Shannon–Hartley theorem

In information theory, the Shannon–Hartley theorem tells the maximum rate at which information can be transmitted over a communications channel of a specified bandwidth in the presence of noise. It is an application of the noisy-channel coding theorem to the archetypal case of a continuous-time analog communications channel subject to Gaussian noise. The theorem establishes Shannon's channel capacity for such a communication link, a bound on the maximum amount of error-free information per time unit that can be transmitted with a specified bandwidth in the presence of the noise interference, assuming that the signal power is bounded, and that the Gaussian noise process is characterized by a known power or power spectral density. The law is named after Claude Shannon and Ralph Hartley. The Shannon–Hartley theorem states the channel capacity , meaning the theoretical tightest upper bound on the information rate of data that can be communicated at an arbitrarily low error rate using an average received signal power through an analog communication channel subject to additive white Gaussian noise (AWGN) of power : where is the channel capacity in bits per second, a theoretical upper bound on the net bit rate (information rate, sometimes denoted ) excluding error-correction codes; is the bandwidth of the channel in hertz (passband bandwidth in case of a bandpass signal); is the average received signal power over the bandwidth (in case of a carrier-modulated passband transmission, often denoted C), measured in watts (or volts squared); is the average power of the noise and interference over the bandwidth, measured in watts (or volts squared); and is the signal-to-noise ratio (SNR) or the carrier-to-noise ratio (CNR) of the communication signal to the noise and interference at the receiver (expressed as a linear power ratio, not as logarithmic decibels). During the late 1920s, Harry Nyquist and Ralph Hartley developed a handful of fundamental ideas related to the transmission of information, particularly in the context of the telegraph as a communications system.