Information Theory by Claude Shannon
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Questions and Answers

What was the primary aim of Claude Shannon's theories in information theory?

  • To measure the loss of information in broadcasting systems.
  • To quantify how much information can be communicated between two components. (correct)
  • To design more effective communication equipment.
  • To explore the relationship between physics and information.

Which of the following accurately describes Shannon's contribution to the understanding of information?

  • Information primarily refers to qualitative data in communication.
  • Information is a vague concept that lacks quantitative measurement.
  • Information is a well-defined and measurable quantity. (correct)
  • Information can be treated as a variable dependent on natural phenomena.

In the context of Shannon's information theory, what does a bit represent?

  • A measurement that only applies to electronic devices.
  • An increment of data capable of two equally probable alternatives. (correct)
  • The highest level of complexity in data communication.
  • The total amount of noise introduced in a communication channel.

What essential concept did Shannon propose regarding information?

<p>Information can be quantified similarly to physical quantities such as mass. (D)</p> Signup and view all the answers

What is the significance of the paper 'A Mathematical Theory of Communication'?

<p>It provided a statistical model for analyzing communication systems. (B)</p> Signup and view all the answers

In Shannon's model of communication, what occurs after a message is encoded?

<p>Noise is added to the message during transmission. (B)</p> Signup and view all the answers

What aspect of communication systems did Shannon focus on analyzing?

<p>The ability to convey information through different channels. (C)</p> Signup and view all the answers

Which of the following statements about the publication of Shannon's work is true?

<p>It was one of the most cited works, with nearly 150,000 citations. (D)</p> Signup and view all the answers

How many possible destinations result from the 3 forks from Point A to Point D?

<p>8 (B)</p> Signup and view all the answers

What does the variable 'n' represent in the context of logarithmic calculations?

<p>The base 2 logarithm of 'm' (D)</p> Signup and view all the answers

What is the main distinction between a bit and a binary digit?

<p>A bit measures information, while a binary digit is a value (D)</p> Signup and view all the answers

In the context of entropy, what does the term 'measure of disorder' refer to?

<p>The uncertainty associated with a system (C)</p> Signup and view all the answers

How is entropy commonly represented in the context of random variables?

<p>By H(x), the entropy of variable x (D)</p> Signup and view all the answers

If a coin flip has a probability of heads at 90%, what is the probability of tails?

<p>0.1 (B)</p> Signup and view all the answers

What is an example of entropy in a practical scenario?

<p>The result of a biased coin flip (C)</p> Signup and view all the answers

What does 'weighted average' refer to in the context of calculating entropy?

<p>An average that accounts for the likelihood of each outcome (D)</p> Signup and view all the answers

What is the entropy of a fair coin?

<p>1 bit (D)</p> Signup and view all the answers

When flipping a biased coin that lands on heads 90% of the time, how much Shannon information is gained when flipping heads?

<p>0.15 bits (B)</p> Signup and view all the answers

What is the Shannon information gained when flipping tails if a biased coin lands on tails 10% of the time?

<p>3.32 bits (A)</p> Signup and view all the answers

Which of the following statements about entropy is correct?

<p>Entropy reflects the average Shannon information of a probabilistic event. (D)</p> Signup and view all the answers

How does the entropy of a biased coin compare to that of a fair coin?

<p>It can be greater or less depending on the bias. (C)</p> Signup and view all the answers

What outcome yields more Shannon information when flipping a biased coin landing heads 90% of the time?

<p>Tails yield more information. (D)</p> Signup and view all the answers

What would be the Shannon information gained for a fair coin resulting in heads or tails?

<p>1 bit for both outcomes (C)</p> Signup and view all the answers

In terms of predictability, how does a fair coin differ from a biased coin?

<p>A fair coin is less predictable than a biased coin. (C)</p> Signup and view all the answers

What happens to channel capacity as noise increases?

<p>Channel capacity decreases (D)</p> Signup and view all the answers

How is channel capacity mathematically expressed?

<p>$C = 2 log_2 (1 + \frac{S}{N})$ (A)</p> Signup and view all the answers

If the signal power is 10 mW and noise power is 1 mW, what is the signal-to-noise ratio (S/N)?

<p>10 (B)</p> Signup and view all the answers

Given a signal voltage of 4.2 mV and noise voltage of 0.4 mV, what is the computed S/N ratio?

<p>10.5 (D)</p> Signup and view all the answers

Using the computed S/N of 10.5, what is the channel capacity C?

<p>1.0 bits (C)</p> Signup and view all the answers

Which formula correctly defines the role of signal-to-noise ratio in channel capacity?

<p>It quantitatively influences the logarithmic computation of channel capacity. (C)</p> Signup and view all the answers

How is Shannon’s Source Coding Theorem related to channel capacity?

<p>It suggests encoding can allow reaching the limit of channel capacity. (C)</p> Signup and view all the answers

If the channel capacity is 1 bit per usage, what does this imply about the maximum information communicated?

<p>Only one bit of information can be communicated. (A)</p> Signup and view all the answers

What does the equation $H(y) = H(x) + H(η)$ represent?

<p>The relationship between input entropy, output entropy, and noise entropy (B)</p> Signup and view all the answers

What effect does increasing noise have on channel capacity?

<p>It reduces the maximum amount of information communicated (B)</p> Signup and view all the answers

If there are 2 equiprobable values for channel noise, what is the noise entropy $H(η)$?

<p>$1$ bit (A)</p> Signup and view all the answers

In the context of information transmission, which variable directly influences the potential for information transfer in the channel?

<p>All of the above (D)</p> Signup and view all the answers

How is input entropy $H(x)$ calculated with known equiprobable input states?

<p>Using the formula $H(x) = log_2(mx)$ (A)</p> Signup and view all the answers

When noise is added in an additive channel, what is the relationship between the input and output?

<p>The output is a combination of input and noise (D)</p> Signup and view all the answers

Given that there are 3 equiprobable input states, what is the input entropy $H(x)$?

<p>$1.58$ bits (D)</p> Signup and view all the answers

Which of the following statements about noise in information transmission is correct?

<p>Noise negatively affects information transmission (C)</p> Signup and view all the answers

What is the output value for 𝑦1 given the equation 𝑦1 = 𝑥1 + 𝜂1 and that 𝑥1 = 200 and 𝜂1 = 20?

<p>220 (B)</p> Signup and view all the answers

How many equiprobable output states are possible given three input states and two noise values?

<p>6 (B)</p> Signup and view all the answers

What is the output entropy H(𝑦) when 𝑚𝑦 = 6?

<p>2.58 bits (D)</p> Signup and view all the answers

Shannon’s Source Coding Theorem particularly applies to which type of channel?

<p>Noiseless channels (B)</p> Signup and view all the answers

According to Shannon's Fundamental Coding Theorem, what is the relationship between channel capacity C and entropy H?

<p>C &gt; H (A)</p> Signup and view all the answers

What limit does Shannon's Source Coding Theorem define for the encoding of data?

<p>Limits of efficient data re-packaging (A)</p> Signup and view all the answers

If 𝑥3 = 300 and 𝜂1 = 40, what is the value of 𝑦6?

<p>340 (D)</p> Signup and view all the answers

What does ε represent in Shannon's theorem regarding the transmission rate?

<p>The smallest error margin (D)</p> Signup and view all the answers

Flashcards

Information Theory

A mathematical framework for quantifying and analyzing information transmission.

Claude Shannon

A mathematician and computer scientist who pioneered Information Theory.

Bit

The smallest unit of information, representing two equally likely choices or possibilities.

Communication Channel

The medium through which a message is transmitted, potentially introducing noise.

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Information as a physical quantity

Information, like mass or energy, can be measured and quantified.

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Equiprobable Alternatives

Possible outcomes with equal likelihood.

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Communication System

The components (transmitter, channel, receiver) involved in sending and receiving a message.

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Noise

Distortion or interference that can affect the signal being communicated.

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Bits vs. Binary Digits

A bit represents an amount of information, while a binary digit is a value of a binary variable.

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Entropy (Information Theory)

The expected number of bits of information in a message, considering all possible messages.

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Information in a Tournament

A message conveying a tournament result has varying information depending on the probability associated with each potential outcome.

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Entropy and Coin Flip

The entropy of a coin flip accounts for the average surprise in the outcomes (head or tail), considering the probability distribution of each outcome.

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Entropy = Average Shannon Information

Entropy, in Information Theory, is the average information contained in a random variable, quantified using Shannon's formula.

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Random Variable (𝑥)

A variable whose value is a result of a random phenomenon.

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Probability Distribution (𝑝(𝑥))

A function describing the likelihood of different outcomes of a random variable.

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𝑛 = log₂ 𝑚 (Fork Problem)

In a branching process, the number of bits (n) needed to represent the possible outcomes is related to the total number of possibilities (m) by this logarithmic relationship; it is the base-2 logarithm of the total number of options available.

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Entropy of a fair coin

The average amount of information gained from flipping an unbiased coin, which is 1 bit.

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Probability of heads (p(xh))

The likelihood of getting a head in a coin flip. For a fair coin, p(xh) = 0.5.

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Probability of tails (p(xt))

The likelihood of getting a tail in a coin flip. For a fair coin, it's 0.5.

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Shannon information

The amount of information gained from an event, calculated using the formula: log₂(1/probability).

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Unbiased coin

A coin where the probability of heads and tails are exactly 50% each.

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Biased coin

A coin where the probability of heads or tails is not 50%.

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Information Gain

The difference in uncertainty before and after an event occurs. Higher surprise yields more information.

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Entropy

The average amount of information content in a message or outcome.

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Channel Capacity Law

The principle stating that noise limits the maximum amount of information a channel can transmit.

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Additive Channel

A communication channel where noise is added to the signal, resulting in a noisy output.

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Input Entropy (H(x))

The uncertainty or randomness associated with the information at the channel's input.

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Output Entropy (H(y))

The uncertainty or randomness associated with the information at the channel's output.

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Noise Entropy (H(η))

The uncertainty or randomness associated with the noise in the channel.

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How does noise affect output entropy?

Noise increases output entropy, making it more difficult to extract the original information.

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High Output Entropy (H(y))

Indicates a high potential for information transmission, but only if input entropy and noise are favorable.

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Relation between Input, Output, and Noise Entropy

In an additive channel, output entropy equals the sum of input entropy and noise entropy: H(y) = H(x) + H(η).

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Signal-to-Noise Ratio (S/N)

A measure of the strength of the signal compared to the noise level in a communication channel.

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How to calculate S/N?

The signal-to-noise ratio (S/N) can be calculated by dividing the signal power (Ps) or voltage (Vs) by the noise power (Pn) or voltage (Vn).

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Shannon's Source Coding Theorem

This theorem states that by efficiently encoding or packaging data, we can almost reach the maximum channel capacity.

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Relationship between S/N and Channel Capacity

As the signal-to-noise ratio (S/N) increases, the channel capacity (C) also increases. This means a higher signal strength relative to noise allows for more information transmission.

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What happens when noise increases?

As noise levels increase, the channel capacity decreases. This means less information can be transmitted reliably.

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What is the formula for Channel Capacity (C)?

Channel Capacity (C) = log2 (1 + S/N/2), where S/N is the signal-to-noise ratio.

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How to improve channel capacity?

  • Increase signal strength (S) - Reduce noise (N) - Use efficient coding techniques (Shannon's Theorem)
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Output Entropy

The amount of uncertainty or information associated with the possible outputs of a system.

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Equiprobable States

Possible outcomes or states that have an equal chance of occurring.

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First Fundamental Coding Theorem

Another name for Shannon's Source Coding Theorem, emphasizing its significance in information theory.

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Noiseless Channel

A communication channel where no errors or distortions occur during transmission.

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Channel Capacity

The maximum rate at which information can be reliably transmitted over a channel.

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Encoding Data

The process of converting data into a different format for efficient transmission or storage.

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Sensory Data Repackaging

The process of efficiently organizing and encoding sensory information received by biological systems.

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Study Notes

Information Theory

  • Claude Shannon, a mathematician and computer scientist, developed the foundation for today's electronic communications networks.
  • His 1948 paper, "A Mathematical Theory of Communication," redefined information as a measurable quantity, significantly advancing scientific understanding.
  • The paper boasts nearly 150,000 citations.
  • This work was later published as a book in 1949.

Information Theory Details

  • Information theory defines fundamental limitations on the amount of information communicable between systems (man-made or natural).
  • The goal is to transmit messages effectively from a transmitter to a receiver, considering channel noise.

Communication Channel

  • A communication channel is used to transmit messages, but noise affects the message transmission.
  • Information is treated like a physical quantity, such as energy, allowing for mathematical analysis.

Bit by Bit

  • A bit is the smallest increment of data, allowing for two or more equiprobable alternatives.
  • Information is measured in bits.

Entropy

  • Entropy is a scientific concept measuring the disorder of a system.
  • In communications, entropy represents the expected number of bits of information in a message.
  • Exemplified using a coin flip, demonstrating how surprising outcomes affect entropy.
  • Entropy is average Shannon information, which is measured in bits.

Calculating Shannon Information

  • Formula to calculate Shannon information: -log₂p(x)
  • Output of the log₂ function is measured in bits.
  • This is the average surprise outcome in a variable.

Entropy of Fair and Unfair Coin

  • For an unbiased (fair) coin, the entropy is 1 bit.
  • For a biased coin, the average entropy is calculated using the weighted average of the probabilities.

Channel Capacity

  • Channel capacity is the maximum information transfer rate through a channel.
  • Capacity is affected by noise, with higher noise reducing capacity.
  • Channel capacity is expressed in bits per unit of time (e.g., bits per second).

Shannon's Source Coding Theorem

  • This theorem applies to noiseless channels.
  • It focuses on the efficient encoding of data before transmission for maximum information potential.
  • The rate of data transmission is limited by the entropy of the signal.

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Explore the fundamentals of information theory as established by Claude Shannon. This quiz covers key concepts from his groundbreaking 1948 paper, which defines information as a measurable quantity and discusses the impact of noise on communication channels. Test your understanding of the principles that underpin modern electronic communication.

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