is the nominal spectral efficiency in (bits/s)/Hz. is also commonly used in the analysis of digital modulation schemes. The two quotients are related to each other according to the following: where is the number of alternative modulation symbols, e.g. for QPSK and for 8PSK. This is the energy per bit, not the energy per information bit. can further be expressed as: where is the carrier-to-noise ratio or signal-to-noise ratio, is the channel bandwidth in hertz, and is the symbol rate in baud or symbols per second. Shannon limit The Shannon–Hartley theorem says that the limit of reliable information rate (data rate exclusive of error-correcting codes) of a channel depends on bandwidth and signal-to-noise ratio according to: where is the information rate in bits per second excluding error-correcting codes, is the bandwidth of the channel in hertz, is the total signal power (equivalent to the carrier power ), and is the total noise power in the bandwidth. This equation can be used to establish a bound on for any system that achieves reliable communication, by considering a gross bit rate equal to the net bit rate and therefore an average energy per bit of , with noise spectral density of . For this calculation, it is conventional to define a normalized rate , a bandwidth utilization parameter of bits per second per half hertz, or bits per dimension (a signal of bandwidth can be encoded with dimensions, according to the Nyquist–Shannon sampling theorem). Making appropriate substitutions, the Shannon limit is: Which can be solved to get the Shannon-limit bound on : When the data rate is small compared to the bandwidth, so that is near zero, the bound, sometimes called the ultimate Shannon limit, is: which corresponds to −1.59dB. This often-quoted limit of −1.59 dB applies only to the theoretical case of infinite bandwidth. The Shannon limit for finite-bandwidth signals is always higher. Cutoff rate For any given system of coding and decoding, there exists what is known as a cutoff rate , typically corresponding to an about 2 dB above the Shannon capacity limit. The cutoff rate used to be thought of as the limit on practical error correction codes without an unbounded increase in processing complexity, but has been rendered largely obsolete by the more recent discovery of turbo codes and low-density parity-check (LDPC) codes. References External links Explained. An introductory article on Noise (electronics) Signal processing Engineering ratios