A Mathematical Theory of Communication.

Shannon, C. E. (1948). A Mathematical Theory of Communication. Bell System Technical Journal, 27(3), 379-423.

Type

Journal article

Intellectual base

Primary research

Year

1948

DOI

10.1002/j.1538-7305.1948.tb01338.x

Link

https://doi.org/10.1002/j.1538-7305.1948.tb01338.x

Cited from

computer_science, engineering_design, information_theory, mathematics, philosophy, physical_sciences, physics, statistics_experimental_design, synthesized, systems_cybernetics

Cited by

28 citations across 27 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Aggregation
  • Aggregation names the moment when multiple distinct entities are deliberately collapsed into a unified measure or category—a designed moment of information loss whose generality is captured by Shannon's (1948) information-theoretic framing of the channel between source and summary.

    This sourceInformation-theoretic framing of the source-to-summary channel and compression; supports naming the deliberate information loss that defines aggregation.

  • The first structural tension—the irreversibility of aggregation as an operation that destroys information—follows directly from Shannon's (1948) data-processing inequality: no post-hoc transformation of the summary y can recover information about the inputs x₁, …, xₙ that was discarded in forming y.

    This sourceEstablishes the data-processing inequality; supports the irreversibility tension — no transformation of a summary can recover the inputs' discarded information.

• Amplification
  • Information-theoretic limits
• Channel
  • There is (1) a two-endpoint coupling (source to receiver) with definite directionality; (2) a capacity — finite, usually quantifiable — that bounds how much can cross per unit time; (3) an alphabet or codebook — the channel's admissible input set, often more constrained than what the source would prefer; (4) a noise or distortion profile — the channel transforms the input probabilistically, making the receiver's reconstruction non-trivial; and (5) a medium — the substrate through which transmission occurs, whose physics determines the other four.

    This sourceFounding paper of information theory; defines the channel, channel capacity, the noisy-channel coding theorem, and the binary symmetric channel.

• Channel Capacity
  • Fourth, a capacity: a maximum reliable rate, increasing with bandwidth and with the logarithm of the signal-to-noise ratio, that no encoding can exceed.

    This sourceDefines channel capacity as the maximum mutual information over input distributions, and proves the achievability-below / impossibility-above coding theorem.

• Compression
  • The construct has roots in Shannon's 1948 foundational information theory

    This sourceFounds information theory; introduces entropy H(X), redundancy, and the source-coding theorem fixing the entropy lower bound on lossless representation. SUPPORTS marker 171 (Shannon's 1948 foundational information theory) and the Shannon-limit framing throughout. DOI (Wiley) verified; also archived at archive.org/details/bstj27-3-379.

• Conditional Entropy
  • Shannon introduced this conditional measure as part of the mathematical architecture of communication, and standard information-theory treatments retain the same definition and chain laws.
• Conditional Probability
  • reinforcement-learning policies are conditional distributions over actions given states. Game theory and information economics. A player's belief about others' types after a signal is a conditional distribution; sequential equilibria, Bayesian games, and signalling models all rest on conditioning and the discipline of updating it. Forecasting and engineering. Forecasts are conditional on currently available data, with the conditional reduction in uncertainty being the forecast's information content; fault-tree analysis is structured by \(P(\text{failure}\mid\text{component failed})\). Communication theory. Mutual information \(I(X;Y) = H(X) - H(X\mid Y)\) is a comparison of unconditional and conditional uncertainties

    This sourceFounds information theory; mutual information I(X;Y) = H(X) − H(X|Y) is built from conditional distributions as a comparison of unconditional and conditional uncertainty.

• Data Integrity
  • The essential commitment is that data without explicit integrity protection is progressively corrupted by bit rot, transmission errors, software bugs, operator mistakes, and adversarial manipulation; that detecting corruption requires redundancy or cryptographic verification; and that different threats require different mechanisms

    This sourceFounds information theory; establishes redundancy and channel coding as the basis for reliable transmission over noisy channels. Anchors D30-106, D30-113, D30-114 as the information-theoretic foundation for detecting corruption via redundancy. Verified. (See flag on D30-106: Shannon grounds redundancy-for-detection but not the specific 'different threats require different mechanisms' phrasing.)

• Discrete vs. Continuous (Quantization)
  • Classical examples: position in Newtonian mechanics, temperature, voltage (pre-quantization), real-valued economic indicators.

    This sourceFounds information theory; formally treats discrete vs continuous information sources/channels and states the sampling theorem (Theorem 13) — a band-limited signal is fully determined by samples at twice its highest frequency.

• Encoding And Decoding
  • In information theory, the Shannon origin, messages are encoded into channel symbols, transmitted, and decoded back, and the entire apparatus of error-correcting codes, source coding, and channel coding lives inside this pair.

    This sourceFounds the source–encoder–channel–decoder framework and the redundancy-versus-recoverability trade governing source and channel coding.

• Entropy (Thermodynamic Sense)
  • Information-theoretic entropy H = −Σ p_i log p_i shares mathematical form with Gibbs entropy

    This sourceIntroduces information entropy H = −Σ p_i log p_i as a measure of uncertainty; the functional form shared with Gibbs entropy.

• Logarithmic Perception and Encoding
  • Engineering displays — Bode plots, log-log plots for power laws, and semi-log axes wherever data span many decades. Information theory — bits are log of the alphabet; entropy is built on log-probability; the log base reflects the measurement scale.

    This sourceDefines information and entropy on the logarithm of probability, with the log base setting the measurement unit (bits).

• Measurement Uncertainty and Observational Noise
  • The same principles apply to radar, sonar, medical ultrasound, and remote sensing.
• Multiplexing
  • It separates two layers (the logical channels the users see and the single physical channel that actually carries them) and names the reversible combining operation that bridges them.
• Perturbation
  • Reducing perturbation to noise loses the diagnostic use.
• Realized vs Possible Outcomes
  • Software testing — full execution paths are the possibility set; paths exercised by the suite are the realized set; coverage reports on the gap. Information theory — channel capacity is the possibility set; the rate achieved is the realized; the gap is engineering inefficiency.

    This sourceDefines channel capacity as the possibility ceiling against which an achieved transmission rate is measured.

• Redundancy
  • Listed in the references but not attached to a specific claim.
• Selective Propagation
  • Once they are separated the channel becomes an object in its own right, with parameters to be measured rather than a transparency to be assumed, and a disagreement between a near measurement and a far one stops being a discrepancy to be resolved and becomes a datum about the path between them.
• Signal Decay and Fadeout
  • Alternatively, SNR may degrade during propagation if the signal weakens faster than noise (or if noise grows independently), in which case SNR and signal decay are correlated but not identical, a separation Shannon (1948) makes precise in his foundational information-theoretic framework, where channel capacity depends on the SNR ratio rather than on absolute signal magnitude.
• Subadditivity
  • Shannon introduced the governing entropy quantities, and Stanford's information-theory notes present the inequality, equality condition, and mutual-information identity together.

    This sourceFoundational primary source for entropy and joint/conditional information quantities.

• Vantage-Induced Omission
  • The unreachable region is everything outside that subspace — a frequency component above the array's Nyquist limit does not appear attenuated in the output, it appears aliased or absent, which is exactly the prime's load-bearing relation that what the apparatus cannot reach is not recorded as low, it is not recorded at all.
• Yield Loss
  • This connects to a family of impossibility results across fields: no real engine reaches Carnot efficiency, no real code reaches channel capacity, no real market reaches Pareto optimality, no real controller eliminates error.

    This sourceEstablishes channel capacity as the maximum reliable transmission rate — the information-theoretic limit no real code can exceed, one member of the family of physical-limit results the claim groups.

Domain-specific

• Bit Rate
  • Shannon's information rate and channel capacity use bits per second as an information-theoretic unit, but that does not make every counted code bit an independent bit of information.
• Digital Data
  • Shannon's finite-alphabet channel model explains how symbol probabilities, noise, coding, and capacity govern reliable communication.
• Shannon–Weaver Communication Model
  • Claude Shannon's 1948 paper introduced this schematic while developing a mathematical theory of communication capacity, entropy, coding, and reliable transmission.
• Units of Information
  • Shannon made the principle explicit: the choice of logarithmic base corresponds to the choice of a unit.
• Variable-Length Encoding
  • Shannon's source-coding framework relates achievable average length to entropy rather than to the mere number of alphabet symbols.

Verification

This reference passed the adversarial substantiation pipeline: it was checked to exist and to support the claim it is attached to. See how references were verified.

Links previously used in the corpus

Before the registry existed this work was also linked 4 other ways.

• https://doi.org/10.1002/j.1538-7305.1948.tb00917.x ×3
• https://onlinelibrary.wiley.com/doi/10.1002/j.1538-7305.1948.tb01338.x ×1
• https://people.math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdf ×1
• https://www.press.uillinois.edu/books/?id=p725487 ×1

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