ShannonHartley

Shannon–Hartley theorem, also known as the Shannon–Hartley capacity formula, states the maximum data rate at which information can be transmitted over a bandwidth-limited communication channel in the presence of additive white Gaussian noise. For a channel with bandwidth B (in hertz), transmitted power P, and noise power spectral density N0/2, the channel capacity C is given by C = B log2(1 + P/(N0 B)) bits per second. Equivalently, defining SNR as P/(N0 B), the capacity can be written as C = B log2(1 + SNR).

The theorem relies on idealized conditions: a linear, time-invariant channel with additive white Gaussian noise, perfect synchronization, and the use of optimal coding with arbitrarily long block lengths. Under these assumptions, C represents an ultimate limit on reliable communication; no scheme can exceed this rate for the specified bandwidth and power.

Interpretation and implications are central to information theory. The capacity increases with bandwidth and with signal-to-noise ratio, but for fixed power and noise level it approaches a finite limit as B grows, specifically P/(N0 ln 2) bits per second. This framework guides the design of practical communication systems by establishing the theoretical boundary between achievable and unattainable data rates, and informs trade-offs among bandwidth, power, and coding complexity.

Historically, the theorem is named for Claude E. Shannon and Ralph Hartley and was introduced in the 1948 landmark work A Mathematical Theory of Communication. It remains a foundational concept for digital communications, coding theory, and network design.