Glossary

1.1 Fundamentals of Information Theory

4 min read•august 7, 2024

Information theory forms the backbone of coding theory, providing tools to measure and manipulate information. It introduces key concepts like , which quantifies uncertainty in messages, and , which measures shared information between variables.

These fundamentals lay the groundwork for understanding , which compresses data, and , which protects against transmission errors. Together, they enable efficient and reliable communication systems, crucial for modern digital technologies.

Information Measures

Entropy and Information Content

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  • Entropy quantifies the amount of uncertainty or randomness in a message or system
    • Measured in bits, with higher entropy indicating more uncertainty and less predictability
    • For a discrete random variable XX with probability distribution p(x)p(x), entropy is defined as: H(X)=−∑xp(x)log⁡2p(x)H(X) = -\sum_{x} p(x) \log_2 p(x)
  • Information content measures the amount of information gained when a particular event or symbol is observed
    • Also measured in bits, with less likely events having higher information content
    • For an event xx with probability p(x)p(x), information content is defined as: I(x)=−log⁡2p(x)I(x) = -\log_2 p(x)
  • The relationship between entropy and information content
    • Entropy is the expected value (average) of the information content across all possible events or in a system
    • Higher entropy systems have a more uniform distribution of information content across events, while lower entropy systems have information content concentrated in fewer events

Mutual Information and Bits

  • Mutual information measures the amount of information shared between two random variables
    • Quantifies the reduction in uncertainty about one variable given knowledge of the other
    • For random variables XX and YY, mutual information is defined as: I(X;Y)=∑x,yp(x,y)log⁡2p(x,y)p(x)p(y)I(X;Y) = \sum_{x,y} p(x,y) \log_2 \frac{p(x,y)}{p(x)p(y)}
    • High mutual information indicates a strong relationship or dependency between variables, while low mutual information suggests variables are more independent
  • Bit, a fundamental unit of information
    • Binary digit, representing one of two possible states (0 or 1)
    • Used to measure entropy, information content, and mutual information
    • Bits can be combined to represent more complex information (8 bits in a byte, representing 256 possible values)

Coding Fundamentals

Source and Channel Coding

  • Source coding involves representing information from a source efficiently
    • Aims to reduce in the source message while preserving its essential content
    • Examples include techniques like or
  • Channel coding adds redundancy to the message to protect against errors during transmission
    • Introduces controlled redundancy to enable and correction
    • Examples include error-correcting codes like Hamming codes or convolutional codes
  • The interplay between source and channel coding
    • Source coding removes redundancy to minimize the amount of data to be transmitted
    • Channel coding adds redundancy to ensure reliable transmission over a noisy channel
    • The goal is to find an optimal balance between compression and error protection based on the characteristics of the source and channel

Data Compression, Redundancy, and Symbols

  • Data compression reduces the size of data by removing redundancy
    • removes redundancy without losing information (Huffman coding, Lempel-Ziv algorithms)
    • removes both redundancy and some less important information (JPEG for images, MP3 for audio)
  • Redundancy refers to the presence of duplicate or unnecessary information in a message
    • Can be inherent in the source (e.g., repeated patterns in an image) or introduced intentionally for error protection
    • Removing redundancy is key to efficient data compression
  • Symbols are the basic units of information in a message
    • Can be bits, characters, or more complex entities depending on the context
    • The choice of symbol representation affects the efficiency of source coding and the effectiveness of channel coding
    • Examples include binary symbols (0 and 1), ASCII characters (8-bit symbols), or QAM symbols in digital modulation

Communication Challenges

Noise and Its Impact on Communication

  • Noise refers to any unwanted disturbances or interference that can corrupt a signal during transmission
    • Can be external (e.g., electromagnetic interference) or internal (e.g., thermal noise in electronic components)
    • Introduces errors in the received message, leading to a mismatch between the transmitted and received information
  • Types of noise
    • (AWGN): common model for thermal noise, adds random values to the signal
    • : short, high-amplitude disturbances (e.g., switching noise, lightning)
    • : variations in signal strength due to changes in the transmission medium (e.g., multipath fading in wireless channels)
  • Techniques to mitigate the impact of noise
    • Channel coding: adding redundancy to the message to enable error detection and correction
    • Modulation: choosing a modulation scheme that is more robust to noise (e.g., PSK over QAM in high-noise environments)
    • Equalization: compensating for the effects of the channel on the signal (e.g., adaptive equalization in wireless communications)
    • Diversity: using multiple copies of the signal to increase the chances of correct reception (e.g., time, frequency, or spatial diversity)

Key Terms to Review (21)

Additive White Gaussian Noise: Additive white Gaussian noise (AWGN) is a basic noise model used to mimic the effect of random processes that affect communication channels. It is characterized by its power spectral density being constant across all frequencies, which means it adds noise uniformly across the spectrum. AWGN is crucial in information theory because it provides a simplified model for analyzing the performance of communication systems under noisy conditions.
Bit rate: Bit rate refers to the number of bits that are processed or transmitted in a given amount of time, usually expressed in bits per second (bps). It is a crucial factor in determining the quality and performance of digital data transmission, influencing how much information can be sent over a communication channel. A higher bit rate generally means better quality, especially in contexts like audio and video streaming, but it also requires more bandwidth.
Channel Capacity: Channel capacity is the maximum rate at which information can be transmitted over a communication channel without error. It reflects the inherent limits of a channel’s ability to transmit data, influenced by factors like noise, bandwidth, and signal strength. Understanding channel capacity is crucial for designing efficient communication systems that minimize errors and maximize data throughput.
Channel Coding: Channel coding is a technique used to protect information during transmission over noisy channels by adding redundancy, allowing the original data to be recovered even in the presence of errors. This process involves encoding data before transmission and decoding it upon reception, making it essential for reliable communication in various systems. The effectiveness of channel coding can be enhanced through methods such as interleaving and iterative decoding, which work together to improve error correction capabilities.
Convolutional code: A convolutional code is a type of error-correcting code used in digital communication that encodes data by passing it through a series of shift registers and applying a set of mathematical operations. This coding technique helps ensure reliable data transmission by adding redundancy to the data stream, which can be used to detect and correct errors that occur during transmission. Convolutional codes are particularly important in improving the performance of communication systems, especially when combined with algorithms for decoding like the Viterbi Algorithm.
Data Compression: Data compression is the process of reducing the size of a data file to save storage space or transmission time. This technique is essential in optimizing resource usage in digital communications and storage, allowing for faster data transfer and reduced costs. Effective data compression methods can significantly improve efficiency in various applications, linking closely to concepts such as coding techniques, information theory, and channel capacity.
Entropy: Entropy is a measure of uncertainty or randomness in a set of data, reflecting the amount of information that is missing when predicting an outcome. In the context of coding and information theory, it quantifies the expected value of the information produced by a stochastic source of data. The higher the entropy, the more unpredictability there is, which has critical implications for encoding information efficiently and understanding how well a communication channel can transmit messages without errors.
Error Correction: Error correction is the process of detecting and correcting errors that occur during data transmission or storage. This method ensures the integrity and reliability of data by enabling systems to identify mistakes and recover the original information through various techniques.
Error detection: Error detection is the process of identifying errors in transmitted or stored data to ensure the integrity and accuracy of information. It plays a crucial role in various systems by allowing the detection of discrepancies between the sent and received data, which can be essential for maintaining reliable communication and storage.
Fading: Fading refers to the variation in signal strength over time and space in wireless communication channels due to environmental factors, leading to potential data loss or degradation of the signal quality. This phenomenon is crucial in understanding how information is transmitted and received, impacting the reliability of communication systems and the design of error-correcting codes.
Hamming Code: Hamming Code is a method of error detection and correction that can identify and correct single-bit errors in transmitted data. It achieves this by adding redundancy through parity bits, allowing the receiver to determine which bit may have been corrupted during transmission, making it essential in various coding techniques used to ensure reliable data communication and storage.
Huffman Coding: Huffman coding is a lossless data compression algorithm that assigns variable-length codes to input characters based on their frequencies, aiming to minimize the overall length of the encoded data. By utilizing shorter codes for more frequent characters and longer codes for less frequent ones, Huffman coding efficiently reduces the amount of storage space required and optimizes the transmission of data.
Impulse noise: Impulse noise refers to a type of signal distortion characterized by sudden and short-duration spikes in amplitude, which can disrupt the transmission of data and lead to errors in communication systems. This noise is often caused by environmental factors such as electrical interference or lightning strikes, making it important to understand in the context of data transmission and error correction techniques.
Lossless compression: Lossless compression is a data encoding technique that reduces file size without any loss of information, ensuring that the original data can be perfectly reconstructed from the compressed data. This method is crucial in various applications where preserving the exact quality of the data is essential, such as in text, images, and audio files. Lossless compression techniques often rely on algorithms that eliminate redundancy while maintaining the integrity of the original content.
Lossy compression: Lossy compression is a data encoding method that reduces file size by permanently eliminating some information, resulting in a decrease in quality that may be acceptable for certain applications. This technique is commonly used in multimedia files, such as images, audio, and video, where a perfect reproduction is not essential. It allows for significant reductions in data size, making it easier to store and transmit files efficiently.
Mutual information: Mutual information is a measure of the amount of information that one random variable contains about another random variable. It quantifies the reduction in uncertainty about one variable given knowledge of the other, highlighting the dependency between them. This concept is crucial in understanding data compression, coding techniques, and evaluating the efficiency of communication channels.
Redundancy: Redundancy in coding theory refers to the intentional inclusion of extra bits in a message to ensure that errors can be detected and corrected. This additional information provides a safety net that helps maintain the integrity of data during transmission or storage, enhancing the reliability of communication systems.
Run-length encoding: Run-length encoding is a simple form of data compression that replaces sequences of the same data value occurring in consecutive runs with a single value and a count. This technique effectively reduces the size of data by eliminating redundancy, which is particularly useful for data that contains many consecutive repeated characters or values. By converting long runs into shorter representations, run-length encoding enhances storage efficiency and speeds up transmission.
Source coding: Source coding is the process of converting information into a format suitable for efficient transmission or storage, minimizing redundancy while preserving the integrity of the original data. This method is crucial for optimizing data compression and is foundational in both digital communication systems and information theory, enabling more effective data representation and transmission. Understanding source coding helps to grasp how information can be efficiently encoded to utilize bandwidth effectively, which is especially relevant when dealing with iterative decoding and error correction.
Symbols: In coding theory, symbols refer to the basic units of information that are used in the encoding and decoding processes. These symbols can represent data in various forms, such as binary digits, characters, or mathematical representations, and are crucial for conveying messages accurately and efficiently. Understanding symbols is key to grasping how data is structured, encoded, and transmitted within different coding systems.
Throughput: Throughput refers to the rate at which data is successfully transmitted over a communication channel in a given amount of time. This key performance indicator reflects the efficiency of a system, affecting both error detection and correction capabilities and overall system performance. Higher throughput can lead to better error correction mechanisms and effective digital communication, ultimately enhancing the performance of coding strategies and information transmission.
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