Definitions

# Eb/N0

Eb/N0 (the energy per bit to noise power spectral density ratio) is an important parameter in digital communication or data transmission. It 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.

Eb/N0 is equal to the SNR divided by the "gross" link spectral efficiency in (bit/s)/Hz, where the bits in this context are transmitted data bits, inclusive of error correction information and other protocol overhead. It should be noted that when forward error correction is being discussed, Eb/N0 is routinely used to refer to the energy per information bit (i.e. the energy per bit net of FEC overhead bits). In this context, Es/N0 is generally used to relate actual transmitted power to noise.

The noise spectral density N0, usually expressed in units of watts per hertz, that is, joules-per-second per cycles-per-second, can also be seen as having dimensions of energy, or units of joules, or joules per cycle. Eb/N0 is therefore a non-dimensional ratio.

Eb/N0 is commonly used with modulation and coding designed for power-limited, rather than bandwidth-limited communications. Examples of power-limited communications include deep-space and spread spectrum, and is optimized by using large bandwidths relative to the bit rate.

## Relation to carrier-to-noise ratio

Eb/N0 is closely related to the carrier-to-noise ratio (CNR or C/N), i.e. the signal-to-noise ratio (SNR) of the received signal, after the receiver filter but before detection:

$C/N=E_b/N_0cdotfrac\left\{f_b\right\}\left\{B\right\}$

where

fb is the channel data rate (gross bitrate), and
B is the channel bandwidth

The equivalent expression in logarithmic form (dB):

$CNR_\left\{dB\right\} = 10log_\left\{10\right\}\left(E_b/N_0\right) + 10log_\left\{10\right\}\left(frac\left\{f_b\right\}\left\{B\right\}\right)$,

## Relation to Es/N0

Eb/N0 can be seen as a normalized measure of the energy per symbol per noise power spectral density (Es/N0), where Es is the Energy per symbol in Joules. This measure is also commonly used in the analysis of digital modulation schemes. The two quotients are related to each other according to the following:

$frac\left\{E_s\right\}\left\{N_0\right\} =frac\left\{E_b\right\}\left\{N_0\right\}log_2 M$,

where

M is the number of alternative modulation symbols.

Es/N0 can further be expressed as:

$frac\left\{E_s\right\}\left\{N_0\right\} = frac\left\{C\right\}\left\{N\right\}frac\left\{B\right\}\left\{f_s\right\}$,

where

C/N is the carrier-to-noise ratio or signal-to-noise ratio.
B is the channel bandwidth in Hertz.
fs is the symbol rate in baud or symbols/second.

For a PSK, ASK or QAM modulation with pulse shaping such as raised cosine shaping, the B/fs ratio is usually slightly larger than 1, depending of the pulse shaping filter.

## Shannon limit

The Shannon–Hartley theorem says that the limit of reliable data rate of a channel depends on bandwidth and signal-to-noise ratio according to:

$R < B log_2 left\left(1+frac\left\{S\right\}\left\{N\right\} right\right)$

where

R is an information rate in bits per second;
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 Eb/N0 for any system that achieves reliable communication, by considering a bit rate equal to R and therefore an average energy per bit of Eb = S/R, with noise spectral density of N0 = N/B. For this calculation, it is conventional to define a normalized rate Rl = 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 2B dimensions, according to the Nyquist–Shannon sampling theorem). Making appropriate substitutions, the Shannon limit is:

$\left\{R over B\right\} = 2 R_l < log_2 left\left(1 + 2R_lfrac\left\{E_b\right\}\left\{N_0\right\} right\right)$

Which can be solved to get the Shannon-limit bound on Eb/N0:

$frac\left\{E_b\right\}\left\{N_0\right\} > frac\left\{2^\left\{2R_l\right\}-1\right\}\left\{2R_l\right\}$

When the data rate is small compared to the bandwidth, so that Rl is near zero, the bound, sometimes called the ultimate Shannon limit, is:

$frac\left\{E_b\right\}\left\{N_0\right\} > ln\left(2\right)$

which corresponds to –1.59 dB.

## Cutoff rate

For any given system of coding and decoding, there exists what is known as a cutoff rate R_0, typically corresponding to an Eb/N0 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.

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