Saturday with Math (Jul 13th)
We live in the age of bits, but few know about the person who coined the term "bit" and pioneered the digital world. This week, we bring you some interesting facts and results in honor of the 75/76th anniversary of Claude Shannon's groundbreaking paper, “A Mathematical Theory of Communication,” [3] which revolutionized telecommunications and initiated modern communications. Despite his monumental contributions, as highlighted in the recent documentary The Bit Player [1], he remains relatively unknown. Shannon never received a Nobel Prize and wasn't a celebrated figure like Albert Einstein or Richard Feynman, even after his death in 2001. [1], [2]
Transforming electricity in digital communication
Claude Shannon's theory centers on a simple yet profound model of communication: a transmitter encodes information into a signal, which is then corrupted by noise and decoded by the receiver. His model introduced two critical insights: separating information and noise sources from the communication system and modeling them probabilistically. Shannon envisioned the information source generating multiple possible messages, each with a specific probability, while noise added randomness for the receiver to decode.
Prior to Shannon, communication was viewed as a deterministic problem of signal reconstruction. Shannon's key insight was that communication revolves around uncertainty. This shifted the focus from physical signal reconstruction to abstract modeling of uncertainty using probability, a concept that astonished contemporary communication engineers.
"A Mathematical Theory of Communication," a seminal article by Claude E. Shannon published in 1948 in the Bell System Technical Journal, and after in the 1949 book of the same name, laid the foundation for the field of information theory. This groundbreaking work earned accolades such as being dubbed the "Magna Carta of the Information Age" by Scientific American. With tens of thousands of citations, the article introduced key concepts like channel capacity and the noisy channel coding theorem. Shannon detailed the basic elements of communication, including an information source, a transmitter, a channel, a receiver, and a destination. His work also developed the ideas of information entropy and redundancy and introduced the term "bit" as a unit of information.
Shannon's landmark paper established the fundamental limit of communication through three main contributions, centered on the concept of an "information bit." First, he formulated the minimum number of bits per second needed to represent information, called the entropy rate (H), which quantifies the uncertainty of the message source. Second, he derived the maximum number of bits per second that can be reliably communicated despite noise, known as the system's capacity (C). Finally, he demonstrated that reliable communication is possible if and only if H < C, comparing information flow to water through a pipe.
Entropy in information theory, introduced by Claude Shannon, quantifies the unpredictability of outcomes from a random variable. It gauges the average information conveyed by these outcomes and plays a critical role in optimizing communication systems and understanding the limits of data processing. Entropy's influence spans across disciplines such as communication, data science, and cryptography, where managing and interpreting data uncertainty is essential.
Shannon's theorems led to counterintuitive conclusions, such as the inefficiency of repeating messages in noisy environments. Instead, he showed that using sophisticated coding techniques allows for fast and reliable communication up to the system's capacity. This groundbreaking work laid the foundation for modern information theory, earning Shannon the title "father of information theory."
Information Theory
Information theory, developed by Claude Shannon and further advanced by subsequent researchers, is a fundamental discipline encompassing the quantification, storage, and communication of information. At its core lies the concept of entropy, which measures the uncertainty or unpredictability of a random variable. Entropy quantifies the amount of information required to describe the variable: for instance, a fair coin flip has lower entropy (1 bit) compared to a fair six-sided die roll (2.58 bits), reflecting varying levels of uncertainty. [6], [7] ?
Mutual information, another key concept, quantifies the amount of information gained about one random variable through another variable. This metric is crucial in understanding how much knowledge of one variable reduces uncertainty about another. Channel capacity is also critical; it defines the maximum rate at which information can be transmitted reliably over a communication channel, considering factors such as channel noise and bandwidth. [6], [7]
Coding theory, a vital component of information theory, focuses on efficient encoding methods for transmitting and storing information. Techniques such as Huffman coding and arithmetic coding play pivotal roles in minimizing errors and maximizing data compression efficiency. Moreover, the Kullback–Leibler (KL) divergence measures the difference between two probability distributions, quantifying how one distribution diverges from another—typically a "true" distribution. This concept is pivotal in Bayesian inference, statistical modeling, and machine learning, helping to approximate one distribution with another and thereby enhancing model accuracy and efficiency. [6], [7]
Applications of information theory abound in modern technology. It underpins the design of efficient communication systems, ensuring reliable data transmission over noisy channels. In data compression, methods derived from information theory—like Huffman coding and arithmetic coding—enable efficient compression while retaining essential information. In cryptography, information theory provides the theoretical basis for secure communication and encryption methods, ensuring data confidentiality and integrity in digital communications. [6], [7].
Information theory underpins data compression technology, enhancing efficiency in data storage, transmission, and processing by minimizing redundancy and maximizing the utilization of data storage and communication bandwidth. Some examples of entropy coders besides Huffman coding are listed below:
Huffman Coding: Huffman coding is widely used in data compression applications, such as in file formats like JPEG (for image compression) and MP3 (for audio compression). It efficiently reduces the size of text, image, and audio files by assigning shorter codes to more frequently occurring symbols or pixel values, resulting in significant compression ratios while maintaining lossless or lossy quality, depending on the application.
Arithmetic Coding: Arithmetic coding is utilized in high-performance compression algorithms, including in video coding standards like H.264 and H.265 (HEVC), where it achieves superior compression efficiency compared to simpler methods like Huffman coding. It's also used in data transmission protocols and archival systems where maximizing compression ratios is crucial.
Lempel-Ziv (LZ) Compression: LZ compression algorithms, such as LZ77 and LZ78, are the basis for popular compression formats like ZIP (file compression), gzip (Unix compression tool), and DEFLATE (used in PNG image format). These formats leverage LZ techniques to efficiently reduce redundancy in data, making them suitable for file compression, archival storage, and data transmission over networks.
Burrows-Wheeler Transform (BWT): The Burrows-Wheeler Transform is a key component in the BZIP2 compression algorithm, used for compressing files and data streams efficiently. It rearranges data to facilitate better compression by subsequent coding methods like move-to-front coding and Huffman coding, making it a core part of many modern compression tools and formats.
Run-Length Encoding (RLE): RLE finds applications in scenarios where consecutive runs of identical data occur frequently, such as in bitmap graphics (BMP) and simple text files. It's also used in fax transmission protocols and in data compression techniques for streaming data or where rapid compression and decompression are needed, albeit typically for scenarios with predictable patterns or data types.
Shannon's theorems, fundamental results in information theory, include the source coding theorem (Huffman coding) and the noisy channel coding theorem (capacity of a communication channel). These theorems have been instrumental in the development of digital communication technologies, facilitating advancements in data storage, processing systems, and artificial intelligence. Quantum information theory extends these principles to quantum mechanics, exploring phenomena like quantum entanglement and quantum communication.
In practical terms, information theory has revolutionized communication technologies, shaping the digital age by enabling rapid advancements in data transmission, storage, and processing. It continues to drive innovation in fields such as machine learning, where efficient information processing and transmission are critical. A solid understanding of information theory provides a foundational framework for addressing challenges in data science, network security, and beyond, ensuring that modern technological landscapes evolve with robust and efficient information handling capabilities.
Applications
Information theory, originating from the works of Harry Nyquist, Ralph Hartley, and Claude Shannon, provides a fundamental framework for various applications across diverse fields:
Communication Systems: They rely on fundamental concepts from information theory, such as channel capacity and the noisy-channel coding theorem, which are essential for modern telecommunications, including mobile networks like 4G and 5G. These theories play a critical role in evaluating spectrum requirements and enhancing spectral efficiency. Channel capacity, defined by Shannon's principles, determines the maximum data transmission rate with minimal errors over a communication channel. This application of information theory not only supports current telecommunications but also informs strategic planning for future network expansions, ensuring efficient and robust communication infrastructures capable of meeting evolving technological demands.
Data Compression: Information theory guides the compression of data, evident in technologies like ZIP files, where it helps in reducing the amount of data required for storage and transmission. Please see above.
Cryptography: The field leverages concepts from information theory for secure communication, ensuring that data can be transmitted without the risk of interception. Shannon's work laid the foundation for modern cryptographic methods.
Neurobiology and Cognitive Science: Information theory is applied to understand neural processing and cognitive functions, analyzing how the brain encodes and processes information.
Quantum Computing: The theory provides a framework for understanding and developing quantum communication and computing technologies.
Statistical Inference: In statistics, information theory helps in making inferences about populations based on sample data, enhancing decision-making processes.
Bioinformatics: It aids in understanding genetic and molecular complexities by analyzing the information processes underlying biological systems.
Economics and Financial Markets: The theory assists in analyzing market dynamics and information flow within financial systems, helping in risk assessment and decision-making. [8]
Art and Music: Information theory has found applications in analyzing patterns in art and music, helping in the creation of algorithmic art and music composition.
Environmental Science: Researchers apply information theoretic approaches to model ecological dynamics and predict environmental changes.
These applications demonstrate the broad impact of information theory, not only in technology and communications but also in natural sciences, arts, and humanities, reflecting its universal relevance in decoding and understanding information in various forms.
Equation in Focus
The equation in focus is a slight modification of the Shannon-Hartley theorem, incorporating the influence of MIMO through the parameter M, and the effect of frequency reuse due to the cellsites intercell interference (hetnet environment) and users through the variable n.
The Shannon-Hartley formulation is classic for determining the capacity limit of a channel, such as the air interface in mobile communications. It depends on the channel's bandwidth and spectral efficiency, which reflects the encoding capability given a certain interference level, or the relationship between the power of the modulated and encoded signal and interference plus noise, i.e., Signal-to-Interference-plus-Noise Ratio (SINR).
Actually, the Shannon-Hartley expression had significant contributions that preceded it. [5] Before Shannon's work, in the late 1920s, Harry Nyquist and Ralph Hartley made significant contributions to the theory of information transmission, initially focusing on telegraph communication. Nyquist demonstrated that the maximum pulse rate, known as the Nyquist rate, through a channel is limited to twice its bandwidth, a fundamental concept for digital communication. Hartley introduced Hartley's law, which quantifies the rate of information transmission by relating pulse rate and amplitude precision to data signaling capacity. These foundational insights were later synthesized by Claude Shannon in the 1940s into a comprehensive theory of information and channel capacity. Shannon's groundbreaking noisy-channel coding theorem established the limits of error correction and defined channel capacity, transforming our understanding of how information can be reliably transmitted despite noise and bandwidth limitations in communication channels. Shannon's work integrated and expanded upon Nyquist's and Hartley's foundational ideas, providing rigorous mathematical frameworks that underlie modern telecommunications and data transmission systems.
Nyquist's Approach: Focuses on the maximum number of independent pulses per second that a channel can handle, establishing the Nyquist rate 2B pulses per second for a channel of bandwidth B.
Hartley's Approach: Quantifies the information transmission capacity based on the amplitude levels and precision of the receiver, leading to Hartley's law R ≤ 2B log2(M).
Shannon's Approach: Introduces the concept of channel capacity considering noise, demonstrating that the maximum data rate for reliable communication is determined by the channel capacity C, incorporating both bandwidth and noise limitations. Shannon’s theorem bridges Nyquist’s and Hartley’s ideas by showing the theoretical limit of data transmission with error correction.
Mobile Network Capacity Discussion
The design of a mobile network considers two dimensions: coverage and capacity. Coverage ensures service continuity, where the network grid is dimensioned based on SLA at the cell border and associated Link Budget [15]. Capacity, on the other hand, relates to the demand for data traffic (Mbps-Gbps) in simultaneous sessions within the same sector/cell. When demand exceeds the capacity offered by the sector/cell, there are three possibilities for infrastructure adjustment: more frequency bandwidth, increased spectral efficiency (e.g., MIMO or higher-order modulation), or cellsite split, where a new cellsite is built. Each of these options has its pros and cons, as outlined above.
To analyze the adequacy of a mobile network's capacity, Shannon's equation can help, but coverage area must be considered. Therefore, instead of using aggregated traffic in Mbps-Gbps, its density per unit area, Gbps/km2, provides a better measure for capacity evaluation, as outlined below:
The demand density D/A, where D is the aggregated traffic demand in Gbps for a street, neighborhood, or city, and A is the corresponding area in km2, can be compared with the capacity density of a cellsite. Considering the cellsite's traffic density given by Ct/Ac, where Ct is the sector/cell capacity in Gbps (from Shannon: spectral efficiency × bandwidth) and Ac is its corresponding area in km2 of cell range, if D/A < Ct/Ac, the network is adequate to handle the offered traffic. However, if D/A > Ct/Ac, site expansion is necessary. Thus, Ct/Ac represents the network's capacity threshold for traffic density.
An important aspect of this expression is that one can determine, through a simple calculation, the strategy for utilizing a given technology, spectral efficiency, bandwidth, and frequency based on the coverage radius and area Ac. Therefore, by calculating Ct/Ac as a threshold, it becomes possible to promptly assess whether the demand density D/A can be met.
Finally, considering the entire coverage area of the network, the network's total capacity is: Ct/Ac * Ac * #CellSites. Since Ct = BW * Spectral Efficiency, the network capacity is given by: BW * Spectral Efficiency * #CellSites.
?About Shannon [9]
Claude Elwood Shannon (1916–2001) was an American mathematician, electrical engineer, and cryptographer, renowned as the "father of information theory" and pivotal to the Information Age. He graduated from the University of Michigan with dual degrees in electrical engineering and mathematics in 1936, followed by a master's degree from MIT where his thesis laid the groundwork for digital circuits using Boolean algebra. During World War II, Shannon's work in cryptanalysis advanced secure telecommunications and marked the transition to modern cryptography. His seminal 1948 paper on information theory revolutionized communication, influencing the Internet and digital technology. Shannon's contributions to artificial intelligence, including early work on computer chess, further cement his legacy as a foundational figure in modern science and technology.
About Hartley [10]
Ralph Vinton Lyon Hartley (1888–1970) was an influential American electronics researcher known for his contributions to information theory. Born in Nevada, he studied at the University of Utah and later as a Rhodes Scholar at Oxford University. Hartley's career spanned significant innovations at Bell Labs, where he developed the Hartley oscillator and fundamental concepts in radio technology. His 1928 paper on transmission of information laid crucial groundwork for Claude Shannon's later work on information theory. Despite health challenges, Hartley led pioneering research on nonlinear circuits and parametric amplifiers, impacting fields from telecommunications to radar technology. He retired in 1950, leaving a lasting legacy in the advancement of electronic communications.
About Nyquist [11]
Harry Nyquist (1889 –1976) was a Swedish-American physicist and electronic engineer pivotal in communication theory. He emigrated from Sweden to the U.S. in 1907, earning his Ph.D. in physics from Yale University in 1917. Nyquist worked at AT&T and later at Bell Telephone Laboratories until his retirement in 1954. His research laid foundational work for digital communication and control systems, influencing the development of information theory. Notable for his work on thermal noise and feedback amplifiers, Nyquist received several prestigious awards, including the IRE Medal of Honor and the National Academy of Engineering's Founder's Medal. He spent his final years in Texas, where he passed away.
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In the next edition, we will explore Stokes', Gauss', Green's theorems and others in their most significant application: Maxwell's Equations, and, of course, their important implications in telecommunications.
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References
[4] https://www.quantamagazine.org/how-claude-shannons-information-theory-invented-the-future-20201222/
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Keywords: #saturdaywithmath, #informationtheory, #shannon; #systemcapacity; #signalcoding; #entropycoding; #4G, #5G”, #6G
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