- Eb/N0
"E"b/"N"0 (the energy per bit to noise power spectral density ratio) is an important parameter in
digital communication ordata transmission . It is a normalizedsignal-to-noise ratio (SNR) measure, also known as the "SNR per bit". It is especially useful when comparing thebit error rate (BER) performance of different digitalmodulation schemes without taking bandwidth into account."E"b/"N"0 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 whenforward error correction is being discussed, "E"b/"N"0 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 "N"0, usually expressed in units ofwatt s perhertz , that is, joules-per-second per cycles-per-second, can also be seen as having dimensions of energy, or units ofjoule s, or joules per cycle. "E"b/"N"0 is therefore a non-dimensional ratio."E"b/"N"0 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. thesignal-to-noise ratio (SNR) of the received signal, after the receiver filter but before detection::
where:"f"b is the channel data rate (
gross bitrate ), and:"B" is the channel bandwidthThe equivalent expression in logarithmic form (dB):
:,
Relation to Es/N0
"E"b/"N"0 can be seen as a normalized measure of the energy per symbol per noise power spectral density ("E"s/"N"0), where "E"s 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:
:,
where
:"M" is the number of alternative modulation symbols.
"E"s/"N"0 can further be expressed as:
:,
where:"C/N" is the
carrier-to-noise ratio orsignal-to-noise ratio .:"B" is the channel bandwidth in Hertz.:"f"s is the symbol rate inbaud or symbols/second.For a PSK, ASK or
QAM modulation withpulse shaping such asraised cosine shaping, the "B"/"f"s ratio is usually slightly larger than 1, depending of the pulse shaping filter.hannon 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::
where:"R" is an information rate in
bits per second ;:"B" is the bandwidth of the channel inhertz ; : "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/"(2"B"), a bandwidth utilization parameter of bits per second per half hertz, or bits per dimension (a signal of bandwidth "B" can be encoded with 2"B" 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", [cite book | title = Algorithms for Communications Systems and Their Applications | author = Nevio Benvenuto and Giovanni Cherubini | ISBN 0470843896 | year = 2002 | page = 508 | publisher = John Wiley & Sons] is:
:
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 ofturbo codes .References
External links
* [http://www.sss-mag.com/ebn0.html Eb/N0 Explained.] An introductory article on Eb/N0
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