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.
The equivalent expression in logarithmic form (dB):
Es/N0 can further be expressed as:
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:
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:
Which can be solved to get the Shannon-limit bound on Eb/N0:
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:
which corresponds to –1.59 dB.
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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