
telegraph channel one could save about 50% in time by proper encoding of the messages.
This is not done
and most of the redundancy of English remains in the channel symbols.
This has the advantage, however,
of allowing considerable noise in the channel.
A sizable fraction of the letters can be received incorrectly
and still reconstructed by the context.
In fact this is probably not a bad approximation to the ideal in many
cases,
since the statistical structure of English is rather involved and the reasonable English sequences are
not too far (in the sense required for the theorem) from a random selection.
24
As
in
the
noiseless
case
a
delay
is
generally
required
to
approach
the
ideal
encoding.
It
now
has
the
additional
function
of
allowing
a
large
sample
of
noise
to
affect
the
signal
before
any
judgment
is
made
at the receiving point as
to
the original message.
Increasing the sample size always
sharpens the possible
statistical assertions.
The content of Theorem 11 and its proof can be formulated in a somewhat different way which exhibits
the connection with the noiseless case more clearly. Consider the possible signals of duration
T
and suppose
a subset of them is selected to be used. Let those in the subset all be used with equal probability, and suppose
the receiver is constructed to select, as the original signal, the most probable cause from the subset, when a
perturbed signal is received. We define
N
(
T
;
q
)
to be the maximum number of signals we can choose for the
subset such that the probability of an incorrect interpretation is less than or equal to
q
.
Theorem 12:
Lim
T
!
∞
log
N
(
T
;
q
)
T
=
C
, where
C
is the channel capacity, provided that
q
does not equal 0 or
1.
In
other
words,
no
matter
how
we
set
out
limits
of
reliability,
we
can
distinguish
reliably
in
time
T
enough messages to correspond to about
CT
bits, when
T
is sufficiently large. Theorem 12 can be compared
with the definition of the capacity of a noiseless channel given in Section 1.
15.
E
XAMPLE
OF
A
D
ISCRETE
C
HANNEL
AND
ITS
C
APACITY
A simple example of a discrete channel is indicated in Fig. 11. There are three possible symbols. The first is
never affected by noise.
The second and third each have probability
p
of coming through undisturbed, and
q
of being changed into the other of the pair.
We have (letting
=
[
p
log
p
+
q
log
q
]
and
P
and
Q
be the
p
p
q
q
TRANSMITTED
SYMBOLS
RECEIVED
SYMBOLS
Fig. 11—Example of a discrete channel.
probabilities of using the first and second symbols)
H
(
x
)
=
P
log
P
2
Q
log
Q
H
y
(
x
)
=
2
Q
:
We wish to choose
P
and
Q
in such a way as to maximize
H
(
x
)
H
y
(
x
)
, subject to the constraint
P
+
2
Q
=
1.
Hence we consider
U
=
P
log
P
2
Q
log
Q
2
Q
+
(
P
+
2
Q
)
∂
U
∂
P
=
1
log
P
+
=
0
∂
U
∂
Q
=
2
2log
Q
2
+
2
=
0
:
Eliminating
log
P
=
log
Q
+
P
=
Qe
=
Q
25
P
=
+
2
Q
=
1
+
2
:
The channel capacity is then
C
=
log
+
2
:
Note how this checks the obvious values in the cases
p
=
1 and
p
=
1
2
.
In the first,
=
1 and
C
=
log3,
which
is
correct
since
the
channel
is
then
noiseless
with
three
possible
symbols.
If
p
=
1
2
,
=
2
and
C
=
log2.
Here
the
second
and
third
symbols
cannot
be
distinguished
at
all
and
act
together
like
one
symbol.
The first symbol is used with probability
P
=
1
2
and the second and third together with probability
1
2
.
This may be distributed between them in any desired way and still achieve the maximum capacity.
For
intermediate
values
of
p
the
channel
capacity
will
lie
between
log2
and
log3.
The
distinction
between the second and third symbols conveys some information but not as much as in the noiseless case.
The first symbol is used somewhat more frequently than the other two because of its freedom from noise.
16.
T
HE
C
HANNEL
C
APACITY
IN
C
ERTAIN
S
PECIAL
C
ASES
If
the
noise
affects
successive
channel
symbols
independently
it
can
be
described
by
a
set
of
transition
probabilities
p
ij
.
This is the probability, if symbol
i
is sent, that
j
will be received.
The maximum channel
rate is then given by the maximum of
∑
i
;
j
P
i
p
ij
log
∑
i
P
i
p
ij
+
∑
i
;
j
P
i
p
ij
log
p
ij
where we vary the
P
i
subject to
∑
P
i
=
1.
This leads by the method of Lagrange to the equations,
∑
j
p
sj
log
p
sj
∑
i
P
i
p
ij
=
s
=
1
;
2
;
:
:
:
:
Multiplying
by
P
s
and
summing
on
s
shows
that
=
C
.
Let
the
inverse of
p
sj
(if
it
exists)
be
h
st
so
that
∑
s
h
st
p
sj
=
t j
.
Then:
∑
s
;
j
h
st
p
sj
log
p
sj
log
∑
i
P
i
p
it
=
C
∑
s
h
st
:
Hence:
∑
i
P
i
p
it
=
exp
h
C
∑
s
h
st
+
∑
s
;
j
h
st
p
sj
log
p
sj
i
or,
P
i
=
∑
t
h
it
exp
h
C
∑
s
h
st
+
∑
s
;
j
h
st
p
sj
log
p
sj
i
:
This is the system of equations for determining the maximizing values of
P
i
, with
C
to be determined so
that
∑
P
i
=
1.
When this is done
C
will be the channel capacity, and the
P
i
the proper probabilities for the
channel symbols to achieve this capacity.
If each input symbol has the same set of probabilities on the lines emerging from it, and the same is true
of each output symbol, the capacity can be easily calculated. Examples are shown in Fig. 12.
In such a case
H
x
(
y
)
is independent of the distribution of probabilities on the input symbols, and is given by
∑
p
i
log
p
i
where the
p
i
are the values of the transition probabilities from any input symbol. The channel capacity is
Max
H
(
y
)
H
x
(
y
)
=
Max
H
(
y
)
+
∑
p
i
log
p
i
:
The maximum of
H
(
y
)
is clearly log
m
where
m
is the number of output symbols, since it is possible to make
them all equally probable by making the input symbols equally probable. The channel capacity is therefore
C
=
log
m
+
∑
p
i
log
p
i
:
26
a
b
c
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
Fig. 12—Examples of discrete channels with the same transition probabilities for each input and for each output.
In Fig. 12a it would be
C
=
log4
log2
=
log2
:
This could be achieved by using only the 1st and 3d symbols.
In Fig. 12b
C
=
log4
2
3
log3
1
3
log6
=
log4
log3
1
3
log2
=
log
1
3
2
5
3
:
In Fig. 12c we have
C
=
log3
1
2
log2
1
3
log3
1
6
log6
=
log
3
2
1
2
3
1
3
6
1
6
:
Suppose the symbols fall into several groups such that the noise never causes a symbol in one group to
be
mistaken
for
a
symbol in
another
group.
Let
the
capacity
for
the
n
th
group be
C
n
(in
bits
per
second)
when we use only the symbols in this group.
Then it is easily shown that, for best use of the entire set, the
total probability
P
n
of all symbols in the
n
th group should be
P
n
=
2
C
n
∑
2
C
n
:
Within a group the probability is distributed just as it would be if these were the only symbols being used.
The channel capacity is
C
=
log
∑
2
C
n
:
17.
A
N
E
XAMPLE
OF
E
FFICIENT
C
ODING
The following example, although somewhat unrealistic, is a case in which exact matching to a noisy channel
is possible.
There are two channel symbols, 0 and 1, and the noise affects them in blocks of seven symbols.
A block of seven is either transmitted without error, or exactly one symbol of the seven is incorrect.
These
eight possibilities are equally likely. We have
C
=
Max
H
(
y
)
H
x
(
y
)
=
1
7
7
+
8
8
log
1
8
=
4
7
bits/symbol
:
An
efficient
code,
allowing
complete
correction
of
errors
and
transmitting
at
the
rate
C
,
is
the
following
(found by a method due to R. Hamming):
27
Let
a
block
of
seven
symbols
be
X
1
;
X
2
;
:
:
:
;
X
7
.
Of
these
X
3
,
X
5
,
X
6
and
X
7
are
message symbols
and
chosen arbitrarily by the source. The other three are redundant and calculated as follows:
X
4
is chosen to make
=
X
4
+
X
5
+
X
6
+
X
7
even
X
2
“
“
“
“
=
X
2
+
X
3
+
X
6
+
X
7
“
X
1
“
“
“
“
=
X
1
+
X
3
+
X
5
+
X
7
“
When a block of seven is received
;
and
are calculated and if even called zero, if odd called one.
The
binary number
then gives the subscript of the
X
i
that is incorrect (if 0 there was no error).
APPENDIX 1
T
HE
G
ROWTH
OF
THE
N
UMBER
OF
B
LOCKS
OF
S
YMBOLS
WITH
A
F
INITE
S
TATE
C
ONDITION
Let
N
i
(
L
)
be the number of blocks of symbols of length
L
ending in state
i
.
Then we have
N
j
(
L
)
=
∑
i
;
s
N
i
L
b
(
s
)
ij
where
b
1
ij
;
b
2
ij
;
:
:
:
;
b
m
ij
are the length of the symbols which may be chosen in state
i
and lead to state
j
.
These
are linear difference equations and the behavior as
L
!
∞
must be of the type
N
j
=
A
j
W
L
:
Substituting in the difference equation
A
j
W
L
=
∑
i
;
s
A
i
W
L
b
(
s
)
ij
or
A
j
=
∑
i
;
s
A
i
W
b
(
s
)
ij
∑
i
∑
s
W
b
(
s
)
ij
ij
A
i
=
0
:
For this to be possible the determinant
D
(
W
)
=
j
a
ij
j
=
∑
s
W
b
(
s
)
ij
ij
must vanish and this determines
W
, which is, of course, the largest real root of
D
=
0.
The quantity
C
is then given by
C
=
Lim
L
!
∞
log
∑
A
j
W
L
L
=
log
W
and we also note that the same growth properties result if we require that all blocks start in the same (arbi-
trarily chosen) state.
APPENDIX 2
D
ERIVATION
OF
H
=
∑
p
i
log
p
i
Let
H
1
n
;
1
n
;
:
:
:
;
1
n
=
A
(
n
)
.
From condition (3) we can decompose a choice from
s
m
equally likely possi-
bilities into a series of
m
choices from
s
equally likely possibilities and obtain
A
(
s
m
)
=
mA
(
s
)
:
28
Similarly
A
(
t
n
)
=
nA
(
t
)
:
We can choose
n
arbitrarily large and find an
m
to satisfy
s
m
t
n
<
s
(
m
+
1
)
:
Thus, taking logarithms and dividing by
n
log
s
,
m
n
log
t
log
s
m
n
+
1
n
or
m
n
log
t
log
s
<
where
is arbitrarily small.
Now from the monotonic property of
A
(
n
)
,
A
(
s
m
)
A
(
t
n
)
A
(
s
m
+
1
)
mA
(
s
)
nA
(
t
)
(
m
+
1
)
A
(
s
)
:
Hence, dividing by
nA
(
s
)
,
m
n
A
(
t
)
A
(
s
)
m
n
+
1
n
or
m
n
A
(
t
)
A
(
s
)
<
A
(
t
)
A
(
s
)
log
t
log
s
<
2
A
(
t
)
=
K
log
t
where
K
must be positive to satisfy (2).
Now suppose we have a choice from
n
possibilities with commeasurable probabilities
p
i
=
n
i
∑
n
i
where
the
n
i
are
integers.
We
can
break
down
a
choice from
∑
n
i
possibilities
into
a
choice
from
n
possibilities
with probabilities
p
1
;
:
:
:
;
p
n
and then, if the
i
th was chosen, a choice from
n
i
with equal probabilities. Using
condition (3) again, we equate the total choice from
∑
n
i
as computed by two methods
K
log
∑
n
i
=
H
(
p
1
;
:
:
:
;
p
n
)
+
K
∑
p
i
log
n
i
:
Hence
H
=
K
h
∑
p
i
log
∑
n
i
∑
p
i
log
n
i
i
=
K
∑
p
i
log
n
i
∑
n
i
=
K
∑
p
i
log
p
i
:
If the
p
i
are incommeasurable, they may be approximated by rationals and the same expression must hold
by our continuity assumption.
Thus the expression holds in general.
The choice of coefficient
K
is a matter
of convenience and amounts to the choice of a unit of measure.
APPENDIX 3
T
HEOREMS
ON
E
RGODIC
S
OURCES
If it is possible to go from any state with
P
>
0 to any other along a path of probability
p
>
0, the system is
ergodic and the strong law of large numbers can be applied.
Thus the number of times a given path
p
ij
in
the network is traversed in a long sequence of length
N
is about proportional to the probability of being at
i
, say
P
i
, and then choosing this path,
P
i
p
ij
N
.
If
N
is large enough the probability of percentage error
in
this is less than
so that for all but a set of small probability the actual numbers lie within the limits
(
P
i
p
ij
)
N
:
Hence nearly all sequences have a probability
p
given by
p
=
∏
p
(
P
i
p
ij
)
N
ij
29
and
log
p
N
is limited by
log
p
N
=
∑
(
P
i
p
ij
)
log
p
ij
or
log
p
N
∑
P
i
p
ij
log
p
ij
<
:
This proves Theorem 3.
Theorem 4 follows immediately from this on calculating upper and lower bounds for
n
(
q
)
based on the
possible range of values of
p
in Theorem 3.
In the mixed (not ergodic) case if
L
=
∑
p
i
L
i
and the entropies of the components are
H
1
H
2
H
n
we have the
Theorem:
Lim
N
!
∞
log
n
(
q
)
N
=
'
(
q
)
is a decreasing step function,
'
(
q
)
=
H
s
in the interval
s
1
∑
1
i
<
q
<
s
∑
1
i
:
To
prove
Theorems
5
and
6
first
note
that
F
N
is
monotonic
decreasing
because
increasing
N
adds
a
subscript to a conditional entropy. A simple substitution for
p
B
i
(
S
j
)
in the definition of
F
N
shows that
F
N
=
NG
N
(
N
1
)
G
N
1
and summing this for all
N
gives
G
N
=
1
N
∑
F
n
.
Hence
G
N
F
N
and
G
N
monotonic decreasing.
Also they
must approach the same limit.
By using Theorem 3 we see that
Lim
N
!
∞
G
N
=
H
.
APPENDIX 4
M
AXIMIZING
THE
R
ATE
FOR
A
S
YSTEM
OF
C
ONSTRAINTS
Suppose
we
have
a
set
of
constraints
on
sequences
of
symbols
that
is
of
the
finite
state
type
and
can
be
represented
therefore
by
a
linear
graph.
Let
`
(
s
)
ij
be
the
lengths
of
the
various
symbols
that
can
occur
in
passing
from
state
i
to
state
j
.
What
distribution
of
probabilities
P
i
for
the
different
states
and
p
(
s
)
ij
for
choosing symbol
s
in state
i
and going to state
j
maximizes the rate of generating information under these
constraints?
The constraints define a discrete channel and the maximum rate must be less than or equal to
the capacity
C
of this channel, since if all blocks of large length were equally likely, this rate would result,
and if possible this would be best. We will show that this rate can be achieved by proper choice of the
P
i
and
p
(
s
)
ij
.
The rate in question is
∑
P
i
p
(
s
)
ij
log
p
(
s
)
ij
∑
P
i
p
(
s
)
ij
`
(
s
)
ij
=
N
M
:
Let
`
ij
=
∑
s
`
(
s
)
ij
.
Evidently for a maximum
p
(
s
)
ij
=
k
exp
`
(
s
)
ij
.
The constraints on maximization are
∑
P
i
=
1,
∑
j
p
ij
=
1,
∑
P
i
(
p
ij
ij
)
=
0.
Hence we maximize
U
=
∑
P
i
p
ij
log
p
ij
∑
P
i
p
ij
`
ij
+
∑
i
P
i
+
∑
i
p
ij
+
∑
j
P
i
(
p
ij
ij
)
∂
U
∂
p
ij
=
MP
i
(
1
+
log
p
ij
)
+
NP
i
`
ij
M
2
+
+
i
+
i
P
i
=
0
:
30
Solving for
p
ij
p
ij
=
A
i
B
j
D
`
ij
:
Since
∑
j
p
ij
=
1
;
A
1
i
=
∑
j
B
j
D
`
ij
p
ij
=
B
j
D
`
ij
∑
s
B
s
D
`
is
:
The correct value of
D
is the capacity
C
and the
B
j
are solutions of
B
i
=
∑
B
j
C
`
ij
for then
p
ij
=
B
j
B
i
C
`
ij
∑
P
i
B
j
B
i
C
`
ij
=
P
j
or
∑
P
i
B
i
C
`
ij
=
P
j
B
j
:
So that if
i
satisfy
∑
i
C
`
ij
=
j
P
i
=
B
i
i
:
Both the sets of equations for
B
i
and
i
can be satisfied since
C
is such that
j
C
`
ij
ij
j
=
0
:
In this case the rate is
∑
P
i
p
ij
log
B
j
B
i
C
`
ij
∑
P
i
p
ij
`
ij
=
C
∑
P
i
p
ij
log
B
j
B
i
∑
P
i
p
ij
`
ij
but
∑
P
i
p
ij
(
log
B
j
log
B
i
)
=
∑
j
P
j
log
B
j
∑
P
i
log
B
i
=
0
Hence the rate is
C
and as this could never be exceeded this is the maximum, justifying the assumed solution.
31
PART III: MATHEMATICAL PRELIMINARIES
In
this
final
installment
of
the
paper
we
consider
the
case
where
the
signals
or
the
messages
or
both
are
continuously variable, in contrast with the discrete nature assumed heretofore.
To a considerable extent the
continuous case can be obtained through a limiting process from the discrete case by dividing the continuum
of messages and signals into a large but finite number of small regions and calculating the various parameters
involved on a discrete basis.
As the size of the regions is decreased these parameters in general approach as
limits the proper values for the continuous case.
There are, however, a few new effects that appear and also
a general change of emphasis in the direction of specialization of the general results to particular cases.
We
will
not
attempt,
in
the
continuous case,
to
obtain
our
results
with
the
greatest
generality,
or
with
the
extreme
rigor
of
pure
mathematics,
since
this
would
involve
a
great
deal
of
abstract
measure
theory
and would obscure the main thread of the analysis.
A preliminary study, however, indicates that the theory
can be formulated in a completely axiomatic and rigorous manner which includes both the continuous and
discrete cases and many others. The occasional liberties taken with limiting processes in the present analysis
can be justified in all cases of practical interest.
18.
S
ETS
AND
E
NSEMBLES
OF
F
UNCTIONS
We
shall
have
to
deal
in
the
continuous case
with
sets
of
functions and
ensembles
of
functions.
A
set
of
functions, as the name implies, is merely a class or collection of functions, generally of one variable, time.
It can
be specified
by giving an
explicit representation of the various functions in
the set,
or implicitly
by
giving a property which functions in the set possess and others do not.
Some examples are:
1.
The set of functions:
f
(
t
)
=
sin
(
t
+
)
:
Each particular value of
determines a particular function in the set.
2.
The set of all functions of time containing no frequencies over
W
cycles per second.
3.
The set of all functions limited in band to
W
and in amplitude to
A
.
4.
The set of all English speech signals as functions of time.
An
ensemble
of
functions
is
a
set
of
functions
together
with
a
probability
measure
whereby
we
may
determine the probability of a function in the set having certain properties.
1
For example with the set,
f
(
t
)
=
sin
(
t
+
)
;
we may give a probability distribution for
,
P
(
)
.
The set then becomes an ensemble.
Some further examples of ensembles of functions are:
1.
A finite set of functions
f
k
(
t
)
(
k
=
1
;
2
;
:
:
:
;
n
) with the probability of
f
k
being
p
k
.
2.
A finite dimensional family of functions
f
(
1
;
2
;
:
:
:
;
n
;
t
)
with a probability distribution on the parameters
i
:
p
(
1
;
:
:
:
;
n
)
:
For example we could consider the ensemble defined by
f
(
a
1
;
:
:
:
;
a
n
;
1
;
:
:
:
;
n
;
t
)
=
n
∑
i
=
1
a
i
sin
i
(
!
t
+
i
)
with the amplitudes
a
i
distributed normally and independently, and the phases
i
distributed uniformly
(from 0 to 2
) and independently.
1
In mathematical terminology the functions belong to a measure space whose total measure is unity.
32
3.
The ensemble
f
(
a
i
;
t
)
=
+
∞
∑
n
=
∞
a
n
sin
(
2
Wt
n
)
(
2
Wt
n
)
with the
a
i
normal and independent all with the same standard deviation
p
N
.
This is a representation
of “white” noise, band limited to the band from 0 to
W
cycles per second and with average power
N
.
2
4.
Let points be distributed on the
t
axis according to a Poisson distribution.
At each selected point the
function
f
(
t
)
is placed and the different functions added, giving the ensemble
∞
∑
k
=
∞
f
(
t
+
t
k
)
where the
t
k
are the points of the Poisson distribution.
This ensemble can be considered as a type of
impulse or shot noise where all the impulses are identical.
5.
The set of English speech functions with the probability measure given by the frequency of occurrence
in ordinary use.
An ensemble of functions
f
(
t
)
is
stationary
if the same ensemble results when all functions are shifted
any fixed amount in time.
The ensemble
f
(
t
)
=
sin
(
t
+
)
is stationary if
is distributed uniformly from 0 to 2
.
If we shift each function by
t
1
we obtain
f
(
t
+
t
1
)
=
sin
(
t
+
t
1
+
)
=
sin
(
t
+
'
)
with
'
distributed
uniformly
from
0
to
2
.
Each
function
has
changed
but
the
ensemble
as
a
whole
is
invariant under the translation. The other examples given above are also stationary.
An
ensemble
is
ergodic
if
it
is
stationary,
and
there
is
no
subset
of
the
functions
in
the
set
with
a
probability different from 0 and 1 which is stationary. The ensemble
sin
(
t
+
)
is ergodic.
No subset of these functions of probability
=
0
;
1 is transformed into itself under all time trans-
lations.
On the other hand the ensemble
a
sin
(
t
+
)
with
a
distributed normally and
uniform is stationary but not ergodic.
The subset of these functions with
a
between 0 and 1 for example is stationary.
Of
the
examples
given,
3
and
4
are
ergodic,
and
5
may
perhaps
be
considered
so.
If
an
ensemble
is
ergodic we
may
say
roughly that
each
function in
the
set
is
typical
of
the
ensemble.
More
precisely
it
is
known that with an ergodic ensemble an average of any statistic over the ensemble is equal (with probability
1)
to
an
average
over
the
time
translations
of
a
particular
function
of
the
set.
3
Roughly
speaking,
each
function can be expected, as time progresses, to go through, with the proper frequency, all the convolutions
of any of the functions in the set.
2
This
representation
can
be
used
as
a
definition
of
band
limited
white
noise.
It
has
certain
advantages
in
that
it
involves
fewer
limiting
operations
than
do
definitions
that
have
been
used
in
the
past.
The
name
“white
noise,”
already
firmly
entrenched
in
the
literature,
is
perhaps
somewhat
unfortunate.
In
optics
white
light
means
either
any
continuous
spectrum
as
contrasted
with
a
point
spectrum, or a spectrum which is flat with
wavelength
(which is not the same as a spectrum flat with frequency).
3
This
is
the
famous
ergodic
theorem
or
rather
one
aspect
of
this
theorem
which
was
proved
in
somewhat
different
formulations
by
Birkoff,
von
Neumann,
and
Koopman,
and
subsequently
generalized
by
Wiener,
Hopf,
Hurewicz
and
others.
The
literature
on
ergodic
theory
is
quite
extensive
and
the
reader
is
referred
to
the
papers
of
these
writers
for
precise
and
general
formulations;
e.g.,
E. Hopf, “Ergodentheorie,”
Ergebnisse der Mathematik und ihrer Grenzgebiete,
v. 5; “On Causality Statistics and Probability,”
Journal
of Mathematics
and Physics,
v. XIII, No. 1, 1934; N. Wiener, “The Ergodic Theorem,”
Duke Mathematical
Journal,
v. 5, 1939.
33
Just as we may perform various operations on numbers or functions to obtain new numbers or functions,
we
can
perform
operations
on
ensembles
to
obtain
new
ensembles.
Suppose,
for
example,
we
have
an
ensemble
of
functions
f
(
t
)
and
an
operator
T
which
gives
for
each
function
f
(
t
)
a
resulting
function
g
(
t
)
:
g
(
t
)
=
T f
(
t
)
:
Probability measure is defined for the set
g
(
t
)
by means of that for the set
f
(
t
)
. The probability of a certain
subset of the
g
(
t
)
functions is equal to that of the subset of the
f
(
t
)
functions which produce members of
the given subset of
g
functions under the operation
T
.
Physically this corresponds to passing the ensemble
through
some
device,
for
example,
a
filter,
a
rectifier
or
a
modulator.
The
output
functions
of
the
device
form the ensemble
g
(
t
)
.
A device or operator
T
will be called invariant if shifting the input merely shifts the output, i.e., if
g
(
t
)
=
T f
(
t
)
implies
g
(
t
+
t
1
)
=
T f
(
t
+
t
1
)
for all
f
(
t
)
and all
t
1
.
It is
easily
shown (see Appendix 5
that if
T
is invariant and
the input ensemble is
stationary
then
the
output
ensemble
is
stationary.
Likewise
if
the
input
is
ergodic the
output
will
also
be
ergodic.
A filter or a rectifier is invariant under all time translations. The operation of modulation is not since the
carrier phase gives a certain time structure.
However, modulation is invariant under all translations which
are multiples of the period of the carrier.
Wiener
has
pointed
out
the
intimate
relation
between
the
invariance
of
physical
devices
under
time
translations and Fourier theory.
4
He has shown, in fact, that if a device is linear as well as invariant Fourier
analysis is then the appropriate mathematical tool for dealing with the problem.
An ensemble of functions is the appropriate mathematical representation of the messages produced by
a continuous source (for example, speech),
of the signals produced by a transmitter, and of the perturbing
noise. Communication theory is properly concerned, as has been emphasized by Wiener, not with operations
on particular functions, but with operations on ensembles of functions. A communication system is designed
not for a particular speech function and still less for a sine wave, but for the ensemble of speech functions.
19.
B
AND
L
IMITED
E
NSEMBLES
OF
F
UNCTIONS
If a function of time
f
(
t
)
is limited to the band from 0 to
W
cycles per second it is completely determined
by giving its ordinates at a series of discrete points spaced
1
2
W
seconds apart in the manner indicated by the
following result.
5
Theorem 13:
Let
f
(
t
)
contain no frequencies over
W
.
Then
f
(
t
)
=
∞
∑
∞
X
n
sin
(
2
Wt
n
)
(
2
Wt
n
)
where
X
n
=
f
n
2
W
:
4
Communication
theory
is
heavily
indebted
to
Wiener
for
much
of
its
basic
philosophy
and
theory.
His
classic
NDRC
report,
The
Interpolation,
Extrapolation
and
Smoothing
of
Stationary
Time
Series
(Wiley,
1949),
contains
the
first
clear-cut
formulation
of
communication
theory as a statistical
problem, the study of operations
on time series.
This work, although chiefly concerned
with the
linear
prediction
and
filtering
problem,
is
an
important
collateral
reference
in
connection
with
the
present
paper.
We may
also
refer
here to Wiener’s
Cybernetics
(Wiley, 1948), dealing with the general problems of communication and control.
5
For a proof of this theorem
and further discussion see the author’s paper “Communication
in the Presence of Noise” published in
the
Proceedings of the Institute of Radio Engineers,
v. 37, No. 1, Jan., 1949, pp. 10–21.
34
In this expansion
f
(
t
)
is represented as a sum of orthogonal functions. The coefficients
X
n
of the various
terms
can
be
considered
as
coordinates
in
an
infinite
dimensional
“function
space.”
In
this
space
each
function corresponds to precisely one point and each point to one function.
A function can be considered to be substantially limited to a time
T
if all the ordinates
X
n
outside this
interval of time are zero.
In this case all but 2
TW
of the coordinates will be zero.
Thus functions limited to
a band
W
and duration
T
correspond to points in a space of 2
TW
dimensions.
A subset of the functions of band
W
and duration
T
corresponds to a region in this space.
For example,
the functions whose total energy is less than or equal to
E
correspond to points in a 2
TW
dimensional sphere
with radius
r
=
p
2
WE
.
An
ensemble
of functions of limited duration and band will be represented by a probability distribution
p
(
x
1
;
:
:
:
;
x
n
)
in the corresponding
n
dimensional space. If the ensemble is not limited in time we can consider
the 2
TW
coordinates in a given interval
T
to represent substantially the part of the function in the interval
T
and the probability distribution
p
(
x
1
;
:
:
:
;
x
n
)
to give the statistical structure of the ensemble for intervals of
that duration.
20.
E
NTROPY
OF
A
C
ONTINUOUS
D
ISTRIBUTION
The entropy of a discrete set of probabilities
p
1
;
:
:
:
;
p
n
has been defined as:
H
=
∑
p
i
log
p
i
:
In
an
analogous
manner
we
define
the
entropy
of
a
continuous
distribution
with
the
density
distribution
function
p
(
x
)
by:
H
=
Z
∞
∞
p
(
x
)
log
p
(
x
)
dx
:
With an
n
dimensional distribution
p
(
x
1
;
:
:
:
;
x
n
)
we have
H
=
Z
Z
p
(
x
1
;
:
:
:
;
x
n
)
log
p
(
x
1
;
:
:
:
;
x
n
)
dx
1
dx
n
:
If
we
have two
arguments
x
and
y
(which
may
themselves be multidimensional) the
joint and
conditional
entropies of
p
(
x
;
y
)
are given by
H
(
x
;
y
)
=
Z
Z
p
(
x
;
y
)
log
p
(
x
;
y
)
dxdy
and
H
x
(
y
)
=
Z
Z
p
(
x
;
y
)
log
p
(
x
;
y
)
p
(
x
)
dxdy
H
y
(
x
)
=
Z
Z
p
(
x
;
y
)
log
p
(
x
;
y
)
p
(
y
)
dxdy
where
p
(
x
)
=
Z
p
(
x
;
y
)
dy
p
(
y
)
=
Z
p
(
x
;
y
)
dx
:
The entropies of continuous distributions have most (but not all)
of the properties of the discrete case.
In particular we have the following:
1.
If
x
is limited to a certain volume
v
in its space, then
H
(
x
)
is a maximum and equal to log
v
when
p
(
x
)
is constant (1
=
v
) in the volume.
35
2.
With any two variables
x
,
y
we have
H
(
x
;
y
)
H
(
x
)
+
H
(
y
)
with equality if (and only if)
x
and
y
are independent, i.e.,
p
(
x
;
y
)
=
p
(
x
)
p
(
y
)
(apart possibly from a
set of points of probability zero).
3.
Consider a generalized averaging operation of the following type:
p
0
(
y
)
=
Z
a
(
x
;
y
)
p
(
x
)
dx
with
Z
a
(
x
;
y
)
dx
=
Z
a
(
x
;
y
)
dy
=
1
;
a
(
x
;
y
)
0
:
Then
the
entropy
of
the
averaged
distribution
p
0
(
y
)
is
equal
to
or
greater
than
that
of
the
original
distribution
p
(
x
)
.
4.
We have
H
(
x
;
y
)
=
H
(
x
)
+
H
x
(
y
)
=
H
(
y
)
+
H
y
(
x
)
and
H
x
(
y
)
H
(
y
)
:
5.
Let
p
(
x
)
be a one-dimensional distribution. The form of
p
(
x
)
giving a maximum entropy subject to the
condition that the standard deviation of
x
be fixed at
is Gaussian.
To show this we must maximize
H
(
x
)
=
Z
p
(
x
)
log
p
(
x
)
dx
with
2
=
Z
p
(
x
)
x
2
dx
and
1
=
Z
p
(
x
)
dx
as constraints. This requires, by the calculus of variations, maximizing
Z
p
(
x
)
log
p
(
x
)
+
p
(
x
)
x
2
+
p
(
x
)
dx
:
The condition for this is
1
log
p
(
x
)
+
x
2
+
=
0
and consequently (adjusting the constants to satisfy the constraints)
p
(
x
)
=
1
p
2
e
(
x
2
=
2
2
)
:
Similarly in
n
dimensions, suppose the second order moments of
p
(
x
1
;
:
:
:
;
x
n
)
are fixed at
A
ij
:
A
ij
=
Z
Z
x
i
x
j
p
(
x
1
;
:
:
:
;
x
n
)
dx
1
dx
n
:
Then the maximum entropy occurs (by a similar calculation) when
p
(
x
1
;
:
:
:
;
x
n
)
is the
n
dimensional
Gaussian distribution with the second order moments
A
ij
.
36
6.
The entropy of a one-dimensional Gaussian distribution whose standard deviation is
is given by
H
(
x
)
=
log
p
2
e
:
This is calculated as follows:
p
(
x
)
=
1
p
2
e
(
x
2
=
2
2
)
log
p
(
x
)
=
log
p
2
+
x
2
2
2
H
(
x
)
=
Z
p
(
x
)
log
p
(
x
)
dx
=
Z
p
(
x
)
log
p
2
dx
+
Z
p
(
x
)
x
2
2
2
dx
=
log
p
2
+
2
2
2
=
log
p
2
+
log
p
e
=
log
p
2
e
:
Similarly the
n
dimensional Gaussian distribution with associated quadratic form
a
ij
is given by
p
(
x
1
;
:
:
:
;
x
n
)
=
j
a
ij
j
1
2
(
2
)
n
=
2
exp
1
2
∑
a
ij
x
i
x
j
and the entropy can be calculated as
H
=
log
(
2
e
)
n
=
2
j
a
ij
j
1
2
where
j
a
ij
j
is the determinant whose elements are
a
ij
.
7.
If
x
is limited to a half line (
p
(
x
)
=
0 for
x
0) and the first moment of
x
is fixed at
a
:
a
=
Z
∞
0
p
(
x
)
xdx
;
then the maximum entropy occurs when
p
(
x
)
=
1
a
e
(
x
=
a
)
and is equal to log
ea
.
8.
There is one important difference between the continuous and discrete entropies.
In the discrete case
the
entropy measures in
an
absolute
way
the
randomness of
the chance variable.
In
the continuous
case the measurement is
relative to the coordinate system
.
If we change coordinates the entropy will
in general change. In fact if we change to coordinates
y
1
y
n
the new entropy is given by
H
(
y
)
=
Z
Z
p
(
x
1
;
:
:
:
;
x
n
)
J
x
y
log
p
(
x
1
;
:
:
:
;
x
n
)
J
x
y
dy
1
dy
n
where
J
x
y
is the Jacobian of the coordinate transformation. On expanding the logarithm and chang-
ing the variables to
x
1
x
n
, we obtain:
H
(
y
)
=
H
(
x
)
Z
Z
p
(
x
1
;
:
:
:
;
x
n
)
log
J
x
y
dx
1
:
:
:
dx
n
:
37
Thus the new entropy is the old entropy less the expected logarithm of the Jacobian. In the continuous
case the entropy can be considered a measure of randomness
relative to an assumed standard
, namely
the coordinate system chosen with each small volume element
dx
1
dx
n
given equal weight.
When
we change the coordinate system the entropy in the new system measures the randomness when equal
volume elements
dy
1
dy
n
in the new system are given equal weight.
In spite of this dependence on the coordinate system the entropy concept is as important in the con-
tinuous case as the discrete case.
This is due to the fact that the derived concepts of information rate
and channel capacity depend on the
difference
of two
entropies and this difference
does not
depend
on the coordinate frame, each of the two terms being changed by the same amount.
The entropy of a continuous distribution can be negative. The scale of measurements sets an arbitrary
zero correspondingto a uniform distribution over a unit volume. A distribution which is more confined
than this has less entropy and will be negative. The rates and capacities will, however, always be non-
negative.
9.
A particular case of changing coordinates is the linear transformation
y
j
=
∑
i
a
ij
x
i
:
In this case the Jacobian is simply the determinant
j
a
ij
j
1
and
H
(
y
)
=
H
(
x
)
+
log
j
a
ij
j
:
In the case of a rotation of coordinates (or any measure preserving transformation)
J
=
1 and
H
(
y
)
=
H
(
x
)
.
21.
E
NTROPY
OF
AN
E
NSEMBLE
OF
F
UNCTIONS
Consider an ergodic ensemble of functions limited to a certain band of width
W
cycles per second. Let
p
(
x
1
;
:
:
:
;
x
n
)
be the density distribution function for
amplitudes
x
1
;
:
:
:
;
x
n
at
n
successive sample points.
We
define the
entropy of the ensemble per degree of freedom by
H
0
=
Lim
n
!
∞
1
n
Z
Z
p
(
x
1
;
:
:
:
;
x
n
)
log
p
(
x
1
;
:
:
:
;
x
n
)
dx
1
:
:
:
dx
n
:
We
may
also
define
an
entropy
H
per
second
by
dividing,
not
by
n
,
but
by
the
time
T
in
seconds
for
n
samples.
Since
n
=
2
TW
,
H
=
2
WH
0
.
With white thermal noise
p
is Gaussian and we have
H
0
=
log
p
2
eN
;
H
=
W
log2
eN
:
For
a
given
average power
N
,
white
noise
has
the
maximum
possible
entropy.
This
follows
from
the
maximizing properties of the Gaussian distribution noted above.
The entropy for a continuous stochastic process has many properties analogous to that for discrete pro-
cesses.
In
the
discrete case the
entropy was
related to
the logarithm of
the
probability
of
long sequences,
and to the
number
of reasonably probable sequences of long length.
In the continuous case it is related in
a similar fashion to the logarithm of the
probability density
for a long series of samples, and the
volume
of
reasonably high probability in the function space.
More precisely, if we assume
p
(
x
1
;
:
:
:
;
x
n
)
continuous in all the
x
i
for all
n
, then for sufficiently large
n
log
p
n
H
0
<
38
for all choices of
(
x
1
;
:
:
:
;
x
n
)
apart from a set whose total probability is less than
, with
and
arbitrarily
small.
This follows form the ergodic property if we divide the space into a large number of small cells.
The
relation
of
H
to
volume
can
be
stated
as
follows:
Under
the
same
assumptions
consider
the
n
dimensional
space
corresponding
to
p
(
x
1
;
:
:
:
;
x
n
)
.
Let
V
n
(
q
)
be
the
smallest
volume
in
this
space
which
includes in its interior a total probability
q
.
Then
Lim
n
!
∞
log
V
n
(
q
)
n
=
H
0
provided
q
does not equal 0 or 1.
These results show that for large
n
there is a rather well-defined volume (at least in the logarithmic sense)
of
high
probability,
and
that
within
this
volume the
probability density
is
relatively
uniform (again
in
the
logarithmic sense).
In the white noise case the distribution function is given by
p
(
x
1
;
:
:
:
;
x
n
)
=
1
(
2
N
)
n
=
2
exp
1
2
N
∑
x
2
i
:
Since this depends only on
∑
x
2
i
the surfaces of
equal probability density are spheres and the entire distri-
bution
has
spherical symmetry.
The region of
high probability is
a
sphere of
radius
p
nN
.
As
n
!
∞
the
probability of being outside a sphere of radius
p
n
(
N
+
)
approaches zero and
1
n
times the logarithm of the
volume of the sphere approaches log
p
2
eN
.
In the continuous case it is convenient to work not with the entropy
H
of an ensemble but with a derived
quantity which we will call the entropy power.
This is defined as the power in a white noise limited to the
same band as the original ensemble and having the same entropy.
In other words if
H
0
is the entropy of an
ensemble its entropy power is
N
1
=
1
2
e
exp2
H
0
:
In the geometrical picture this amounts to measuring the high probability volume by the squared radius of a
sphere having the same volume. Since white noise has the maximum entropy for a given power, the entropy
power of any noise is less than or equal to its actual power.
22.
E
NTROPY
L
OSS
IN
L
INEAR
F
ILTERS
Theorem 14:
If an ensemble having an entropy
H
1
per degree of freedom in band
W
is passed through a
filter with characteristic
Y
(
f
)
the output ensemble has an entropy
H
2
=
H
1
+
1
W
Z
W
log
j
Y
(
f
)
j
2
d f
:
The operation of the filter is essentially a linear transformation of coordinates. If we think of the different
frequency components as the original coordinate system, the new frequency components are merely the old
ones multiplied by factors.
The coordinate transformation matrix is thus essentially
diagonalized in terms
of these coordinates. The Jacobian of the transformation is (for
n
sine and
n
cosine components)
J
=
n
∏
i
=
1
j
Y
(
f
i
)
j
2
where the
f
i
are equally spaced through the band
W
.
This becomes in the limit
exp
1
W
Z
W
log
j
Y
(
f
)
j
2
d f
:
Since
J
is constant its average value is the same quantity and applying the theorem on the change of entropy
with a change of coordinates, the result follows.
We may also phrase it in terms of the entropy power.
Thus
if the entropy power of the first ensemble is
N
1
that of the second is
N
1
exp
1
W
Z
W
log
j
Y
(
f
)
j
2
d f
:
39
TABLE I
ENTROPY
ENTROPY
GAIN
POWER
POWER
GAIN
IMPULSE RESPONSE
FACTOR
IN
DECIBELS
0
!
1
!
1
e
2
8
:
69
sin
2
(
t
=
2
)
t
2
=
2
0
!
1
!
2
2
e
4
5
:
33
2
sin
t
t
3
cos
t
t
2
0
!
1
!
3
0
:
411
3
:
87
6
cos
t
1
t
4
cos
t
2
t
2
+
sin
t
t
3
0
!
p
1
!
2
2
e
2
2
:
67
2
J
1
(
t
)
t
0
!
1
e
2
8
:
69
1
t
2
cos
(
1
)
t
cos
t
The final entropy power is the initial entropy power multiplied by the geometric mean gain of the filter.
If
the gain is measured in
db
, then the output entropy power will be increased by the arithmetic mean
db
gain
over
W
.
In Table I the entropy power loss has been calculated (and also expressed in
db
) for a number of ideal
gain characteristics. The impulsive responses of these filters are also given for
W
=
2
, with phase assumed
to be 0.
The
entropy
loss
for
many
other
cases
can
be
obtained
from
these
results.
For
example
the
entropy
power factor 1
=
e
2
for the first case also
applies to any gain characteristic obtain from 1
!
by a measure
preserving transformation of the
!
axis.
In particular a linearly increasing gain
G
(
!
)
=
!
, or a “saw tooth”
characteristic
between
0
and
1
have
the
same
entropy
loss.
The
reciprocal gain
has
the
reciprocal factor.
Thus 1
=!
has the factor
e
2
.
Raising the gain to any power raises the factor to this power.
23.
E
NTROPY
OF
A
S
UM
OF
T
WO
E
NSEMBLES
If we have two ensembles of functions
f
(
t
)
and
g
(
t
)
we can form a new ensemble by “addition.” Suppose
the first ensemble has the probability density function
p
(
x
1
;
:
:
:
;
x
n
)
and the second
q
(
x
1
;
:
:
:
;
x
n
)
.
Then the
40
density function for the sum is given by the convolution:
r
(
x
1
;
:
:
:
;
x
n
)
=
Z
Z
p
(
y
1
;
:
:
:
;
y
n
)
q
(
x
1
y
1
;
:
:
:
;
x
n
y
n
)
dy
1
dy
n
:
Physically this corresponds to
adding the noises or signals represented by the original ensembles of
func-
tions.
The following result is derived in Appendix 6.
Theorem 15:
Let the average power of two ensembles be
N
1
and
N
2
and let their entropy powers be
N
1
and
N
2
.
Then the entropy power of the sum,
N
3
, is bounded by
N
1
+
N
2
N
3
N
1
+
N
2
:
White
Gaussian
noise
has
the
peculiar
property
that
it
can
absorb
any
other
noise
or
signal
ensemble
which may be added to it with a resultant entropy power approximately equal to the sum of the white noise
power and the signal power (measured from the average signal value, which is normally zero), provided the
signal power is small, in a certain sense, compared to noise.
Consider
the
function
space
associated
with
these
ensembles
having
n
dimensions.
The
white
noise
corresponds to the spherical Gaussian distribution in this space. The signal ensemble corresponds to another
probability distribution, not necessarily Gaussian or spherical.
Let the second moments of this distribution
about its center of gravity be
a
ij
.
That is, if
p
(
x
1
;
:
:
:
;
x
n
)
is the density distribution function
a
ij
=
Z
Z
p
(
x
i
i
)(
x
j
j
)
dx
1
dx
n
where the
i
are the coordinates of the center of gravity.
Now
a
ij
is a positive definite quadratic form, and
we can rotate our coordinate system to align it with the principal directions of this form.
a
ij
is then reduced
to diagonal form
b
ii
.
We require that each
b
ii
be small compared to
N
,
the squared radius of the spherical
distribution.
In this case the convolution of the noise and signal produce approximately a Gaussian distribution whose
corresponding quadratic form is
N
+
b
ii
:
The entropy power of this distribution is
h
∏
(
N
+
b
ii
)
i
1
=
n
or approximately
=
h
(
N
)
n
+
∑
b
ii
(
N
)
n
1
i
1
=
n
:
=
N
+
1
n
∑
b
ii
:
The last term is the signal power, while the first is the noise power.
PART IV: THE CONTINUOUS CHANNEL
24.
T
HE
C
APACITY
OF
A
C
ONTINUOUS
C
HANNEL
In a continuous channel the input or transmitted signals will be continuous functions of time
f
(
t
)
belonging
to
a
certain
set,
and
the
output
or
received
signals
will
be
perturbed
versions
of
these.
We
will
consider
only the case where both transmitted and
received signals are limited
to
a certain band
W
.
They can
then
be specified, for a time
T
, by 2
TW
numbers, and their statistical structure by finite dimensional distribution
functions. Thus the statistics of the transmitted signal will be determined by
P
(
x
1
;
:
:
:
;
x
n
)
=
P
(
x
)
41
and those of the noise by the conditional probability distribution
P
x
1
;:::;
x
n
(
y
1
;
:
:
:
;
y
n
)
=
P
x
(
y
)
:
The rate of transmission of information for a continuous channel is defined in a way analogous to that
for a discrete channel, namely
R
=
H
(
x
)
H
y
(
x
)
where
H
(
x
)
is the entropy of the input and
H
y
(
x
)
the equivocation. The channel capacity
C
is defined as the
maximum of
R
when we vary the input over all possible ensembles.
This means that in a finite dimensional
approximation we must vary
P
(
x
)
=
P
(
x
1
;
:
:
:
;
x
n
)
and maximize
Z
P
(
x
)
log
P
(
x
)
dx
+
ZZ
P
(
x
;
y
)
log
P
(
x
;
y
)
P
(
y
)
dxdy
:
This can be written
Z
Z
P
(
x
;
y
)
log
P
(
x
;
y
)
P
(
x
)
P
(
y
)
dxdy
using the fact that
Z
Z
P
(
x
;
y
)
log
P
(
x
)
dxdy
=
Z
P
(
x
)
log
P
(
x
)
dx
.
The channel capacity is thus expressed as
follows:
C
=
Lim
T
!
∞
Max
P
(
x
)
1
T
Z
Z
P
(
x
;
y
)
log
P
(
x
;
y
)
P
(
x
)
P
(
y
)
dxdy
:
It
is
obvious in
this
form
that
R
and
C
are
independent of
the
coordinate system
since
the
numerator
and
denominator in
log
P
(
x
;
y
)
P
(
x
)
P
(
y
)
will
be multiplied
by
the same
factors
when
x
and
y
are
transformed in
any one-to-one way.
This integral expression for
C
is more general than
H
(
x
)
H
y
(
x
)
.
Properly interpreted
(see Appendix 7) it will always exist while
H
(
x
)
H
y
(
x
)
may assume an indeterminate form
∞
∞
in some
cases.
This occurs, for example, if
x
is limited to a surface of fewer dimensions than
n
in its
n
dimensional
approximation.
If
the
logarithmic
base
used
in
computing
H
(
x
)
and
H
y
(
x
)
is
two
then
C
is
the
maximum
number
of
binary
digits
that
can
be
sent
per
second
over
the
channel
with
arbitrarily
small
equivocation,
just
as
in
the
discrete
case.
This
can
be
seen
physically
by
dividing
the
space
of
signals
into
a
large
number
of
small cells,
sufficiently small so that the probability density
P
x
(
y
)
of signal
x
being perturbed to
point
y
is
substantially constant over a cell (either of
x
or
y
). If the cells are considered as distinct points the situation is
essentially the same as a discrete channel and the proofs used there will apply. But it is clear physically that
this quantizing of the volume into individual points cannot in any practical situation alter
the final answer
significantly, provided the regions are sufficiently small.
Thus the capacity will be the limit of the capacities
for the discrete subdivisions and this is just the continuous capacity defined above.
On the mathematical side it can be shown first (see Appendix 7) that if
u
is the message,
x
is the signal,
y
is the received signal (perturbed by noise) and
v
is the recovered message then
H
(
x
)
H
y
(
x
)
H
(
u
)
H
v
(
u
)
regardless of what operations are performed on
u
to
obtain
x
or on
y
to
obtain
v
.
Thus no
matter how we
encode the binary digits to obtain the signal, or how we decode the received signal to recover the message,
the discrete rate for the binary digits does not exceed the channel capacity we have defined.
On the other
hand, it is possible under very general conditions to find a coding system for transmitting binary digits at the
rate
C
with as small an equivocation or frequency of errors as desired. This is true, for example, if, when we
take a finite dimensional approximating space for the signal functions,
P
(
x
;
y
)
is continuous in both
x
and
y
except at a set of points of probability zero.
An important special case occurs when the noise is added to the signal and is independent of it (in the
probability sense).
Then
P
x
(
y
)
is a function only of the difference
n
=
(
y
x
)
,
P
x
(
y
)
=
Q
(
y
x
)
42
and
we
can
assign
a
definite
entropy
to
the
noise
(independent of
the
statistics
of
the
signal),
namely
the
entropy of the distribution
Q
(
n
)
.
This entropy will be denoted by
H
(
n
)
.
Theorem 16:
If the signal and noise are independent and the received signal is the sum of the transmitted
signal and the noise then the rate of transmission is
R
=
H
(
y
)
H
(
n
)
;
i.e., the entropy of the received signal less the entropy of the noise.
The channel capacity is
C
=
Max
P
(
x
)
H
(
y
)
H
(
n
)
:
We have, since
y
=
x
+
n
:
H
(
x
;
y
)
=
H
(
x
;
n
)
:
Expanding the left side and using the fact that
x
and
n
are independent
H
(
y
)
+
H
y
(
x
)
=
H
(
x
)
+
H
(
n
)
:
Hence
R
=
H
(
x
)
H
y
(
x
)
=
H
(
y
)
H
(
n
)
:
Since
H
(
n
)
is independent of
P
(
x
)
, maximizing
R
requires maximizing
H
(
y
)
, the entropy of the received
signal.
If
there
are certain
constraints on
the
ensemble of
transmitted
signals,
the entropy of
the
received
signal must be maximized subject to these constraints.
25.
C
HANNEL
C
APACITY
WITH
AN
A
VERAGE
P
OWER
L
IMITATION
A simple application of Theorem 16 is the case when the noise is a white thermal noise and the transmitted
signals
are limited
to
a
certain
average power
P
.
Then
the received signals
have an
average power
P
+
N
where
N
is the average noise power.
The maximum entropy for the received signals occurs when they also
form a white noise ensemble since this is the greatest possible entropy for a power
P
+
N
and can be obtained
by
a
suitable
choice of
transmitted
signals,
namely
if
they
form
a
white
noise ensemble of
power
P
.
The
entropy (per second) of the received ensemble is then
H
(
y
)
=
W
log2
e
(
P
+
N
)
;
and the noise entropy is
H
(
n
)
=
W
log2
eN
:
The channel capacity is
C
=
H
(
y
)
H
(
n
)
=
W
log
P
+
N
N
:
Summarizing we have the following:
Theorem 17:
The capacity of a channel of band
W
perturbed by white thermal noise power
N
when the
average transmitter power is limited to
P
is given by
C
=
W
log
P
+
N
N
:
This
means
that
by
sufficiently
involved
encoding
systems
we
can
transmit
binary
digits
at
the
rate
W
log
2
P
+
N
N
bits per second,
with arbitrarily small frequency of errors.
It is not possible to
transmit at a
higher rate by any encoding system without a definite positive frequency of errors.
To approximate this limiting rate of transmission the transmitted signals must approximate, in statistical
properties,
a
white
noise.
6
A
system
which
approaches
the
ideal
rate
may
be
described
as
follows:
Let
6
This
and
other
properties
of
the
white
noise
case
are
discussed
from
the
geometrical
point
of
view
in
“Communication
in
the
Presence of Noise,”
loc. cit.
43
M
=
2
s
samples of white noise be constructed each of duration
T
.
These are assigned binary numbers from
0
to
M
1.
At
the
transmitter
the
message
sequences
are
broken
up
into
groups of
s
and
for
each
group
the corresponding noise sample is transmitted as the signal.
At the receiver the
M
samples are known and
the actual received signal (perturbed by noise) is compared with
each of them.
The sample which has the
least R.M.S. discrepancy from the received signal is chosen as the transmitted signal and the corresponding
binary
number
reconstructed.
This
process
amounts
to
choosing
the
most
probable
(
a
posteriori
)
signal.
The number
M
of noise samples used will depend on the tolerable frequency
of errors, but for almost all
selections of samples we have
Lim
!
0
Lim
T
!
∞
log
M
(
;
T
)
T
=
W
log
P
+
N
N
;
so that no matter how small
is chosen, we can, by taking
T
sufficiently large, transmit as near as we wish
to
TW
log
P
+
N
N
binary digits in the time
T
.
Formulas
similar
to
C
=
W
log
P
+
N
N
for
the
white
noise
case
have
been
developed
independently
by
several
other
writers,
although
with
somewhat
different interpretations.
We
may
mention
the
work
of
N. Wiener,
7
W. G. Tuller,
8
and H. Sullivan in this connection.
In the case of an arbitrary perturbing noise (not necessarily white thermal noise) it does not appear that
the maximizing problem involved in determining the channel capacity
C
can be solved explicitly. However,
upper and lower bounds can be set for
C
in terms of the average noise power
N
the noise entropy power
N
1
.
These bounds are sufficiently close together in most practical cases to furnish a satisfactory solution to the
problem.
Theorem 18:
The
capacity
of
a
channel of
band
W
perturbed by
an
arbitrary
noise
is
bounded by
the
inequalities
W
log
P
+
N
1
N
1
C
W
log
P
+
N
N
1
where
P
=
average transmitter power
N
=
average noise power
N
1
=
entropy power of the noise.
Here
again
the
average power
of
the
perturbed signals
will
be
P
+
N
.
The
maximum entropy
for
this
power
would
occur
if
the
received
signal
were
white
noise
and
would
be
W
log2
e
(
P
+
N
)
.
It
may
not
be possible to
achieve this;
i.e., there may not be any ensemble of transmitted signals which, added to the
perturbing noise, produce a white thermal noise at the receiver, but at least this sets an upper bound to
H
(
y
)
.
We have, therefore
C
=
Max
H
(
y
)
H
(
n
)
W
log2
e
(
P
+
N
)
W
log2
eN
1
:
This is the upper limit given in the theorem.
The lower limit can be obtained by considering the rate if we
make the transmitted signal a white noise, of power
P
.
In this case the entropy power of the received signal
must
be
at
least
as
great
as
that
of
a
white
noise
of
power
P
+
N
1
since
we
have
shown
in
in
a
previous
theorem
that
the
entropy
power
of
the
sum
of
two
ensembles
is
greater
than
or
equal
to
the
sum
of
the
individual entropy powers. Hence
Max
H
(
y
)
W
log2
e
(
P
+
N
1
)
7
Cybernetics, loc. cit.
8
“Theoretical
Limitations
on
the
Rate
of
Transmission
of
Information,”
Proceedings
of
the
Institute
of
Radio
Engineers,
v.
37,
No. 5, May, 1949, pp. 468–78.
44
and
C
W
log2
e
(
P
+
N
1
)
W
log2
eN
1
=
W
log
P
+
N
1
N
1
:
As
P
increases, the upper and lower bounds approach each other, so we have as an asymptotic rate
W
log
P
+
N
N
1
:
If the noise is itself white,
N
=
N
1
and the result reduces to the formula proved previously:
C
=
W
log
1
+
P
N
:
If the noise is Gaussian but with a spectrum which is not necessarily flat,
N
1
is the geometric mean of
the noise power over the various frequencies in the band
W
.
Thus
N
1
=
exp
1
W
Z
W
log
N
(
f
)
d f
where
N
(
f
)
is the noise power at frequency
f
.
Theorem 19:
If we set the capacity for a given transmitter power
P
equal to
C
=
W
log
P
+
N
N
1
then
is monotonic decreasing as
P
increases and approaches 0 as a limit.
Suppose that for a given power
P
1
the channel capacity is
W
log
P
1
+
N
1
N
1
:
This
means
that
the
best
signal
distribution,
say
p
(
x
)
,
when
added
to
the
noise
distribution
q
(
x
)
,
gives
a
received distribution
r
(
y
)
whose entropy power is
(
P
1
+
N
1
)
.
Let us increase the power to
P
1
+
P
by
adding a white noise of power
P
to the signal.
The entropy of the received signal is now at least
H
(
y
)
=
W
log2
e
(
P
1
+
N
1
+
P
)
by
application
of
the
theorem
on
the
minimum
entropy
power
of
a
sum.
Hence,
since
we
can
attain
the
H
indicated, the entropy of the maximizing distribution must be at least as great and
must be monotonic
decreasing.
To show that
!
0 as
P
!
∞
consider a signal which is white noise with a large
P
.
Whatever
the perturbing noise, the received signal will be approximately a white noise, if
P
is sufficiently large, in the
sense of having an entropy power approaching
P
+
N
.
26.
T
HE
C
HANNEL
C
APACITY
WITH
A
P
EAK
P
OWER
L
IMITATION
In some applications the transmitter is limited not by the average power output but by the peak instantaneous
power.
The
problem
of
calculating
the
channel
capacity
is
then
that
of
maximizing
(by
variation
of
the
ensemble of transmitted symbols)
H
(
y
)
H
(
n
)
subject to the constraint that all the functions
f
(
t
)
in the ensemble be less than or equal to
p
S
,
say, for all
t
.
A constraint of this type does not work out as well mathematically as the average power limitation.
The
most we have obtained for this case is a lower bound valid for all
S
N
,
an “asymptotic” upper bound (valid
for large
S
N
) and an asymptotic value of
C
for
S
N
small.
45
Theorem 20:
The
channel
capacity
C
for
a
band
W
perturbed
by
white
thermal
noise
of
power
N
is
bounded by
C
W
log
2
e
3
S
N
;
where
S
is the peak allowed transmitter power.
For sufficiently large
S
N
C
W
log
2
e
S
+
N
N
(
1
+
)
where
is arbitrarily small.
As
S
N
!
0 (and provided the band
W
starts at 0)
C
.
W
log
1
+
S
N
!
1
:
We wish to maximize the entropy of the received signal.
If
S
N
is large this will occur very nearly when
we maximize the entropy of the transmitted ensemble.
The asymptotic upper bound is obtained by relaxing the conditions on the ensemble. Let us suppose that
the power is limited to
S
not at every instant of time, but only at the sample points. The maximum entropy of
the transmitted ensemble under these weakened conditions is certainly greater than or equal to that under the
original conditions. This altered problem can be solved easily.
The maximum entropy occurs if the different
samples are independent and have a distribution function which is constant from
p
S
to
+
p
S
.
The entropy
can be calculated as
W
log4
S
:
The received signal will then have an entropy less than
W
log
(
4
S
+
2
eN
)(
1
+
)
with
!
0
as
S
N
!
∞
and the channel capacity is obtained by subtracting the entropy of
the white noise,
W
log2
eN
:
W
log
(
4
S
+
2
eN
)(
1
+
)
W
log
(
2
eN
)
=
W
log
2
e
S
+
N
N
(
1
+
)
:
This is the desired upper bound to the channel capacity.
To obtain a lower bound consider the same ensemble of functions. Let these functions be passed through
an
ideal
filter
with
a
triangular
transfer
characteristic.
The
gain
is
to
be
unity
at
frequency 0
and
decline
linearly
down
to
gain
0
at
frequency
W
.
We
first
show
that
the
output functions of
the
filter
have
a
peak
power limitation
S
at all times (not just the sample points).
First we note that a pulse
sin2
Wt
2
Wt
going into
the filter produces
1
2
sin
2
Wt
(
Wt
)
2
in the output.
This function is never negative.
The input function (in the general case) can be thought of as
the sum of a series of shifted functions
a
sin2
Wt
2
Wt
where
a
, the amplitude of the sample, is not greater than
p
S
.
Hence the output is the sum of shifted functions
of the non-negative form above with the same coefficients. These functions being non-negative, the greatest
positive value for any
t
is obtained when all the coefficients
a
have their maximum positive values, i.e.,
p
S
.
In this case the input function was a constant of amplitude
p
S
and since the filter has unit gain for D.C., the
output is the same.
Hence the output ensemble has a peak power
S
.
46
The
entropy
of
the
output
ensemble
can
be
calculated
from
that
of
the
input
ensemble
by
using
the
theorem dealing with such a situation. The output entropy is equal to the input entropy plus the geometrical
mean gain of the filter:
Z
W
0
log
G
2
d f
=
Z
W
0
log
W
f
W
2
d f
=
2
W
:
Hence the output entropy is
W
log4
S
2
W
=
W
log
4
S
e
2
and the channel capacity is greater than
W
log
2
e
3
S
N
:
We now wish to show that, for small
S
N
(peak signal power over average white noise power), the channel
capacity is approximately
C
=
W
log
1
+
S
N
:
More precisely
C
.
W
log
1
+
S
N
!
1 as
S
N
!
0.
Since the average signal power
P
is less than or equal
to the peak
S
, it follows that for all
S
N
C
W
log
1
+
P
N
W
log
1
+
S
N
:
Therefore, if we can find an ensemble of functions such that they correspond to a rate nearly
W
log
1
+
S
N
and are limited to band
W
and peak
S
the result will be proved.
Consider the ensemble of functions of the
following type. A series of
t
samples have the same value, either
+
p
S
or
p
S
, then the next
t
samples have
the same value, etc.
The value for a series is chosen at random, probability
1
2
for
+
p
S
and
1
2
for
p
S
.
If
this ensemble be passed through a filter with triangular gain characteristic (unit gain at D.C.), the output is
peak limited to
S
. Furthermore the average power is nearly
S
and can be made to approach this by taking
t
sufficiently large. The entropy of the sum of this and the thermal noise can be found by applying the theorem
on the sum of a noise and a small signal. This theorem will apply if
p
t
S
N
is sufficiently small.
This can be ensured by taking
S
N
small enough (after
t
is chosen).
The entropy power
will be
S
+
N
to as close an approximation as desired, and hence the rate of transmission as near as we wish
to
W
log
S
+
N
N
:
PART V: THE RATE FOR A CONTINUOUS SOURCE
27.
F
IDELITY
E
VALUATION
F
UNCTIONS
In
the
case
of
a
discrete
source
of
information
we
were
able
to
determine
a
definite
rate
of
generating
information, namely the entropy of the underlying stochastic process. With a continuous source the situation
is
considerably
more
involved.
In
the
first
place
a
continuously
variable
quantity
can
assume
an
infinite
number of
values and
requires,
therefore,
an
infinite
number of
binary digits for
exact specification.
This
means that to transmit the output of a continuous source with
exact recovery
at the receiving point requires,
47
in
general,
a
channel
of
infinite
capacity
(in
bits
per
second).
Since,
ordinarily,
channels
have
a
certain
amount of noise, and therefore a finite capacity, exact transmission is impossible.
This,
however, evades the real issue.
Practically,
we
are not interested in
exact transmission when
we
have a
continuous source,
but
only
in
transmission
to
within
a
certain
tolerance.
The question
is,
can
we
assign a definite rate to a continuous source when we require only a certain fidelity of recovery, measured in
a suitable way.
Of course, as the fidelity requirements are increased the rate will increase.
It will be shown
that
we
can,
in
very
general cases,
define
such
a
rate,
having
the
property that
it
is
possible,
by
properly
encoding the information, to transmit it over a channel whose capacity is equal to the rate in question, and
satisfy the fidelity requirements. A channel of smaller capacity is insufficient.
It
is
first
necessary
to
give
a
general
mathematical formulation of
the
idea
of
fidelity
of
transmission.
Consider
the
set
of
messages
of
a
long
duration,
say
T
seconds.
The
source
is
described
by
giving
the
probability density, in the associated space, that the source will select the message in question
P
(
x
)
.
A given
communication system is described (from the external point of view) by giving the conditional probability
P
x
(
y
)
that if message
x
is produced by the source the recovered message at the receiving point will be
y
.
The
system as a whole (including source and transmission system) is described by the probability function
P
(
x
;
y
)
of having message
x
and final output
y
.
If this function is known, the complete characteristics of the system
from
the
point
of
view
of
fidelity
are
known.
Any
evaluation
of
fidelity
must
correspond mathematically
to an operation applied to
P
(
x
;
y
)
.
This operation must at least have the properties of a simple ordering of
systems; i.e., it must be possible to say of two systems represented by
P
1
(
x
;
y
)
and
P
2
(
x
;
y
)
that, according to
our fidelity criterion, either (1) the first has higher fidelity, (2) the second has higher fidelity, or (3) they have
equal fidelity. This means that a criterion of fidelity can be represented by a numerically valued function:
v
P
(
x
;
y
)
whose argument ranges over possible probability functions
P
(
x
;
y
)
.
We
will
now show that
under very general and
reasonable assumptions the function
v
P
(
x
;
y
)
can
be
written in a seemingly much more specialized form, namely as an average of a function
(
x
;
y
)
over the set
of possible values of
x
and
y
:
v
P
(
x
;
y
)
=
Z
Z
P
(
x
;
y
)
(
x
;
y
)
dxdy
:
To
obtain this we need only assume (1) that the source and system
are ergodic so
that a very long sample
will be, with probability nearly 1, typical of the ensemble, and (2) that the evaluation is “reasonable” in the
sense
that
it
is
possible,
by
observing
a
typical
input
and
output
x
1
and
y
1
,
to
form
a
tentative
evaluation
on
the basis
of
these samples;
and
if
these samples
are increased in
duration the tentative evaluation will,
with
probability
1,
approach
the
exact
evaluation
based
on
a
full
knowledge
of
P
(
x
;
y
)
.
Let
the
tentative
evaluation be
(
x
;
y
)
.
Then the function
(
x
;
y
)
approaches (as
T
!
∞
) a constant for almost all
(
x
;
y
)
which
are in the high probability region corresponding to the system:
(
x
;
y
)
!
v
P
(
x
;
y
)
and we may also write
(
x
;
y
)
!
Z
Z
P
(
x
;
y
)
(
x
;
y
)
dxdy
since
Z
Z
P
(
x
;
y
)
dxdy
=
1
:
This establishes the desired result.
The function
(
x
;
y
)
has the general nature of a “distance” between
x
and
y
.
9
It measures how undesirable
it is (according to our fidelity criterion) to receive
y
when
x
is transmitted.
The general result given above
can be restated as follows: Any reasonable evaluation can be represented as an average of a distance function
over the set
of messages
and recovered messages
x
and
y
weighted according to
the probability
P
(
x
;
y
)
of
getting the pair in question, provided the duration
T
of the messages be taken sufficiently large.
The following are simple examples of evaluation functions:
9
It is not a “metric” in the strict sense, however, since in general it does not satisfy either
(
x
;
y
)
=
(
y
;
x
)
or
(
x
;
y
)
+
(
y
;
z
)
(
x
;
z
)
.
48
1.
R.M.S. criterion.
v
=
x
(
t
)
y
(
t
)
2
:
In this very commonly used measure of fidelity the distance function
(
x
;
y
)
is (apart from a constant
factor)
the
square
of
the
ordinary
Euclidean
distance
between
the
points
x
and
y
in
the
associated
function space.
(
x
;
y
)
=
1
T
Z
T
0
x
(
t
)
y
(
t
)
2
dt
:
2.
Frequency weighted R.M.S. criterion. More generally one can apply different weights to the different
frequency components before using an R.M.S. measure of fidelity.
This is equivalent to
passing the
difference
x
(
t
)
y
(
t
)
through a shaping filter and then determining the average power in the output.
Thus let
e
(
t
)
=
x
(
t
)
y
(
t
)
and
f
(
t
)
=
Z
∞
∞
e
(
)
k
(
t
)
d
then
(
x
;
y
)
=
1
T
Z
T
0
f
(
t
)
2
dt
:
3.
Absolute error criterion.
(
x
;
y
)
=
1
T
Z
T
0
x
(
t
)
y
(
t
)
dt
:
4.
The structure of the ear and brain determine implicitly an evaluation, or rather a number of evaluations,
appropriate in
the
case
of
speech
or
music
transmission.
There
is,
for
example,
an
“intelligibility”
criterion
in
which
(
x
;
y
)
is
equal
to
the
relative
frequency
of
incorrectly
interpreted
words
when
message
x
(
t
)
is received as
y
(
t
)
. Although we cannot give an explicit representation of
(
x
;
y
)
in these
cases it could, in principle, be determined by sufficient experimentation. Some of its properties follow
from well-known experimental results in hearing, e.g., the ear is relatively insensitive to phase and the
sensitivity to amplitude and frequency is roughly logarithmic.
5.
The discrete case can be considered as a specialization in which we have tacitly assumed an evaluation
based on the frequency of errors. The function
(
x
;
y
)
is then defined as the number of symbols in the
sequence
y
differing from the corresponding symbols in
x
divided by the total number of symbols in
x
.
28.
T
HE
R
ATE
FOR
A
S
OURCE
R
ELATIVE
TO
A
F
IDELITY
E
VALUATION
We are now in a position to define a rate of generating information for a continuous source.
We are given
P
(
x
)
for
the
source
and
an
evaluation
v
determined
by
a
distance
function
(
x
;
y
)
which
will
be
assumed
continuous in both
x
and
y
.
With a particular system
P
(
x
;
y
)
the quality is measured by
v
=
Z
Z
(
x
;
y
)
P
(
x
;
y
)
dxdy
:
Furthermore the rate of flow of binary digits corresponding to
P
(
x
;
y
)
is
R
=
Z
Z
P
(
x
;
y
)
log
P
(
x
;
y
)
P
(
x
)
P
(
y
)
dxdy
:
We define the rate
R
1
of generating information for a given quality
v
1
of reproduction to be the minimum of
R
when we keep
v
fixed at
v
1
and vary
P
x
(
y
)
.
That is:
R
1
=
Min
P
x
(
y
)
Z
Z
P
(
x
;
y
)
log
P
(
x
;
y
)
P
(
x
)
P
(
y
)
dxdy
49
subject to the constraint:
v
1
=
Z
Z
P
(
x
;
y
)
(
x
;
y
)
dxdy
:
This
means
that
we
consider,
in
effect,
all
the
communication
systems
that
might
be
used
and
that
transmit
with
the
required
fidelity.
The
rate
of
transmission
in
bits
per
second
is
calculated
for
each
one
and we choose that having the least rate.
This latter rate is the rate we assign the source for the fidelity in
question.
The justification of this definition lies in the following result:
Theorem 21:
If a source has a rate
R
1
for a valuation
v
1
it is possible to encode the output of the source
and transmit it over a channel of capacity
C
with fidelity as near
v
1
as desired provided
R
1
C
.
This is not
possible if
R
1
>
C
.
The last statement in the theorem follows immediately from the definition of
R
1
and previous results.
If
it were not true we could transmit more than
C
bits per second over a channel of capacity
C
.
The first part
of
the
theorem is
proved by a
method analogous to
that
used
for Theorem 11.
We
may,
in
the
first
place,
divide the
(
x
;
y
)
space into a large number of small cells and represent the situation as a discrete case.
This
will not change the evaluation function by more than an arbitrarily small amount (when the cells are very
small)
because of
the
continuity assumed
for
(
x
;
y
)
.
Suppose that
P
1
(
x
;
y
)
is
the particular system
which
minimizes the rate and gives
R
1
.
We choose from the high probability
y
’s a set at random containing
2
(
R
1
+
)
T
members where
!
0 as
T
!
∞
.
With large
T
each chosen point will be connected by a high probability
line (as in Fig. 10) to a set of
x
’s.
A calculation similar to that used in proving Theorem 11 shows that with
large
T
almost all
x
’s are covered by the fans from the chosen
y
points for almost all choices of the
y
’s.
The
communication system
to
be
used
operates
as
follows:
The
selected
points
are
assigned
binary
numbers.
When
a
message
x
is
originated it
will
(with
probability approaching 1
as
T
!
∞
)
lie
within
at
least
one
of the fans.
The corresponding binary number is transmitted (or one of them chosen arbitrarily if there are
several) over the channel by suitable coding means to give a small probability of error.
Since
R
1
C
this is
possible.
At the receiving point the corresponding
y
is reconstructed and used as the recovered message.
The
evaluation
v
0
1
for
this
system
can
be
made
arbitrarily
close
to
v
1
by
taking
T
sufficiently
large.
This is due to the fact that for each long sample of message
x
(
t
)
and recovered message
y
(
t
)
the evaluation
approaches
v
1
(with probability 1).
It is interesting to note that, in this system, the noise in the recovered message is actually produced by a
kind of general quantizing at the transmitter and not produced by the noise in the channel. It is more or less
analogous to the quantizing noise in PCM.
29.
T
HE
C
ALCULATION
OF
R
ATES
The definition of the rate is similar in many respects to the definition of channel capacity. In the former
R
=
Min
P
x
(
y
)
Z
Z
P
(
x
;
y
)
log
P
(
x
;
y
)
P
(
x
)
P
(
y
)
dxdy
with
P
(
x
)
and
v
1
=
Z
Z
P
(
x
;
y
)
(
x
;
y
)
dxdy
fixed. In the latter
C
=
Max
P
(
x
)
Z
Z
P
(
x
;
y
)
log
P
(
x
;
y
)
P
(
x
)
P
(
y
)
dxdy
with
P
x
(
y
)
fixed and possibly one or more other constraints (e.g., an average power limitation) of the form
K
=
R
R
P
(
x
;
y
)
(
x
;
y
)
dxdy
.
A partial solution of the general maximizing problem for determining the rate of a source can be given.
Using Lagrange’s method we consider
Z
Z
P
(
x
;
y
)
log
P
(
x
;
y
)
P
(
x
)
P
(
y
)
+
P
(
x
;
y
)
(
x
;
y
)
+
(
x
)
P
(
x
;
y
)
dxdy
:
50
The variational equation (when we take the first variation on
P
(
x
;
y
)
) leads to
P
y
(
x
)
=
B
(
x
)
e
(
x
;
y
)
where
is determined to give the required fidelity and
B
(
x
)
is chosen to satisfy
Z
B
(
x
)
e
(
x
;
y
)
dx
=
1
:
This shows that, with best encoding, the conditional probability of a certain cause for various received
y
,
P
y
(
x
)
will decline exponentially with the distance function
(
x
;
y
)
between the
x
and
y
in question.
In the special case where the distance function
(
x
;
y
)
depends only on the (vector) difference between
x
and
y
,
(
x
;
y
)
=
(
x
y
)
we have
Z
B
(
x
)
e
(
x
y
)
dx
=
1
:
Hence
B
(
x
)
is constant, say
, and
P
y
(
x
)
=
e
(
x
y
)
:
Unfortunately these formal solutions are difficult to evaluate in particular cases and seem to be of little value.
In fact, the actual calculation of rates has been carried out in only a few very simple cases.
If the distance function
(
x
;
y
)
is the mean square discrepancy between
x
and
y
and the message ensemble
is white noise, the rate can be determined. In that case we have
R
=
Min
H
(
x
)
H
y
(
x
)
=
H
(
x
)
Max
H
y
(
x
)
with
N
=
(
x
y
)
2
.
But the Max
H
y
(
x
)
occurs when
y
x
is a white noise, and is equal to
W
1
log2
eN
where
W
1
is the bandwidth of the message ensemble. Therefore
R
=
W
1
log2
eQ
W
1
log2
eN
=
W
1
log
Q
N
where
Q
is the average message power. This proves the following:
Theorem 22:
The rate for a white noise source of power
Q
and band
W
1
relative to an R.M.S. measure
of fidelity is
R
=
W
1
log
Q
N
where
N
is the allowed mean square error between original and recovered messages.
More generally with any message source we can obtain inequalities bounding the rate relative to a mean
square error criterion.
Theorem 23:
The rate for any source of band
W
1
is bounded by
W
1
log
Q
1
N
R
W
1
log
Q
N
where
Q
is the average power of the source,
Q
1
its entropy power and
N
the allowed mean square error.
The lower bound follows from the fact that the Max
H
y
(
x
)
for a given
(
x
y
)
2
=
N
occurs in the white
noise case.
The upper bound results if we place points (used in the proof of Theorem 21) not in the best way
but at random in a sphere of radius
p
Q
N
.
51
A
CKNOWLEDGMENTS
The writer is indebted to his colleagues at the Laboratories, particularly to Dr. H. W. Bode, Dr. J. R. Pierce,
Dr. B. McMillan, and Dr. B. M. Oliver for many helpful suggestions and criticisms during the course of this
work. Credit should also be given to Professor N. Wiener, whose elegant solution of the problems of filtering
and prediction of stationary ensembles has considerably influenced the writer’s thinking in this field.
APPENDIX 5
Let
S
1
be
any
measurable
subset
of
the
g
ensemble,
and
S
2
the
subset
of
the
f
ensemble
which
gives
S
1
under the operation
T
.
Then
S
1
=
TS
2
:
Let
H
be the operator which shifts all functions in a set by the time
.
Then
H
S
1
=
H
TS
2
=
TH
S
2
since
T
is invariant and therefore commutes with
H
.
Hence if
m
[
S
]
is the probability measure of the set
S
m
[
H
S
1
]
=
m
[
TH
S
2
]
=
m
[
H
S
2
]
=
m
[
S
2
]
=
m
[
S
1
]
where
the
second
equality
is
by
definition
of
measure
in
the
g
space,
the
third
since
the
f
ensemble
is
stationary, and the last by definition of
g
measure again.
To
prove that
the
ergodic property is
preserved
under
invariant operations,
let
S
1
be
a
subset
of
the
g
ensemble which is invariant under
H
, and let
S
2
be the set of all functions
f
which transform into
S
1
.
Then
H
S
1
=
H
TS
2
=
TH
S
2
=
S
1
so that
H
S
2
is included in
S
2
for all
.
Now, since
m
[
H
S
2
]
=
m
[
S
1
]
this implies
H
S
2
=
S
2
for all
with
m
[
S
2
]
=
0
;
1. This contradiction shows that
S
1
does not exist.
APPENDIX 6
The upper bound,
N
3
N
1
+
N
2
, is due to the fact that the maximum possible entropy for a power
N
1
+
N
2
occurs when we have a white noise of this power. In this case the entropy power is
N
1
+
N
2
.
To
obtain
the
lower
bound,
suppose
we
have
two
distributions
in
n
dimensions
p
(
x
i
)
and
q
(
x
i
)
with
entropy
powers
N
1
and
N
2
.
What
form
should
p
and
q
have
to
minimize
the
entropy
power
N
3
of
their
convolution
r
(
x
i
)
:
r
(
x
i
)
=
Z
p
(
y
i
)
q
(
x
i
y
i
)
dy
i
:
The entropy
H
3
of
r
is given by
H
3
=
Z
r
(
x
i
)
log
r
(
x
i
)
dx
i
:
We wish to minimize this subject to the constraints
H
1
=
Z
p
(
x
i
)
log
p
(
x
i
)
dx
i
H
2
=
Z
q
(
x
i
)
log
q
(
x
i
)
dx
i
:
52
We consider then
U
=
Z
r
(
x
)
log
r
(
x
)
+
p
(
x
)
log
p
(
x
)
+
q
(
x
)
log
q
(
x
)
dx
U
=
Z
[
1
+
log
r
(
x
)]
r
(
x
)
+
[
1
+
log
p
(
x
)]
p
(
x
)
+
[
1
+
log
q
(
x
)]
q
(
x
)
dx
:
If
p
(
x
)
is varied at a particular argument
x
i
=
s
i
, the variation in
r
(
x
)
is
r
(
x
)
=
q
(
x
i
s
i
)
and
U
=
Z
q
(
x
i
s
i
)
log
r
(
x
i
)
dx
i
log
p
(
s
i
)
=
0
and similarly when
q
is varied. Hence the conditions for a minimum are
Z
q
(
x
i
s
i
)
log
r
(
x
i
)
dx
i
=
log
p
(
s
i
)
Z
p
(
x
i
s
i
)
log
r
(
x
i
)
dx
i
=
log
q
(
s
i
)
:
If we multiply the first by
p
(
s
i
)
and the second by
q
(
s
i
)
and integrate with respect to
s
i
we obtain
H
3
=
H
1
H
3
=
H
2
or solving for
and
and replacing in the equations
H
1
Z
q
(
x
i
s
i
)
log
r
(
x
i
)
dx
i
=
H
3
log
p
(
s
i
)
H
2
Z
p
(
x
i
s
i
)
log
r
(
x
i
)
dx
i
=
H
3
log
q
(
s
i
)
:
Now suppose
p
(
x
i
)
and
q
(
x
i
)
are normal
p
(
x
i
)
=
j
A
ij
j
n
=
2
(
2
)
n
=
2
exp
1
2
∑
A
ij
x
i
x
j
q
(
x
i
)
=
j
B
ij
j
n
=
2
(
2
)
n
=
2
exp
1
2
∑
B
ij
x
i
x
j
:
Then
r
(
x
i
)
will also be normal with quadratic form
C
ij
.
If the inverses of these forms are
a
ij
,
b
ij
,
c
ij
then
c
ij
=
a
ij
+
b
ij
:
We
wish
to
show that these functions satisfy
the minimizing conditions if
and
only if
a
ij
=
Kb
ij
and thus
give the minimum
H
3
under the constraints. First we have
log
r
(
x
i
)
=
n
2
log
1
2
j
C
ij
j
1
2
∑
C
ij
x
i
x
j
Z
q
(
x
i
s
i
)
log
r
(
x
i
)
dx
i
=
n
2
log
1
2
j
C
ij
j
1
2
∑
C
ij
s
i
s
j
1
2
∑
C
ij
b
ij
:
This should equal
H
3
H
1
n
2
log
1
2
j
A
ij
j
1
2
∑
A
ij
s
i
s
j
which requires
A
ij
=
H
1
H
3
C
ij
.
In this case
A
ij
=
H
1
H
2
B
ij
and both equations reduce to identities.
53
APPENDIX 7
The following will indicate a more general and more rigorous approach to the central definitions of commu-
nication theory. Consider a probability measure space whose elements are ordered pairs
(
x
;
y
)
.
The variables
x
,
y
are to be identified as the possible transmitted and received signals of some long duration
T
.
Let us call
the set of all points whose
x
belongs to a subset
S
1
of
x
points the strip over
S
1
, and similarly the set whose
y
belong to
S
2
the strip over
S
2
.
We divide
x
and
y
into a collection of non-overlapping measurable subsets
X
i
and
Y
i
approximate to the rate of transmission
R
by
R
1
=
1
T
∑
i
P
(
X
i
;
Y
i
)
log
P
(
X
i
;
Y
i
)
P
(
X
i
)
P
(
Y
i
)
where
P
(
X
i
)
is the probability measure of the strip over
X
i
P
(
Y
i
)
is the probability measure of the strip over
Y
i
P
(
X
i
;
Y
i
)
is the probability measure of the intersection of the strips
:
A further subdivision can never decrease
R
1
.
For let
X
1
be divided into
X
1
=
X
0
1
+
X
0
0
1
and let
P
(
Y
1
)
=
a
P
(
X
1
)
=
b
+
c
P
(
X
0
1
)
=
b
P
(
X
0
1
;
Y
1
)
=
d
P
(
X
0
0
1
)
=
c
P
(
X
0
0
1
;
Y
1
)
=
e
P
(
X
1
;
Y
1
)
=
d
+
e
:
Then in the sum we have replaced (for the
X
1
,
Y
1
intersection)
(
d
+
e
)
log
d
+
e
a
(
b
+
c
)
by
d
log
d
ab
+
e
log
e
ac
:
It is easily shown that with the limitation we have on
b
,
c
,
d
,
e
,
d
+
e
b
+
c
d
+
e
d
d
e
e
b
d
c
e
and
consequently
the
sum
is
increased.
Thus
the
various
possible
subdivisions
form
a
directed
set,
with
R
monotonic increasing with
refinement of
the subdivision.
We
may
define
R
unambiguously as the
least
upper bound for
R
1
and write it
R
=
1
T
Z
Z
P
(
x
;
y
)
log
P
(
x
;
y
)
P
(
x
)
P
(
y
)
dxdy
:
This integral, understood in the above sense, includes both the continuous and discrete cases and of course
many others which cannot be represented in either form.
It is trivial in this formulation that if
x
and
u
are
in one-to-one correspondence, the rate from
u
to
y
is equal to that from
x
to
y
.
If
v
is any function of
y
(not
necessarily
with
an
inverse) then
the
rate
from
x
to
y
is
greater
than
or
equal to
that
from
x
to
v
since,
in
the calculation of
the approximations, the subdivisions of
y
are essentially
a finer subdivision of those for
v
.
More generally if
y
and
v
are related not functionally but statistically, i.e., we have a probability measure
space
(
y
;
v
)
, then
R
(
x
;
v
)
R
(
x
;
y
)
.
This means that any operation applied to the received signal, even though
it involves statistical elements, does not increase
R
.
Another
notion
which
should
be
defined
precisely
in
an
abstract
formulation
of
the
theory
is
that
of
“dimension
rate,”
that
is
the
average
number
of
dimensions
required
per
second
to
specify
a
member
of
an
ensemble.
In the band limited case 2
W
numbers per second are sufficient.
A general definition can be
framed
as
follows.
Let
f
(
t
)
be
an
ensemble
of
functions
and
let
T
[
f
(
t
)
;
f
(
t
)]
be
a
metric
measuring
54
the “distance” from
f
to
f
over the time
T
(for example the R.M.S.
discrepancy over this interval.)
Let
N
(
;
;
T
)
be
the
least
number of
elements
f
which
can
be
chosen
such
that
all
elements of
the
ensemble
apart from a set of measure
are within the distance
of at least one of those chosen. Thus we are covering
the space to within
apart from a set of small measure
.
We define the dimension rate
for the ensemble
by the triple limit
=
Lim
!
0
Lim
!
0
Lim
T
!
∞
log
N
(
;
;
T
)
T
log
:
This is a generalization of the measure type definitions of dimension in topology, and agrees with the intu-
itive dimension rate for simple ensembles where the desired result is obvious.
55