All Study Guides Information Theory Unit 1
ℹ️ Information Theory Unit 1 – Introduction to Information TheoryInformation 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.
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
Amount of information in a message depends on its probability
Less likely messages convey more information than more predictable ones
Self-information I ( x ) = − log 2 P ( x ) I(x) = -\log_2 P(x) I ( x ) = − log 2 P ( x ) , where P ( x ) P(x) P ( x ) is the probability of event x x x
Measured in bits if logarithm base 2 is used
Joint entropy H ( X , Y ) H(X,Y) H ( X , Y ) measures uncertainty in a pair of random variables X X X and Y Y Y
H ( X , Y ) = − ∑ x , y P ( x , y ) log 2 P ( x , y ) H(X,Y) = -\sum_{x,y} P(x,y) \log_2 P(x,y) H ( X , Y ) = − ∑ x , y P ( x , y ) log 2 P ( x , y )
Conditional entropy H ( Y ∣ X ) H(Y|X) H ( Y ∣ X ) uncertainty in Y Y Y given knowledge of X X X
H ( Y ∣ X ) = − ∑ x , y P ( x , y ) log 2 P ( y ∣ x ) H(Y|X) = -\sum_{x,y} P(x,y) \log_2 P(y|x) H ( Y ∣ X ) = − ∑ x , y P ( x , y ) log 2 P ( y ∣ x )
Mutual information I ( X ; Y ) I(X;Y) I ( X ; Y ) reduction in uncertainty of Y Y Y due to knowledge of X X X
I ( X ; Y ) = H ( Y ) − H ( Y ∣ X ) I(X;Y) = H(Y) - H(Y|X) I ( X ; Y ) = H ( Y ) − H ( Y ∣ X )
Entropy H ( X ) H(X) H ( X ) measures average amount of information or uncertainty in a random variable X X X
H ( X ) = − ∑ x P ( x ) log 2 P ( x ) H(X) = -\sum_{x} P(x) \log_2 P(x) H ( X ) = − ∑ 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, C R = uncompressed size compressed size CR = \frac{\text{uncompressed size}}{\text{compressed size}} CR = compressed size uncompressed 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 C C C maximum rate at which information can be reliably transmitted over a noisy channel
C = max P ( X ) I ( X ; Y ) C = \max_{P(X)} I(X;Y) C = max P ( X ) I ( X ; Y ) , maximized over input distribution P ( X ) 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 k k k information bits into n n n -bit codewords, n > k n > k n > k
Generator matrix G G G used for encoding, parity-check matrix H H H for decoding
Hamming distance measures number of positions in which two codewords differ
Minimum distance d m i n d_{min} d min of a code determines its error-correcting capability
Code can correct up to ⌊ d m i n − 1 2 ⌋ \lfloor \frac{d_{min}-1}{2} \rfloor ⌊ 2 d min − 1 ⌋ 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