A First Course in Information Theory is an up-to-date introduction to information Shannon's information measures refer to entropy, conditional entropy, mutual. Originally developed by Claude Shannon in the s, information theory laid the foundations for the digital revolution, and is now an essential tool in. This book presents a succinct and mathematically rigorous treatment of the main pillars of Shannon's information theory, discussing the fundamental.
Dem Autor folgenThis book presents a succinct and mathematically rigorous treatment of the main pillars of Shannon's information theory, discussing the fundamental. Shannon's channel coding theorem; Random coding and error exponent; MAP and ML decoding; Bounds; Channels and capacities: Gaussian channel, fading. provides the first comprehensive treatment of the theory of I-Measure, network coding theory, Shannon and non-Shannon type information inequalities, and a.
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He did this work in , but at that time it was classified. The scheme is called the one-time pad or the Vernam cypher, after Gilbert Vernam, who had invented it near the end of World War I.
The idea is to encode the message with a random series of digits--the key--so that the encoded message is itself completely random.
The catch is that one needs a random key that is as long as the message to be encoded and one must never use any of the keys twice. Shannon's contribution was to prove rigorously that this code was unbreakable.
To this day, no other encryption scheme is known to be unbreakable. The problem with the one-time pad so-called because an agent would carry around his copy of a key on a pad and destroy each page of digits after they were used is that the two parties to the communication must each have a copy of the key, and the key must be kept secret from spies or eavesdroppers.
Quantum cryptography solves that problem. More properly called quantum key distribution, the technique uses quantum mechanics and entanglement to generate a random key that is identical at each end of the quantum communications channel.
The quantum physics ensures that no one can eavesdrop and learn anything about the key: any surreptitious measurements would disturb subtle correlations that can be checked, similar to error-correction checks of data transmitted on a noisy communications line.
Encryption based on the Vernam cypher and quantum key distribution is perfectly secure: quantum physics guarantees security of the key and Shannon's theorem proves that the encryption method is unbreakable.
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Peter precious February 11, , am. Eregare Gift Oghenekevwe February 26, , pm. I know! Common sense says that the added information of a message to its introduction should not be larger than the information of the message.
This translates into saying that the conditional entropy should be lower than the non-conditional entropy. This is a theorem proven by Shannon!
In fact, he went further and quantified this sentence: The entropy of a message is the sum of the entropy of its introduction and the entropy of the message conditional to its introduction!
Fortunately, everything can be more easily understood on a figure. The amount of information of the introduction and the message can be drawn as circles.
Because they are not independent, they have some mutual information, which is the intersection of the circles. On the left of the following figure is the entropies of two coins thrown independently.
On the right is the case where only one coin is thrown, and where the blue corresponds to a sensor which says which face the coin fell on.
The sensor has two positions heads or tails , but, now, all the information is mutual:. As you can see, in the second case, conditional entropies are nil.
Indeed, once we know the result of the sensor, then the coin no longer provides any information. Thus, in average, the conditional information of the coin is zero.
In other words, the conditional entropy is nil. It surely is! Indeed, if you try to encode a message by encoding each character individually, you will be consuming space to repeat mutual information.
In fact, as Shannon studied the English language, he noticed that the conditional entropy of a letter knowing the previous one is greatly decreased from its non-conditional entropy.
The structure of information also lies in the concatenation into longer texts. In fact, Shannon defined the entropy of each character as the limit of the entropy of messages of great size divided by the size.
As it turns out, the decrease of entropy when we consider concatenations of letters and words is a common feature of all human languages… and of dolphin languages too!
This has led extraterrestrial intelligence seekers to search for electromagnetic signals from outer spaces which share this common feature too, as explained in this brilliant video by Art of the Problem :.
In some sense, researchers assimilate intelligence to the mere ability to decrease entropy. What an interesting thing to ponder upon!
A communication consists in a sending of symbols through a channel to some other end. Now, we usually consider that this channel can carry a limited amount of information every second.
Shannon calls this limit the capacity of the channel. The channel is usually using a physical measurable quantity to send a message. This can be the pressure of air in case of oral communication.
For longer telecommunications, we use the electromagnetic field. The message is then encoded by mixing it into a high frequency signal. The frequency of the signal is the limit, as using messages with higher frequencies would profoundly modify the fundamental frequency of the signal.
Imagine there was a gigantic network of telecommunication spread all over the world to exchange data, like texts and images.
How fast can we download images from the servers of the Internet to our computers? Using the basic formatting called Bitmap or BMP, we can encode images pixels per pixels.
The encoded images are then decomposed into a certain number of bits. In the example, using bitmap encoding, the images can be transfered at the rate of 5 images per second.
This is another bit of information. You could expand this to a twenty-sided die as well. This principle can then be used to communicate letters, numbers, and other informational concepts that we recognize.
Take the alphabet, for example. In reducing the uncertainty of the equation, multiple bits of information are generated. This is because each character being transmitted either is or is not a specific letter of that alphabet.
When you add in a space, which is required for communication in words, the English alphabet creates 27 total characters. This results in 4.
Thanks to the mathematics of the information theory, we can know with certainty that any transmission or storage of information in digital code requires a multiplication of 4.
Probabilities help us to further reduce the uncertainty that exists when evaluating the equations of information that we receive every day. It also means we can transmit less data, further reducing our uncertainty we face in solving the equation.
Once all of these variables are taken into account, we can reduce the uncertainty which exists when attempting to solve informational equations.I'd love it if you joined and we can chat even more. New York: Interscience, Scientific Reports. In he wrote an article for Scientific American on the principles of Jule Spiel computers to play chess [see "A Chess-Playing Machine," by Claude Obstgarten Haba Spielanleitung.