The theorem establishes Shannon's channel capacity for such a communication link. This boundary is 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. The assumption is that the signal power is bounded, and that the Gaussian noise process is characterized by a known power or power spectral density.
The theorem states the channel capacity C, 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 S through an analog communication channel subject to additive white Gaussian noise of power N.

C = B * log₂(1 +

S / N

)

• C is the channel capacity in bits per second, a theoretical upper bound on the net bit rate (information rate, sometimes denoted I) excluding error-correction codes;
• B is the bandwidth of the channel in hertz (passband bandwidth in case of a bandpass signal);
• S 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);
• N is the average power of the noise and interference over the bandwidth, measured in watts (or volts squared); and
• S / N 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).

Nyquist determined that the number of independent pulses that could be put through a telegraph channel per unit time is limited to twice the bandwidth of the channel. In symbols,

f_p = 2B

• f_p is the pulse frequency (in pulses per second) and
• B is the bandwidth (in hertz).

The quantity 2B later came to be called the Nyquist rate, and transmitting at the limiting pulse rate of 2B pulses per second as signalling at the Nyquist rate.

Hartley formulated a way to quantify information and its line rate (also known as data signalling rate R bits per second). This method, later known as Hartley's law, became an important precursor for Shannon's more sophisticated notion of channel capacity.
Hartley argued that the maximum number of distinguishable pulse levels that can be transmitted and received reliably over a communications channel is limited by the dynamic range of the signal amplitude and the precision with which the receiver can distinguish amplitude levels. Specifically, if the amplitude of the transmitted signal is restricted to the range of [-A ... +A] volts, and the precision of the receiver is ±?V volts, then the maximum number of distinct pulses M is given by

M = 1 +

A / ΔV

R = f_p * log ₂( M )

Combining the above quantification with Nyquist's observation that the number of independent pulses that could be put through a channel of bandwidth B hertz was 2B pulses per second, gives rise to a quantitative measure for achievable line rate.
Hartley's law is sometimes quoted as just a proportionality between the analog bandwidth, Bin Hertz and what is called the digital bandwidth, R in bit/s.
On other occasions it is quoted in this more quantitative form, as an achievable line rate of R bits per secon

R ≤ 2B * log₂(M)