Eb/N0

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E_b/N₀ (the Energy per bit per noise power spectral density) is a parameter used in signal processing and telecommunications similar to signal-to-noise ratio (SNR). It defines the SNR per bit and is an important measure to evaluate and compare different digital communication systems.

The noise power 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 the dimension of simply energy, or units of joules, or joules per cycle. The value E_b/N₀ is therefore effectively non-dimensional, allowing comparison across a wide range of different systems.

E_b/N₀ is a measure normally used to compare communication schemes that are limited by power and by noise spectral density, rather than by bandwidth, such as deep-space channels, and is optimized by using bandwidths large compared to the information bit rate.

1 Carrier-to-noise ratio
2 Shannon limit
3 Cutoff rate
4 References
5 External links

[edit] Carrier-to-noise ratio

E_b/N₀ is closely related to carrier-to-noise ratio (CNR):

$CNR=E_b/N_0\cdot\frac{f_b}{B}$

or in logarithmic form (dB):

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

Where

f_b is the channel data rate

B is the channel bandwidth

[edit] 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:

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

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; and

N is the total noise power in the bandwidth.

This equation can be used to establish a bound on E_b/N₀ for any system that achieves reliable communication, by considering a bit rate equal to R and therefore an average energy per bit of E_b = S/R, with noise spectral density of N₀ = N/B. For this calculation, it is conventional to define a normalized rate 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 2B dimensions, according to the Nyquist–Shannon sampling theorem). Making appropriate substitutions, the Shannon limit is:

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

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

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

When the data rate is small compared to the bandwidth, so that R_l is near zero, the bound, sometimes called the ultimate Shannon limit,^[1] is:

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

which corresponds to –1.59 dB.

[edit] 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 E_b/N₀ about 2 dB above the Shannon capacity limit.