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14 de out. de 2002 · Claude E. Shannon: Founder of Information Theory. With the fundamental new discipline of quantum information science now under construction, it's a good time to look back at an...
- y Hy x
- 1 4 W 2 W 4 W 2 W 0
- p i j
- j log pi j
- åfiHi
- Bi log p Bi
- ✪ W Bi
- H H C H.
- PART II: THE DISCRETE CHANNEL WITH NOISE
- E f S N
- p i j
- H z
- Hy x
- Hy x C
- Hy x .
- Cåhst
- AjWL
- Kå pi logni
- Kå pi log pi
- Pipij N
- GN H. N ¥
- BjD ij
- PART III: MATHEMATICAL PRELIMINARIES
- T f t
- T f t
- H W log2 eN
- f 2 d f
- N3 ✪ N1 N2
- Px y
- R H x Hy x
- x y logP x dxdy
- P x P y
- P x y
- P x P y
- Hv u
- H n
- PART V: THE RATE FOR A CONTINUOUS SOURCE
- v P x y
- x y
- P x y dxdy 1
- x y P x y dxdy
- K P x y x y dxdy.
- x y log
- B x e
- Py x e x y
- e elog
- P x P y
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 x x log p x dx with
On expansion this leads to the equation given above for this case. 2. THE DISCRETE SOURCE OF INFORMATION 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...
✪ åj p i j We define the conditional entropy of y, Hx 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
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
where fi is the average frequency (occurrences per second) of state i. Clearly
where the sum is over all sequences Bi containing N symbols. Then GN is a monotonic decreasing function of N and Lim GN H N ¥ Theorem 6: Let p Bi Sj be the probability of sequence Bi followed by symbol Sj and pBi Sj p Bi Sj p Bi be the conditional probability of Sj after Bi. Let
s ij where s ij is the duration of the sth symbol leading from state i to state j and the Bi satisfy Bi åBjW j 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. THE FUNDAMENTAL THEOREM FOR A NOISELESS CHANNEL We will now justi...
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 2RN members (where R is the logarithm of the number of different...
11. REPRESENTATION OF A NOISY DISCRETE CHANNEL 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 pro...
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 probabi...
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 ther...
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 Hyz x 0. ...
H y Hx y x y x y 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 s...
with positive. This contradicts the definition of C as the maximum of H 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 can be more than below the maximum. Since arbitrarily small we can say that almost all the systems are arbitrarily close to the ideal. is 14. DISCUSSION The demonstration of Theorem 11, while not ...
s Hence: åPipit i exp Cåhst s åhst psj log psj s j or, Pi åhit exp t Cåhst s åhst psj log psj s j This is the system of equations for determining the maximizing values of Pi, with C to be determined so that åPi 1. When this is done C will be the channel capacity, and the Pi the proper probabilities for the channel symbols to achieve this capacity. ...
Substituting in the difference equation or AjWL åAiWL i s b s ij Aj åAiW s b ij i s å å W i s ij b s ij
Hence H K å pi logåni å pi logni ni Kå pi log ✪ åni
If the pi 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 THEOREMS ON ERGODIC SOURCES If it is possible to go from any stat...
Hence nearly all sequences have a probability p given by Pipij N p Õ p ij log p and ✪ is limited by N log p ✪ N
APPENDIX 4 MAXIMIZING THE RATE FOR A SYSTEM OF CONSTRAINTS Suppose we have a set of constraints on sequences of symbols that is of the finite state type and can be s represented therefore by a linear graph. Let ij be the lengths of the various symbols that can occur in passing from state i to state j. What distribution of probabilities Pi for the d...
✪ ås BsD is The correct value of D is the capacity C and the Bj are solutions of for then or So that if i satisfy Bi åBjC ij pij Bj C ij Bi Bj åPi C ij Bi Pj
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 int...
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 ex...
implies g t1 f t1 for all f t and all t1. 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 ...
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 probabili...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
- 357KB
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22 de dez. de 2023 · Atualizado em: 22/12/2023. Talvez você não tenha ouvido falar em Claude Elwood Shannon, mas saiba que ele teve grande influência para você estar lendo este texto agora. O matemático estadunidense é conhecido como nada menos que o pai da teoria da informação, e completaria 105 anos hoje (30).
Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American mathematician, electrical engineer, computer scientist and cryptographer known as the "father of information theory". Shannon was the first to describe the Boolean gates (electronic circuits) that are essential to all digital electronic circuits, and also ...
30 de abr. de 2024 · Claude Shannon was an American mathematician and electrical engineer who laid the theoretical foundations for digital circuits and information theory, a mathematical communication model. After graduating from the University of Michigan in 1936 with bachelor’s degrees in mathematics and electrical.
- George Markowsky
Claude Elwood Shannon (30 de abril de 1916 — 24 de fevereiro de 2001) foi um matemático, engenheiro eletrônico e criptógrafo estadunidense, conhecido como "o pai da teoria da informação". [2] [3] De 1932 a 1936, estudou matemática e engenharia elétrica na Universidade de Michigan.
22 de dez. de 2020 · Claude Shannon wrote a master’s thesis that jump-started digital circuit design, and a decade later he wrote his seminal paper on information theory, “A Mathematical Theory of Communication.” Courtesy of MIT Museum. Next, Shannon set his sights on an even bigger target: communication. Communication is one of the most basic human needs.
