ShannonHartleyTheorem

The Shannon–Hartley theorem defines the maximum theoretical data rate, in bits per second, that can be transmitted over a communication channel of a given bandwidth in the presence of additive white Gaussian noise (AWGN). The capacity C is given by C = B log2(1 + S/N), where B is the channel bandwidth in hertz and S/N is the signal-to-noise power ratio at the input of the receiver. An equivalent form using the two-sided noise spectral density N0 is C = B log2(1 + P/(N0 B)), with P representing the average signal power. The base-2 logarithm yields a result in bits per second.

The theorem assumes an ideal coding scheme and long codewords; it states that no method can reliably transmit information faster than C bits per second over the specified channel. For AWGN and fixed B, increasing S/N increases capacity logarithmically. At low SNR, C ≈ (B/ln 2) (S/N); at high SNR, C grows approximately as B log2(S/N). The theorem is named after Claude Shannon and Ralph Hartley, reflecting both the formal capacity concept and the historical development of information measurement.

Applications of the Shannon–Hartley theorem provide a fundamental limit used to assess channel design, coding schemes, and modulation strategies in digital communications. It clarifies why increasing bandwidth or improving noise conditions yields diminishing returns in capacity and serves as a benchmark for evaluating practical systems.