第1页 / 共11页
第2页 / 共11页
第3页 / 共11页
第4页 / 共11页
第5页 / 共11页
第6页 / 共11页
第7页 / 共11页
第8页 / 共11页
shannon communication in the presence of noise.pdf
Communication in the Presence of Noise
CLAUDE E. SHANNON, MEMBER, IRE
Classic Paper
A method is developed for representing any communication
system geometrically. Messages and the corresponding signals are
points in two “function spaces,” and the modulation process is a
mapping of one space into the other. Using this representation, a
number of results in communication theory are deduced concern-
ing expansion and compression of bandwidth and the threshold
effect. Formulas are found for the maximum rate of transmission
of binary digits over a system when the signal is perturbed by
various types of noise. Some of the properties of “ideal” systems
which transmit at this maximum rate are discussed. The equivalent
number of binary digits per second for certain information sources
is calculated.
I.
INTRODUCTION
A general communications system is shown schemati-
cally in Fig. 1. It consists essentially of five elements.
1) An Information Source: The source selects one mes-
sage from a set of possible messages to be transmitted to
the receiving terminal. The message may be of various
types; for example, a sequence of letters or numbers, as
in telegraphy or teletype, or a continuous function of time
, as in radio or telephony.
2) The Transmitter: This operates on the message in
some way and produces a signal suitable for transmission
to the receiving point over the channel. In telephony, this
operation consists of merely changing sound pressure into
a proportional electrical current. In telegraphy, we have
a encoding operation which produces a sequence of dots,
dashes, and spaces corresponding to the letters of the
message. To take a more complex example, in the case of
multiplex PCM telephony the different speech functions
must be sampled, compressed, quantized and encoded, and
finally interleaved properly to construct the signal.
3) The Channel: This is merely the medium used to
transmit the signal from the transmitting to the receiving
point. It may be a pair of wires, a coaxial cable, a band
of radio frequencies, etc. During transmission, or at the
receiving terminal, the signal may be perturbed by noise
or distortion. Noise and distortion may be differentiated on
the basis that distortion is a fixed operation applied to the
signal, while noise involves statistical and unpredictable
This paper is reprinted from the PROCEEDINGS OF THE IRE, vol. 37, no.
1, pp. 10–21, Jan. 1949.
Fig. 1. General communications system.
perturbations. Distortion can, in principle, be corrected by
applying the inverse operation, while a perturbation due to
noise cannot always be removed, since the signal does not
always undergo the same change during transmission.
4) The Receiver: This operates on the received signal
and attempts to reproduce, from it, the original message.
Ordinarily it will perform approximately the mathematical
inverse of the operations of the transmitter, although they
may differ somewhat with best design in order to combat
noise.
5) The Destination: This is the person or thing for whom
the message is intended.
Following Nyquist1 and Hartley,2 it is convenient to use
a logarithmic measure of information. If a device has
possible positions it can, by definition, store log
information. The choice of the base
units of
log
log
. We will use the base
2 and call the resulting units binary digits or bits. A group
of
relays or flip-flop circuits has
possible sets of
positions, and can therefore store log
bits.
If it is possible to distinguish reliably
different signal
on a channel, we can say that the
functions of duration
channel can transmit log
bits in time
. More precisely, the channel
. The rate oftransmission is then log
capacity may be defined as
(1)
1 H. Nyquist, “Certain factors affecting telegraph speed,” Bell Syst. Tech.
J., vol. 3, p. 324, Apr. 1924.
2 R. V. L. Hartley, “The transmission of information,” Bell Syst. Tech.
Publisher Item Identifier S 0018-9219(98)01299-7.
J., vol. 3, p. 535–564, July 1928.
0018–9219/98$10.00 ª
1998 IEEE
PROCEEDINGS OF THE IEEE, VOL. 86, NO. 2, FEBRUARY 1998
447
amounts to a choice
A precise meaning will be given later to the requirement
of reliable resolution of the
signals.
apart. The function can be simply reconstructed from the
samples by using a pulse of the type
II. THE SAMPLING THEOREM
Let us suppose that the channel has a certain bandwidth
in cps starting at zero frequency, and that we are allowed
to use this channel for a certain period of time
. Without
any further restrictions this would mean that we can use
as signal functions any functions of time whose spectra lie
entirely within the band
, and whose time functions lie
within the interval
. Although it is not possible to fulfill
both of these conditions exactly, it is possible to keep the
spectrum within the band
, and to have the time function
. Can we describe in a more
very small outside the interval
useful way the functions which satisfy these conditions?
One answer is the following.
Theorem 1: If a function
contains no frequencies
cps, it is completely determined by giving
seconds
higher than
its ordinates at a series of points spaced 1/2
apart.
This is a fact which is common knowledge in the
communication art. The intuitive justification is that, if
, it cannot change to
contains no frequencies higher than
a substantially new value in a time less than one-half cycle
of the highest frequency, that is, 1/2
. A mathematical
this is not only approximately, but
proof showing that
exactly, true can be given as follows. Let
be the
spectrum of
. Then
(2)
(3)
since
is assumed zero outside the band
. If we let
where
is any positive or negative integer, we obtain
(4)
(5)
at the sampling points. The
On the left are the values of
integral on the right will be recognized as essentially the
th coefficient in a Fourier-series expansion of the function
as a fundamental
, taking the interval
to
, since
. Thus they determine
, and for lower frequencies
period. This means that the values of the samples
determine the Fourier coefficients in the series expansion
of
is zero for
frequencies greater than
is determined if its Fourier coefficients are determined. But
completely,
since a function is determined if its spectrum is known.
Therefore the original samples determine the function
completely. There is one and only one function whose
spectrum is limited to a band
, and which passes through
given values at sampling points separated 1.2
seconds
determines the original function
sin
(6)
and zero at
,
This function is unity at
i.e., at all other sample points. Furthermore, its spectrum is
constant in the band
and zero outside. At each sample
point a pulse of this type is placed whose amplitude is
adjusted to equal that of the sample. The sum of these pulses
is the required function, since it satisfies the conditions on
the spectrum and passes through the sampled values.
Mathematically, this process can be described as follows.
is
th sample. Then the function
Let
be the
represented by
sin
(7)
A similar result is true if the band
does not start
at zero frequency but at some higher value, and can be
proved by a linear translation (corresponding physically to
single-sideband modulation) of the zero-frequency case. In
this case the elementary pulse is obtained from sin
by
single-side-band modulation.
If the function is limited to the time interval
and the
samples are spaced 1/2
seconds apart, there will be a total
of
samples in the interval. All samples outside will
be substantially zero. To be more precise, we can define a
function to be limited to the time interval
if, and only if,
all the samples outside this interval are exactly zero. Then
we can say that any function limited to the bandwidth
and the time interval
numbers.
can be specified by giving
. This given
Theorem 1 has been given previously in other forms
by mathematicians3 but in spite of its evident importance
seems not
to have appeared explicitly in the literature
of communication theory. Nyquist,4, 5 however, and more
recently Gabor,6 have pointed out that approximately
numbers are sufficient, basing their arguments on a Fourier
series expansion of the function over the time interval
cosine terms up to
frequency
. The slight discrepancy is due to the fact
that the functions obtained in this way will not be strictly
limited to the band
but, because of the sudden starting
and stopping of the sine and cosine components, contain
some frequency content outside the band. Nyquist pointed
out the fundamental importance of the time interval
seconds in connection with telegraphy, and we will call this
the Nyquist interval corresponding to the band
and
.
3 J. M. Whittaker, Interpolatory Function Theory, Cambridge Tracts
in Mathematics and Mathematical Physics, no. 33. Cambridge, U.K.:
Cambridge Univ. Press, ch. IV, 1935.
4 H. Nyquist, “Certain topics in telegraph transmission theory,” AIEE
Trans., p. 617, Apr. 1928.
5 W. R. Bennett, “Time division multiplex systems,” Bell Syst. Tech.
J., vol. 20, p. 199, Apr. 1941, where a result similar to Theorem 1 is
established, but on a steady-state basis.
6 D. Gabor, “Theory of communication,” J. Inst. Elect. Eng. (London),
vol. 93, pt. 3, no. 26, p. 429, 1946.
448
PROCEEDINGS OF THE IEEE, VOL. 86, NO. 2, FEBRUARY 1998
The
numbers used to specify the function need not
be the equally spaced samples used above. For example,
the samples can be unevenly spaced, although, if there is
considerable bunching, the samples must be known very
accurately to give a good reconstruction of the function. The
reconstruction process is also more involved with unequal
spacing. One can further show that the value of the function
and its derivative at every other sample point are sufficient.
The value and first and second derivatives at every third
sample point give a still different set of parameters which
uniquely determine the function. Generally speaking, any
set of
independent numbers associated with the
function can be used to describe it.
III. GEOMETRICAL REPRESENTATION OF THE SIGNALS
,
,
A set of three numbers
, regardless of their
source, can always be thought of as coordinates of a point in
three-dimensional space. Similarly, the
evenly spaced
samples of a signal can be thought of as coordinates of
a point in a space of
dimensions. Each particular
selection of these numbers corresponds to a particular point
in this space. Thus there is exactly one point corresponding
to each signal in the band
and with duration
.
The number of dimensions
will be, in general, very
high. A 5-Mc television signal lasting for an hour would be
represented by a point in a space with
dimensions. Needless to say, such a space cannot
be visualized. It is possible, however, to study analytically
the properties of
-dimensional space. To a considerable
extent, these properties are a simple generalization of the
properties of two- and three-dimensional space, and can
often be arrived at by inductive reasoning from these cases.
The advantage of this geometrical representation of the
signals is that we can use the vocabulary and the results of
geometry in the communication problem. Essentially, we
have replaced a complex entity (say, a television signal) in
a simple environment [the signal requires only a plane for
its representation as
] by a simple entity (a point) in a
complex environment (
dimensional space).
If we imagine the
coordinate axes to be at right
angles to each other, then distances in the space have a
simple interpretation. The distance from the origin to a point
is analogous to the two- and three-dimensional cases
where
is the
th sample. Now, since
sin
we have
(8)
(9)
(10)
using the fact that
sin
sin
(11)
times the
Hence, the square of the distance to a point is
energy (more precisely, the energy into a unit resistance)
of the corresponding signal
(12)
is the average power over the time
where
the distance between two points is
discrepancy between the two corresponding signals.
. Similarly,
times the rms
If we consider only signals whose average power is less
, these will correspond to points within a sphere of
than
radius
(13)
If noise is added to the signal in transmission, it means
that the point corresponding to the signal has been moved a
certain distance in the space proportional to the rms value of
the noise. Thus noise produces a small region of uncertainty
about each point in the space. A fixed distortion in the
channel corresponds to a warping of the space, so that each
point is moved, but in a definite fixed way.
In ordinary three-dimensional space it
is possible to
set up many different coordinate systems. This is also
possible in the signal space of
dimensions that we
are considering. A different coordinate system corresponds
to a different way of describing the same signal function.
The various ways of specifying a function given above are
special cases of this. One other way of particular importance
in communication is in terms of frequency components.
The function
can be expanded as a sum of sines and
cosines of frequencies
apart, and the coefficients used
as a different set of coordinates. It can be shown that these
coordinates are all perpendicular to each other and are
obtained by what is essentially a rotation of the original
coordinate system.
Passing a signal through an ideal filter corresponds to
projecting the corresponding point onto a certain region in
the space. In fact, in the frequency-coordinate system those
components lying in the pass band of the filter are retained
and those outside are eliminated, so that the projection is
on one of the coordinate lines, planes, or hyperplanes. Any
filter performs a linear operation on the vectors of the space,
producing a new vector linearly related to the old one.
IV. GEOMETRICAL REPRESENTATION OF MESSAGES
We have associated a space of
dimensions with the
set of possible signals. In a similar way one can associate
a space with the set of possible messages. Suppose we are
considering a speech system and that the messages consist
SHANNON: COMMUNICATION IN THE PRESENCE OF NOISE
449
decreased. A similar effect can occur through probability
considerations. Certain messages may be possible, but so
improbable relative to the others that we can, in a certain
sense, neglect them. In a television image, for example,
successive frames are likely to be very nearly identical.
There is a fair probability of a particular picture element
having the same light intensity in successive frames. If
this is analyzed mathematically, it results in an effective
reduction of dimensionality of the message space when
is large.
We will not go further into these two effects at present,
but let us suppose that, when they are taken into account, the
resulting message space has a dimensionality
, which will,
of course, be less than or equal to
. In many cases,
even though the effects are present, their utilization involves
too much complication in the way of equipment. The
system is then designed on the basis that all functions are
different and that there are no limitations on the information
source. In this case, the message space is considered to have
the full
dimensions.
V. GEOMETRICAL REPRESENTATION OF THE
TRANSMITTER AND RECEIVER
We now consider the function of the transmitter from
this geometrical standpoint. The input to the transmitter is
a message; that is, one point in the message space. Its output
is a signal—one point in the signal space. Whatever form
of encoding or modulation is performed, the transmitter
must establish some correspondence between the points in
the two spaces. Every point in the message space must
correspond to a point in the signal space, and no two
messages can correspond to the same signal. If they did,
there would be no way to determine at the receiver which
of the two messages was intended. The geometrical name
for such a correspondence is a mapping. The transmitter
maps the message space into the signal space.
In a similar way, the receiver maps the signal space back
into the message space. Here, however, it is possible to
have more than one point mapped into the same point.
This means that several different signals are demodulated
or decoded into the same message. In AM, for example,
the phase of the carrier is lost in demodulation. Different
signals which differ only in the phase of the carrier are
demodulated into the same message. In FM the shape of
the signal wave above the limiting value of the limiter
does not affect the recovered message. In PCM considerable
distortion of the received pulses is possible, with no effect
on the output of the receiver.
We have so far established a correspondence between a
communication system and certain geometrical ideas. The
correspondence is summarized in Table 1.
VI. MAPPING CONSIDERATIONS
It is possible to draw certain conclusions of a general
nature regarding modulation methods from the geometrical
picture alone. Mathematically, the simplest types of map-
pings are those in which the two spaces have the same
Fig. 2. Reduction of dimensionality through equivalence classes.
of all possible sounds which contain no frequencies over a
certain limit
and last for a time
.
Just as for the case of the signals, these messages can
be represented in a one-to-one way in a space of
dimensions. There are several points to be noted, however.
In the first place, various different points may represent the
same message, insofar as the final destination is concerned.
For example, in the case of speech, the ear is insensitive
to a certain amount of phase distortion. Messages differing
only in the phases of their components (to a limited extent)
sound the same. This may have the effect of reducing the
number of essential dimensions in the message space. All
the points which are equivalent for the destination can be
grouped together and treated as one point. It may then
require fewer numbers to specify one of these “equivalence
classes” than to specify an arbitrary point. For example, in
Fig. 2 we have a two-dimensional space, the set of points in
a square. If all points on a circle are regarded as equivalent,
it reduces to a one-dimensional space—a point can now be
specified by one number, the radius of the circle. In the
case of sounds, if the ear were completely insensitive to
phase, then the number of dimensions would be reduced
by one-half due to this cause alone. The sine and cosine
for a given frequency would not
components
need to be specified independently, but only
; that
is, the total amplitude for this frequency. The reduction in
frequency discrimination of the ear as frequency increases
indicates that a further reduction in dimensionality occurs.
The vocoder makes use to a considerable extent of these
equivalences among speech sounds, in the first place by
eliminating, to a large degree, phase information, and in
the second place by lumping groups of frequencies together,
particularly at the higher frequencies.
and
In other types of communication there may not be any
equivalence classes of this type. The final destination is
sensitive to any change in the message within the full
dimensions. This appears to be
message space of
the case in television transmission.
A second point to be noted is that the information source
may put certain restrictions on the actual messages. The
dimensions contains a point for every
space of
and of duration
function of time
. The class of messages we wish to transmit may be
only a small subset of these functions. For example, speech
sounds must be produced by the human vocal system. If
we are willing to forego the transmission of any other
sounds, the effective dimensionality may be considerably
limited to the band
450
PROCEEDINGS OF THE IEEE, VOL. 86, NO. 2, FEBRUARY 1998
Table 1
Fig. 3. Mapping similar to frequency modulation.
number of dimensions. Single-sideband amplitude modula-
tion is an example of this type and an especially simple one,
since the coordinates in the signal space are proportional
to the corresponding coordinates in the message space. In
double-sideband transmission the signal space has twice the
number of coordinates, but they occur in pairs with equal
values. If there were only one dimension in the message
space and two in the signal space, it would correspond to
mapping a line onto a square so that the point
on the line
is represented by
in the square. Thus no significant
use is made of the extra dimensions. All the messages go
into a subspace having only
dimensions.
In frequency modulation the mapping is more involved.
The signal space has a much larger dimensionality than
the message space. The type of mapping can be suggested
by Fig. 3, where a line is mapped into a three-dimensional
space. The line starts at unit distance from the origin on the
first coordinate axis, stays at this distance from the origin
on a circle to the next coordinate axis, and then goes to
the third. It can be seen that the line is lengthened in this
mapping in proportion to the total number of coordinates.
It is not, however, nearly as long as it could be if it wound
back and forth through the space, filling up the internal
volume of the sphere it traverses.
This expansion of the line is related to the improved
signal-to-noise ratio obtainable with increased bandwidth.
Since the noise produces a small region of uncertainty about
each point, the effect of this on the recovered message will
be less if the map is in a large scale. To obtain as large
a scale as possible requires that the line wander back and
Fig. 4. Efficient mapping of a line into a square.
forth through the higher dimensional region as indicated
in Fig. 4, where we have mapped a line into a square. It
will be noticed that when this is done the effect of noise is
small relative to the length of the line, provided the noise
is less than a certain critical value. At this value it becomes
uncertain at the receiver as to which portion of the line
contains the message. This holds generally, and it shows
that any system which attempts to use the capacities of a
wider band to the full extent possible will suffer from a
threshold effect when there is noise. If the noise is small,
very little distortion will occur, but at some critical noise
amplitude the message will become very badly distorted.
This effect is well known in PCM.
over a channel with
Suppose, on the other hand, we wish to reduce dimen-
sionality, i.e., to compress bandwidth or time or both. That
is, we wish to send messages of band
and duration
. It has already been
indicated that the effective dimensionality
of the message
space may be less than
due to the properties of the
source and of the destination. Hence we certainly need no
dimension in the signal space for a good
more than
mapping. To make this saving it is necessary, of course, to
isolate the effective coordinates in the message space, and
to send these only. The reduced bandwidth transmission of
speech by the vocoder is a case of this kind.
The question arises, however, as to whether further
reduction is possible. In our geometrical analogy,
is it
possible to map a space of high dimensionality onto one of
lower dimensionality? The answer is that it is possible, with
certain reservations. For example, the points of a square
can be described by their two coordinates which could be
written in decimal notation
From these two numbers we can construct one number by
taking digits alternately from and
(15)
(14)
determines
and . Thus there is a one-to-one correspondence
A knowledge of
both
between the points of a square and the points of a line.
and
determines
, and
SHANNON: COMMUNICATION IN THE PRESENCE OF NOISE
451
This type of mapping, due to the mathematician Cantor,
can easily be extended as far as we wish in the direction of
reducing dimensionality. A space of
dimensions can be
mapped in a one-to-one way into a space of one dimension.
Physically, this means that the frequency-time product can
be reduced as far as we wish when there is no noise, with
exact recovery of the original messages.
In a less exact sense, a mapping of the type shown in
Fig. 4 maps a square into a line, provided we are not too
particular about recovering exactly the starting point, but
are satisfied with a nearby one. The sensitivity we noticed
before when increasing dimensionality now takes a different
form. In such a mapping, to reduce
, there will be a
certain threshold effect when we perturb the message. As
we change the message a small amount, the corresponding
signal will change a small amount, until some critical
value is reached. At this point the signal will undergo a
considerable change. In topology it is shown7 that it is
not possible to map a region of higher dimension into a
region of lower dimension continuously. It is the necessary
discontinuity which produces the threshold effects we have
been describing for communication systems.
This discussion is relevant to the well-known “Hartley
law,” which states that “an upper limit
to the amount
of information which may be transmitted is set by the
sum for the various available lines of the product of the
line-frequency range of each by the time during which
it is available for use.” There is a sense in which this
statement is true, and another sense in which it is false.
It
is not possible to map the message space into the
signal space in a one-to-one, continuous manner (this is
known mathematically as a topological mapping) unless
the two spaces have the same dimensionality; i.e., unless
. Hence, if we limit the transmitter and receiver
to continuous one-to-one operations, there is a lower bound
to the product
in the channel. This lower bound is
determined, not by the product
of message bandwidth
and time, but by the number of essential dimension
, as
indicated in Section IV. There is, however, no good reason
for limiting the transmitter and receiver to topological
mappings. In fact, PCM and similar modulation systems
are highly discontinuous and come very close to the type
of mapping given by (14) and (15). It is desirable, then, to
find limits for what can be done with no restrictions on the
type of transmitter and receiver operations. These limits,
which will be derived in the following sections, depend on
the amount and nature of the noise in the channel, and on
the transmitter power, as well as on the bandwidth-time
product.
It is evident that any system, either to compress
, or
to expand it and make full use of the additional volume,
must be highly nonlinear in character and fairly complex
because of the peculiar nature of the mappings involved.
VII. THE CAPACITY OF A CHANNEL IN THE
PRESENCE OF WHITE THERMAL NOISE
It is not difficult to set up certain quantitative relations
that must hold when we change the product
. Let us
assume, for the present, that the noise in the system is a
white thermal-noise band limited to the band
, and that
it is added to the transmitted signal to produce the received
signal. A white thermal noise has the property that each
sample is perturbed independently of all the others, and the
distribution of each amplitude is Gaussian with standard
deviation
is the average noise power.
How many different signals can be distinguished at the
receiving point in spite of the perturbations due to noise?
A crude estimate can be obtained as follows. If the signal
has a power
, then the perturbed signal will have a power
. The number of amplitudes that can be reasonably
where
well distinguished is
(16)
is a small constant
in the neighborhood of
where
unity depending on how the phrase “reasonably well” is
interpreted. If we require very good separation,
will
be small, while toleration of occasional errors allows
to be larger. Since in time
independent
amplitudes, the total number of reasonably distinct signals
is
there are
The number of bits that can be sent in this time is log
and the rate of transmission is
(17)
,
log
(bits per second)
(18)
The difficulty with this argument, apart from its general
approximate character,
lies in the tacit assumption that
for two signals to be distinguishable they must differ at
some sampling point by more than the expected noise.
The argument presupposes that PCM, or something very
similar to PCM, is the best method of encoding binary
digits into signals. Actually, two signals can be reliably
distinguished if they differ by only a small amount, pro-
vided this difference is sustained over a long period of
time. Each sample of the received signal then gives a small
amount of statistical information concerning the transmitted
signal; in combination, these statistical indications result in
near certainty. This possibility allows an improvement of
about 8 dB in power over (18) with a reasonable definition
of reliable resolution of signals, as will appear later. We
will now make use of the geometrical representation to
determine the exact capacity of a noisy channel.
Theorem 2: Let
be the average transmitter power, and
in the
. By sufficiently complicated encoding systems it
suppose the noise is white thermal noise of power
band
is possible to transmit binary digits at a rate
7 W. Hurewitz and H. Wallman, Dimension Theory. Princeton, NJ:
Princeton Univ. Press, 1941.
log
(19)
452
PROCEEDINGS OF THE IEEE, VOL. 86, NO. 2, FEBRUARY 1998
with as small a frequency of errors as desired. It is not
possible by any encoding method to send at a higher rate
and have an arbitrarily low frequency of errors.
log
This shows that the rate
measures
in a sharply defined way the capacity of the channel for
transmitting information. It is a rather surprising result,
since one would expect that reducing the frequency of
errors would require reducing the rate of transmission, and
that the rate must approach zero as the error frequency
does. Actually, we can send at
but reduce
errors by using more involved encoding and longer delays
at the transmitter and receiver. The transmitter will take
long sequences of binary digits and represent this entire
sequence by a particular signal function of long duration.
The delay is required because the transmitter must wait for
the full sequence before the signal is determined. Similarly,
the receiver must wait for the full signal function before
decoding into binary digits.
the rate
We now prove Theorem 2. In the geometrical represen-
tation each signal point is surrounded by a small region
of uncertainty due to noise. With white thermal noise, the
perturbations of the different samples (or coordinates) are
all Gaussian and independent. Thus the probability of a
perturbation having coordinates
(these are
the differences between the original and received signal
coordinates) is the product of the individual probabilities
for the different coordinates
exp
exp
Since this depends only on
the probability of a given perturbation depends only on the
distance from the original signal and not on the direction.
In other words, the region of uncertainty is spherical in
nature. Although the limits of this region are not sharply
defined for a small number of dimensions
, the
limits become more and more definite as the dimensionality
increases. This is because the square of the distance a
signal is perturbed is equal to
times the average
noise power during the time
this
average noise power must approach
. Thus, for large
, the perturbation will almost certainly be to some point
centered
the original signal point. More precisely, by taking
sufficiently large we can insure (with probability as
lie
near to one as we wish) that
within a sphere of radius
is
arbitrarily small. The noise regions can therefore be thought
of roughly as sharply defined billiard balls, when
is
very large. The received signals have an average power
, and in the same sense must almost all lie on
near the surface of a sphere of radius
at
the perturbation will
increases,
where
. As
the surface of a sphere of radius
. How
many different transmitted signals can be found which will
be distinguishable? Certainly not more than the volume
of the sphere of radius
divided by the
, since overlap of
volume of a sphere of radius
the noise spheres results in confusion as to the message
at the receiving point. The volume of an
-dimensional
sphere8 of radius
is
(20)
Hence, an upper limit for the number
signals is
of distinguishable
Consequently, the channel capacity is bounded by
This proves the last statement in the theorem.
log
(21)
(22)
log
To prove the first part of the theorem, we must show
that there exists a system of encoding which transmits
binary digits per second with a fre-
quency of errors less than when
is arbitrarily small.
The system to be considered operates as follows. A long
sequence of, say,
binary digits is taken in at the trans-
mitter. There are
such sequences, and each corresponds
to a particular signal function of duration
. Thus there
are
different signal functions. When the sequence
of
is completed, the transmitter starts sending the cor-
responding signal. At the receiver a perturbed signal is
received. The receiver compares this signal with each of the
possible transmitted signals and selects the one which
is nearest the perturbed signal (in the sense of rms error)
as the one actually sent. The receiver then constructs, as its
output, the corresponding sequence of binary digits. There
will be, therefore, an overall delay of
seconds.
To insure a frequency of errors less than , the
signal
functions must be reasonably well separated from each
other. In fact, we must choose them in such a way that,
when a perturbed signal is received, the nearest signal
point (in the geometrical representation) is, with probability
greater than
, the actual original signal.
It turns out, rather surprisingly, that it is possible to
choose our
signal functions at random from the points
inside the sphere of radius
, and achieve the most
that is possible. Physically, this corresponds very nearly to
using
different samples of band-limited white noise with
power
as signal functions.
A particular selection of
points in the sphere corre-
sponds to a particular encoding system. The general scheme
of the proof is to consider all such selections, and to show
that the frequency of errors averaged over all the particular
selections is less than . This will show that there are
8 D. M. Y. Sommerville, An Introduction to the Geometry of N
Dimensions. New York: Dutton, 1929, p. 135.
SHANNON: COMMUNICATION IN THE PRESENCE OF NOISE
453
frequency of errors will be less than . This will be true if
Now
positive. Consequently, (25) will be true if
is always greater than
when
is
(25)
Fig. 5. The geometry involved in Theorem 2.
particular selections in the set with frequency of errors less
than . Of course, there will be other particular selections
with a high frequency of errors.
or if
or
log
log
log
(26)
(27)
(28)
, received signal
The geometry is shown in Fig. 5. This is a plane cross
section through the high-dimensional sphere defined by a
typical transmitted signal
, and the
lie very close to
origin 0. The transmitted signal will
the surface of the sphere of radius
, since in
a high-dimensional sphere nearly all the volume is very
close to the surface. The received signal similarly will lie
on the surface of the sphere of radius
.
The high-dimensional lens-shaped region
is the region
, since the
of possible signals that might have caused
distance between the transmitted and received signal is
almost certainly very close to
is of smaller
volume than a sphere of radius
by equating the area of the triangle
different ways
. We can determine
, calculated two
.
) lying in
The probability of any particular signal point (other
therefore,
than the actual cause of
less than the ratio of the volumes of spheres of radii
, since in our ensemble
of coding systems we chose the signal points at random
. This
from the points in the sphere of radius
ratio is
and
is,
(23)
We have
except the actual cause of
signal points. Hence the probability
are outside
that all
is greater than
When these points are outside
correctly. Therefore, if we make
, the signal is interpreted
, the
greater than
(24)
log
For any fixed , we can satisfy this by taking
or log
suffi-
as
ciently large, and also have log
close as desired to
. This shows that, with
a random selection of points for signals, we can obtain an
arbitrarily small frequency of errors and transmit at a rate
arbitrarily close to the rate
. We can also send at the
rate with arbitrarily small
, since the extra binary digits
need not be sent at all, but can be filled in at random at the
receiver. This only adds another arbitrarily small quantity
to . This completes the proof.
VIII. DISCUSSION
We will call a system that transmits without errors at the
rate
an ideal system. Such a system cannot be achieved
with any finite encoding process but can be approximated
as closely as desired. As we approximate more closely to
the ideal, the following effects occur.
1) The rate of transmission of binary digits approaches
log
.
2) The frequency of errors approaches zero.
3) The transmitted signal approaches a white noise in
statistical properties. This is true, roughly speaking,
because the various signal functions used must be dis-
.
tributed at random in the sphere of radius
4) The threshold effect becomes very sharp. If the noise
is increased over the value for which the system
was designed, the frequency of errors increases very
rapidly.
5) The required delays at transmitter and receiver in-
crease indefinitely. Of course, in a wide-band system
a millisecond may be substantially an infinite delay.
log
in dB horizontal and
In Fig. 6 the function
is plotted
with
the number of bits
per cycle of band vertical. The circles represent PCM
systems of the binary, ternary, etc., types, using positive
and negative pulses and adjusted to give one error in about
binary digits. The dots are for a PPM system with two,
454
PROCEEDINGS OF THE IEEE, VOL. 86, NO. 2, FEBRUARY 1998
相关推荐
2023年江西萍乡中考道德与法治真题及答案.doc
2012年重庆南川中考生物真题及答案.doc
2013年江西师范大学地理学综合及文艺理论基础考研真题.doc
2020年四川甘孜小升初语文真题及答案I卷.doc
2020年注册岩土工程师专业基础考试真题及答案.doc
2023-2024学年福建省厦门市九年级上学期数学月考试题及答案.doc
2021-2022学年辽宁省沈阳市大东区九年级上学期语文期末试题及答案.doc
2022-2023学年北京东城区初三第一学期物理期末试卷及答案.doc
2018上半年江西教师资格初中地理学科知识与教学能力真题及答案.doc
2012年河北国家公务员申论考试真题及答案-省级.doc
2020-2021学年江苏省扬州市江都区邵樊片九年级上学期数学第一次质量检测试题及答案.doc
2022下半年黑龙江教师资格证中学综合素质真题及答案.doc
