A topic in the Open Knowledge Graph — a free, open map of 15,290 topics and the order to learn them in.
The additive white Gaussian noise (AWGN) channel models a continuous-valued channel where the received signal Y = X + Z, with Z ~ N(0, N) independent of the input X, and the input has a power constraint E[X2] <= P. Its capacity is C = (1/2) log2(1 + P/N) bits per channel use — the Shannon-Hartley formula. The capacity-achieving input distribution is Gaussian: X ~ N(0, P). Combined with the bandwidth theorem, this gives C = W log2(1 + P/(NW)) bits per second for bandwidth W, establishing the fundamental tradeoff between bandwidth, power, and data rate that governs all modern wireless and wired communication.
The additive white Gaussian noise (AWGN) channel is the most important continuous channel model in information theory. It models any communication system where the dominant impairment is thermal noise: Y = X + Z, where X is the transmitted signal, Z ~ N(0, N) is Gaussian noise, and E[X2] <= P constrains the transmit power. The capacity of this channel — the Shannon-Hartley formula — is one of the most important equations in engineering.
The capacity derivation uses differential entropy. I(X;Y) = h(Y) - h(Y|X). Since Y|X = X + Z and Z is independent of X, h(Y|X) = h(Z) = (1/2) log2(2*pi*e*N), which is fixed. To maximize I(X;Y), we maximize h(Y). The variance of Y = X + Z is Var(X) + N <= P + N. Among all distributions with a given variance, the Gaussian maximizes differential entropy. So h(Y) <= (1/2) log2(2*pi*e*(P+N)), with equality when X ~ N(0, P). The capacity is C = (1/2) log2(2*pi*e*(P+N)) - (1/2) log2(2*pi*e*N) = (1/2) log2(1 + P/N).
The bandwidth extension connects to real-world systems. A band-limited channel of bandwidth W Hz can carry 2W independent real-valued samples per second (Nyquist's theorem). If the noise power spectral density is N_0/2, the total noise power in bandwidth W is N = N_0*W. With total signal power P, the capacity in bits per second is C = W * log2(1 + P/(N_0*W)). This formula captures the fundamental bandwidth-power tradeoff: you can increase data rate by using more bandwidth (but with diminishing returns) or more power (with logarithmic returns). The limit as W goes to infinity with fixed P is C = P/(N_0 * ln 2) — Shannon's ultimate limit, determined by power alone.
Modern communication systems (4G LTE, 5G NR, Wi-Fi 6/7, satellite links) are designed with this formula as the benchmark. The gap between a system's actual throughput and the Shannon-Hartley capacity quantifies how much room for improvement exists. Turbo codes and LDPC codes operate within 0.1 dB of the Gaussian channel capacity, a remarkable engineering achievement that took nearly 50 years after Shannon's theorem to attain.