Information Theory

ℹ️Information Theory Unit 1 – Introduction to Information Theory

Information Theory, developed by Claude Shannon in the 1940s, is a field that quantifies and analyzes information transmission, storage, and processing. It provides a mathematical framework for understanding communication systems, data compression, and cryptography, playing a crucial role in modern digital technologies. Key concepts include entropy, mutual information, and channel capacity. These ideas enable efficient and reliable communication systems, from smartphones to satellite communications. Information Theory's applications span diverse areas, including telecommunications, machine learning, and even neuroscience.

What's Information Theory?

  • Field of study that quantifies, stores, and communicates information
  • Developed by Claude Shannon in the 1940s at Bell Labs
  • Provides mathematical framework for analyzing information transmission, storage, and processing
  • Applies to diverse areas including telecommunications, data compression, cryptography, and machine learning
  • Fundamental concepts include entropy, mutual information, channel capacity, and coding theory
  • Enables efficient and reliable communication systems (internet, smartphones, satellite communications)
  • Plays crucial role in modern digital technologies and infrastructure

Key Concepts and Definitions

  • Information defined as reduction in uncertainty after receiving a message
  • Bit basic unit of information, represents binary choice between two equally likely alternatives
  • Source generates messages or sequences of symbols drawn from an alphabet
  • Encoder converts source messages into codewords for transmission over a channel
  • Channel medium over which encoded information is transmitted (wired, wireless, optical fiber)
  • Decoder reconstructs original message from received codewords
  • Noise any factor that corrupts or interferes with the transmitted signal
  • Distortion measure of dissimilarity between the original and reconstructed messages

Measuring Information

  • Amount of information in a message depends on its probability
  • Less likely messages convey more information than more predictable ones
  • Self-information I(x)=log2P(x)I(x) = -\log_2 P(x), where P(x)P(x) is the probability of event xx
    • Measured in bits if logarithm base 2 is used
  • Joint entropy H(X,Y)H(X,Y) measures uncertainty in a pair of random variables XX and YY
    • H(X,Y)=x,yP(x,y)log2P(x,y)H(X,Y) = -\sum_{x,y} P(x,y) \log_2 P(x,y)
  • Conditional entropy H(YX)H(Y|X) uncertainty in YY given knowledge of XX
    • H(YX)=x,yP(x,y)log2P(yx)H(Y|X) = -\sum_{x,y} P(x,y) \log_2 P(y|x)
  • Mutual information I(X;Y)I(X;Y) reduction in uncertainty of YY due to knowledge of XX
    • I(X;Y)=H(Y)H(YX)I(X;Y) = H(Y) - H(Y|X)

Entropy: The Heart of Information Theory

  • Entropy H(X)H(X) measures average amount of information or uncertainty in a random variable XX
    • H(X)=xP(x)log2P(x)H(X) = -\sum_{x} P(x) \log_2 P(x)
  • Quantifies the minimum number of bits needed to encode a message without loss
  • Maximum entropy achieved when all outcomes are equally likely
  • Joint entropy, conditional entropy, and mutual information are derived from basic entropy concept
  • Entropy used to characterize information sources, channels, and communication limits
  • Plays central role in data compression, coding theory, and cryptography
  • Understanding entropy is key to designing efficient information processing systems

Data Compression Basics

  • Data compression reduces size of data by removing redundancy
  • Lossless compression allows perfect reconstruction of original data (text, executable files)
    • Examples: Huffman coding, arithmetic coding, Lempel-Ziv algorithms
  • Lossy compression achieves higher compression ratios but with some information loss (images, audio, video)
    • Examples: JPEG, MP3, MPEG
  • Compression ratio measures reduction in data size, CR=uncompressed sizecompressed sizeCR = \frac{\text{uncompressed size}}{\text{compressed size}}
  • Theoretical limit for lossless compression is the entropy of the data source
  • Practical compression algorithms balance compression ratio, computational complexity, and data type

Channel Capacity and Noise

  • Channel capacity CC maximum rate at which information can be reliably transmitted over a noisy channel
    • C=maxP(X)I(X;Y)C = \max_{P(X)} I(X;Y), maximized over input distribution P(X)P(X)
  • Shannon's noisy channel coding theorem establishes fundamental limits of communication
  • Noise reduces channel capacity and introduces errors in transmitted messages
  • Signal-to-noise ratio (SNR) measures relative strength of signal compared to noise
    • Higher SNR enables higher channel capacities
  • Channel coding introduces redundancy to combat noise and ensure reliable communication
  • Examples of noisy channels: wireless links, telephone lines, storage media
  • Capacity-achieving codes (Turbo codes, LDPC codes) approach theoretical limits

Coding Theory Fundamentals

  • Coding theory studies design of error-correcting codes for reliable data transmission and storage
  • Error-correcting codes add redundancy to messages to detect and correct errors introduced by noise
  • Linear block codes encode kk information bits into nn-bit codewords, n>kn > k
    • Generator matrix GG used for encoding, parity-check matrix HH for decoding
  • Hamming distance measures number of positions in which two codewords differ
  • Minimum distance dmind_{min} of a code determines its error-correcting capability
    • Code can correct up to dmin12\lfloor \frac{d_{min}-1}{2} \rfloor errors
  • Convolutional codes generate parity bits based on sliding window of input bits
    • Decoded using Viterbi algorithm or MAP (maximum a posteriori) decoding
  • Codes designed to approach channel capacity while minimizing complexity and delay

Real-World Applications

  • Data compression: ZIP files, PNG images, MP3 audio, MPEG video
  • Error correction: CDs, DVDs, QR codes, satellite communications
  • Cryptography: random number generation, key distribution, secure communication protocols
  • Machine learning: feature selection, dimensionality reduction, clustering
  • Genetics: DNA sequence analysis, genome compression, mutation modeling
  • Neuroscience: neural coding, information processing in the brain
  • Financial markets: portfolio optimization, risk assessment, high-frequency trading
  • Physics: entropy in thermodynamics, quantum information theory


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© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.