Bit-error rate (BER) vs ${\displaystyle E_{b}/N_{0}}$ curves for different digital modulation methods is a common application example of ${\displaystyle E_{b}/N_{0}}$. Here an AWGN channel is assumed.

In digital communication or data transmission, ${\displaystyle E_{b}/N_{0}}$ (energy per bit to noise power spectral density ratio) is a normalized signal-to-noise ratio (SNR) measure, also known as the "SNR per bit". It is especially useful when comparing the bit error rate (BER) performance of different digital modulation schemes without taking bandwidth into account.

As the description implies, ${\displaystyle E_{b}}$ is the signal energy associated with each user data bit; it is equal to the signal power divided by the user bit rate (not the channel symbol rate). If signal power is in watts and bit rate is in bits per second, ${\displaystyle E_{b}}$ is in units of joules (watt-seconds). ${\displaystyle N_{0}}$ is the noise spectral density, the noise power in a 1 Hz bandwidth, measured in watts per hertz or joules.

These are the same units as ${\displaystyle E_{b}}$ so the ratio ${\displaystyle E_{b}/N_{0}}$ is dimensionless; it is frequently expressed in decibels. ${\displaystyle E_{b}/N_{0}}$ directly indicates the power efficiency of the system without regard to modulation type, error correction coding or signal bandwidth (including any use of spread spectrum). This also avoids any confusion as to which of several definitions of "bandwidth" to apply to the signal.

But when the signal bandwidth is well defined, ${\displaystyle E_{b}/N_{0}}$ is also equal to the signal-to-noise ratio (SNR) in that bandwidth divided by the "gross" link spectral efficiency in bit/s⋅Hz, where the bits in this context again refer to user data bits, irrespective of error correction information and modulation type.^[1]

${\displaystyle E_{b}/N_{0}}$ must be used with care on interference-limited channels since additive white noise (with constant noise density ${\displaystyle N_{0}}$) is assumed, and interference is not always noise-like. In spread spectrum systems (e.g., CDMA), the interference is sufficiently noise-like that it can be represented as ${\displaystyle I_{0}}$ and added to the thermal noise ${\displaystyle N_{0}}$ to produce the overall ratio ${\displaystyle E_{b}/(N_{0}+I_{0})}$.

Relation to carrier-to-noise ratio

${\displaystyle E_{b}/N_{0}}$ 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.

Relation to E_s/N₀

${\displaystyle E_{b}/N_{0}}$ 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.

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:

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

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.

Cutoff rate

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.

References

External links

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