For a communication channel modeled as an additive white Gaussian noise (AWGN) channel with bandwidth B (Hz) and signal-to-noise ratio SNR (linear), the Shannon–Hartley theorem gives the channel capacity C (maximum achievable reliable information rate in bits per second) as C = B · log2(1 + SNR). The result is an information-theoretic upper bound assuming ideal coding over arbitrarily long blocks and ergodic, stationary AWGN noise.

Principle

Channel capacity increases with available bandwidth and with SNR in a logarithmic manner; for AWGN the tradeoff between power (SNR) and bandwidth is quantitatively captured by C = B log2(1+SNR), which sets an achievable upper limit under the theorem's statistical and coding assumptions.

Demonstration

Illustrative scenario — Ideal AWGN link: Situation: A hypothetic AWGN channel of bandwidth B connects transmitter and receiver; the transmitted signal has average power P and the noise has power spectral density N0/2 so that SNR = P/(N0 B). Recognition: Apply Shannon–Hartley to compute C = B log2(1+SNR). Action: Compare required raw bit rate R to C; if R

Misapplication

Interpreting C as a guaranteed throughput for a particular implementation (without considering finite block length, latency, nonideal coding, channel nonstationarity, interference, or regulatory constraints) is an error of category: capacity is an asymptotic theoretical limit, not a practical instantaneous rate guarantee.

Consequence

Provides a fundamental engineering benchmark that quantifies the tradeoff between bandwidth and power for AWGN channels, guiding system design, spectral efficiency targets, and motivating coding theoretic development; it does not prescribe specific codes or finite-length performance.

Reversal

Channels that deviate from the AWGN, stationary, ergodic assumptions (e.g., fading channels, channels with interference, or channels with memory) have different capacity formulas; practical constraints (finite alphabet, regulatory spectral masks, latency) reduce achievable rates below the Shannon bound and may require alternative capacity analyses.

Boundary

Clearly within: Memoryless AWGN channel with well-defined flat bandwidth B and stationary ergodic noise, allowing asymptotically long block codes. Boundary case: AWGN with additional constraints (finite block length, peak power limits, nonflat spectra) where Shannon capacity approximates but does not exactly predict performance. Clearly outside: channels with strong non-Gaussian interference, nonergodic fading without channel state information, or deterministic lower-layer constraints that make the AWGN capacity model inapplicable.

Semantic Tension

Tension between the theorem's role as an asymptotic, model-dependent upper bound and practical system metrics (throughput, latency, complexity, error probability at finite block length); design must trade off between approaching capacity and satisfying real-world constraints.

Synthesis

The Shannon–Hartley theorem quantifies a fundamental power–bandwidth tradeoff for ideal AWGN channels: it establishes an asymptotic upper limit on reliable communication rate that organizes engineering targets but must be interpreted relative to model assumptions and finite-resource limitations in practice.