Information Theory

ℹ️Information Theory Unit 9 – Noisy Channel Coding Theorem

The Noisy Channel Coding Theorem is a cornerstone of information theory, establishing the maximum rate at which information can be reliably transmitted over a noisy channel. It introduces key concepts like channel capacity, mutual information, and entropy, which are fundamental to understanding communication systems. This theorem has profound implications for designing reliable communication systems, from mobile networks to deep space communications. It also explores encoding and decoding strategies, error probability, and practical applications, providing a framework for achieving efficient and reliable data transmission in noisy environments.

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Key Concepts and Definitions

  • Noisy channel coding theorem establishes the maximum rate at which information can be transmitted over a noisy channel with a specified probability of error
  • Channel capacity (CC) represents the maximum rate at which information can be reliably transmitted over a noisy channel
  • Mutual information (I(X;Y)I(X;Y)) measures the amount of information that can be obtained about one random variable by observing another
  • Entropy (H(X)H(X)) quantifies the average amount of information contained in a random variable
    • Conditional entropy (H(XY)H(X|Y)) measures the remaining uncertainty in XX given knowledge of YY
  • Encoding involves converting the source message into a suitable form for transmission over the noisy channel
  • Decoding aims to reconstruct the original message from the received signal, taking into account the channel's noise characteristics

Shannon's Noisy Channel Model

  • Shannon's noisy channel model consists of an information source, encoder, noisy channel, decoder, and destination
  • The information source generates the message to be transmitted over the noisy channel
  • The encoder converts the message into a form suitable for transmission, adding redundancy for error correction
  • The noisy channel introduces errors or distortions to the transmitted signal (additive white Gaussian noise)
  • The decoder receives the corrupted signal and attempts to reconstruct the original message by exploiting the added redundancy
    • Maximum likelihood decoding and maximum a posteriori decoding are common decoding strategies
  • The destination receives the decoded message, which should closely resemble the original message if the communication system is properly designed

Mutual Information and Channel Capacity

  • Mutual information quantifies the reduction in uncertainty about one random variable by observing another
  • For a noisy channel, mutual information measures the amount of information that can be reliably transmitted
  • Channel capacity is the maximum mutual information over all possible input distributions
    • C=maxp(x)I(X;Y)C = \max_{p(x)} I(X;Y), where p(x)p(x) is the input probability distribution
  • The noisy channel coding theorem states that for any transmission rate R<CR < C, there exists an encoding and decoding scheme that achieves arbitrarily small error probability
  • Achieving channel capacity requires optimal encoding and decoding strategies, which may be computationally complex in practice

Encoding and Decoding Strategies

  • Encoding adds redundancy to the source message to enable error correction at the receiver
  • Linear block codes (Hamming codes) and convolutional codes are widely used encoding schemes
    • Linear block codes add parity bits to fixed-size message blocks
    • Convolutional codes generate encoded bits based on the current and previous input bits
  • Decoding aims to reconstruct the original message from the received signal
  • Maximum likelihood decoding chooses the codeword that is most likely to have been transmitted given the received signal
    • Viterbi algorithm is an efficient implementation of maximum likelihood decoding for convolutional codes
  • Maximum a posteriori decoding incorporates prior knowledge about the message probabilities to improve decoding performance
  • Turbo codes and low-density parity-check (LDPC) codes are powerful encoding and decoding schemes that approach the channel capacity

Error Probability and Reliability

  • Error probability quantifies the likelihood of incorrectly decoding a received message
  • The noisy channel coding theorem guarantees the existence of encoding and decoding schemes that achieve arbitrarily small error probability for rates below the channel capacity
  • Bit error rate (BER) and block error rate (BLER) are common metrics for assessing the reliability of a communication system
    • BER measures the average number of bit errors per transmitted bit
    • BLER measures the fraction of decoded blocks that contain at least one error
  • Forward error correction (FEC) techniques, such as encoding and decoding, help improve reliability by detecting and correcting errors at the receiver
  • Automatic repeat request (ARQ) protocols retransmit erroneously received data to ensure reliable communication

Practical Applications and Examples

  • Noisy channel coding theorem has significant implications for the design of reliable communication systems
  • Error-correcting codes are widely used in digital communication systems (mobile networks, satellite communications)
    • Convolutional codes and turbo codes are employed in cellular networks (3G, 4G, 5G)
    • Reed-Solomon codes are used in compact discs (CDs) and digital video broadcasting (DVB)
  • Deep space communications rely on powerful error-correcting codes to overcome the challenges of long distances and low signal-to-noise ratios
  • Data storage systems (hard drives, solid-state drives) employ error-correcting codes to ensure data integrity
  • Quantum error correction aims to protect quantum information from errors in quantum computing and communication systems

Limitations and Considerations

  • Achieving channel capacity in practice may require complex encoding and decoding schemes, which can be computationally expensive
  • The noisy channel coding theorem assumes memoryless channels, where the noise affects each transmitted symbol independently
    • Channels with memory (burst errors) may require different coding techniques
  • The theorem assumes a fixed and known channel model, but in practice, the channel characteristics may vary over time (fading channels)
    • Adaptive coding and modulation techniques can help mitigate the effects of time-varying channels
  • Feedback from the receiver to the transmitter can improve the performance of error-correcting codes (ARQ protocols)
  • The theorem does not account for the delay introduced by encoding and decoding, which may be a concern in real-time applications
  • Shannon's source coding theorem establishes the minimum average number of bits required to represent a source message without loss of information
  • Joint source-channel coding theorem considers the optimal encoding and decoding strategies when the source and channel coding are combined
  • Network information theory extends the noisy channel coding theorem to multi-user scenarios (multiple access, broadcast, relay channels)
    • Capacity regions characterize the achievable rates for multiple users sharing a common channel
  • Quantum information theory adapts the noisy channel coding theorem to quantum channels and quantum error correction
  • Secure communication over noisy channels can be achieved using techniques from information-theoretic security (one-time pad, wiretap channels)


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.