information theory channel capacity communication systems theory and practice Die Informationstheorie wurde von Claude Elwood Shannon. A First Course in Information Theory is an up-to-date introduction to information Shannon's information measures refer to entropy, conditional entropy, mutual. This book presents a succinct and mathematically rigorous treatment of the main pillars of Shannon's information theory, discussing the fundamental.
Summer Term 2015This book presents a succinct and mathematically rigorous treatment of the main pillars of Shannon's information theory, discussing the fundamental. Shannon's information theory deals with source coding [ ] Claude Shannon established the mathematical basis of information theory and published [ ]. Shannon's channel coding theorem; Random coding and error exponent; MAP and ML decoding; Bounds; Channels and capacities: Gaussian channel, fading.
Shannon Information Theory Shannon’s Bits VideoClaude Shannon: The Ingenious \
Instead of trying to figure out all of the variables in a communication effort like Morse Code, the 0s and 1s of digital coding allow for long strings of digits to be sent without the same levels of informational entropy.
A 0, for example, can be represented by a specific low-voltage signal. A 1 could then be represented by a high voltage signal.
Because there are just two digits and each has a very specific state that can be recognized, even after the signal has experienced extensive entropy, it becomes possible to reconstruct the information with greater accuracy.
Using the information theory, a base 2 is used for the mathematical logarithms so that we can obtain total informational content.
In the instance of a coin flip, the value received is one bit. The same would be true when dice are rolled. 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. One of the key goals for people who use this theory is to identify the causes of noise and try to minimize them to improve the quality of the message.
Examples: Examples of external noise may include the crackling of a poorly tuned radio, a lost letter in the post, an interruption in a television broadcast, or a failed internet connection.
Decoding is the exact opposite of encoding. Shannon and Weaver made this model in reference to communication that happens through devices like telephones.
So, in this model, there usually needs to be a device that decodes a message from binary digits or waves back into a format that can be understood by the receiver.
For example, you might need to decode a secret message, turn written words into something that makes sense in your mind by reading them out loud, or you may need to interpret decode the meaning behind a picture that was sent to you.
Examples: Decoders can include computers that turn binary packets of 1s and 0s into pixels on a screen that make words, a telephone that turns signals such as digits or waves back into sounds, and cell phones that also turn bits of data into readable and listenable messages.
Examples: Examples of a receiver might be: the person on the other end of a telephone, the person reading an email you sent them, an automated payments system online that has received credit card details for payment, etc.
Norbert Weiner came up with the feedback step in response to criticism of the linear nature of the approach. Feedback occurs when the receiver of the message responds to the sender in order to close the communication loop.
They might respond to let the sender know they got the message or to show the sender:. Examples: Feedback does not occur in all situations.
The Shannon-Weaver model of communication was originally proposed for technical communication, such as through telephone communications.
Nonetheless, it has been widely used in multiple different areas of human communication. Sender: The sender is the person who has made the call, and wants to tell the person at the other end of the phone call something important.
This discovery inspired engineers to look for practical techniques to improve performance in signal transmissions that were far from optimal.
Before Shannon, engineers lacked a systematic way of analyzing and solving such problems. Though information theory does not always make clear exactly how to achieve specific results, people now know which questions are worth asking and can focus on areas that will yield the highest return.
They also know which sorts of questions are difficult to answer and the areas in which there is not likely to be a large return for the amount of effort expended.
The section Applications of information theory surveys achievements not only in such areas of telecommunications as data compression and error correction but also in the separate disciplines of physiology, linguistics, and physics.
Unfortunately, many of these purported relationships were of dubious worth. I personally believe that many of the concepts of information theory will prove useful in these other fields—and, indeed, some results are already quite promising—but the establishing of such applications is not a trivial matter of translating words to a new domain, but rather the slow tedious process of hypothesis and experimental verification.
Information theory Article Media Additional Info. In other words, entropy is a measure of the spreading of a probability.
In some sense, the second law of thermodynamics which states that entropy cannot decrease can be reinterpreted as the increasing impossibility of defining precise contexts on a macroscopic level.
It is essential! The most important application probably regards data compression. Indeed, the entropy provides the theoretical limit to the average number of bits to code a message of a context.
It also gives an insight into how to do so. Data compression has been applied to image, audio or file compressing, and is now essential on the Web.
Youtube videos can now be compressed enough to surf all over the Internet! For any given introduction, the message can be described with a conditional probability.
This defines a entropy conditional to the given introduction. Now, the conditional entropy is the average of this entropy conditional to the given introduction, when this given introduction follows the probabilistic distribution of introductions.
Roughly said, the conditional entropy is the average added information of the message given its introduction. 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!
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