E_b/N₀

${\displaystyle E_{b}/N_{0}}$ is closely related to the carrier-to-noise ratio (CNR or ${\displaystyle {\frac {C}{N}}}$), i.e. the signal-to-noise ratio (SNR) of the received signal, after the receiver filter but before detection:

$${\displaystyle {\frac {C}{N}}={\frac {E_{\text{b}}}{N_{0}}}{\frac {f_{\text{b}}}{B}}}$$

where
     ${\displaystyle f_{b}}$ is the channel data rate (net bit rate) and
     B is the channel bandwidth.

The equivalent expression in logarithmic form (dB):

$${\displaystyle {\text{CNR}}_{\text{dB}}=10\log _{10}\left({\frac {E_{\text{b}}}{N_{0}}}\right)+10\log _{10}\left({\frac {f_{\text{b}}}{B}}\right)}$$

Caution: Sometimes, the noise power is denoted by ${\displaystyle N_{0}/2}$ when negative frequencies and complex-valued equivalent baseband signals are considered rather than passband signals, and in that case, there will be a 3 dB difference.

${\displaystyle E_{b}/N_{0}}$ can be seen as a normalized measure of the energy per symbol to noise power spectral density (${\displaystyle E_{s}/N_{0}}$):

$${\displaystyle {\frac {E_{b}}{N_{0}}}={\frac {E_{\text{s}}}{\rho N_{0}}}}$$

where ${\displaystyle E_{s}}$ is the energy per symbol in joules and ρ is the nominal spectral efficiency in (bits/s)/Hz.[2] ${\displaystyle E_{s}/N_{0}}$ is also commonly used in the analysis of digital modulation schemes. The two quotients are related to each other according to the following:

$${\displaystyle {\frac {E_{\text{s}}}{N_{0}}}={\frac {E_{\text{b}}}{N_{0}}}\log _{2}(M)}$$

where M is the number of alternative modulation symbols, e.g. ${\displaystyle M=4}$ for QPSK and ${\displaystyle M=8}$ for 8PSK.

This is the energy per bit, not the energy per information bit.

${\displaystyle E_{s}/N_{0}}$ can further be expressed as:

$${\displaystyle {\frac {E_{\text{s}}}{N_{0}}}={\frac {C}{N}}{\frac {B}{f_{\text{s}}}}}$$

where
     ${\displaystyle {\frac {C}{N}}}$ is the carrier-to-noise ratio or signal-to-noise ratio,
     B is the channel bandwidth in hertz, and
     ${\displaystyle f_{s}}$ is the symbol rate in baud or symbols per second.

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:

$${\displaystyle I<B\log _{2}\left(1+{\frac {S}{N}}\right)}$$

where
     I is the information rate in bits per second excluding error-correcting codes,
     B is the bandwidth of the channel in hertz,
     S is the total signal power (equivalent to the carrier power C), and
     N is the total noise power in the bandwidth.

This equation can be used to establish a bound on ${\displaystyle E_{b}/N_{0}}$ for any system that achieves reliable communication, by considering a gross bit rate R equal to the net bit rate I and therefore an average energy per bit of ${\displaystyle E_{b}=S/R}$, with noise spectral density of ${\displaystyle N_{0}=N/B}$. For this calculation, it is conventional to define a normalized rate ${\displaystyle R_{l}=R/(2B)}$, a bandwidth utilization parameter of bits per second per half hertz, or bits per dimension (a signal of bandwidth B can be encoded with ${\displaystyle 2B}$ dimensions, according to the Nyquist–Shannon sampling theorem). Making appropriate substitutions, the Shannon limit is:

$${\displaystyle {R \over B}=2R_{l}<\log _{2}\left(1+2R_{l}{\frac {E_{\text{b}}}{N_{0}}}\right)}$$

Which can be solved to get the Shannon-limit bound on ${\displaystyle E_{b}/N_{0}}$:

$${\displaystyle {\frac {E_{\text{b}}}{N_{0}}}>{\frac {2^{2R_{l}}-1}{2R_{l}}}}$$

When the data rate is small compared to the bandwidth, so that ${\displaystyle R_{l}}$ is near zero, the bound, sometimes called the ultimate Shannon limit,[3] is:

$${\displaystyle {\frac {E_{\text{b}}}{N_{0}}}>\ln(2)}$$

which corresponds to −1.59 dB.

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.

For any given system of coding and decoding, there exists what is known as a cutoff rate ${\displaystyle R_{0}}$, typically corresponding to an ${\displaystyle E_{b}/N_{0}}$ 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.

• E_b/N₀ Explained. An introductory article on ${\displaystyle E_{b}/N_{0}}$