137.122.205.68: /* Comparison with other polymerization methods */

2014-08-27T19:44:34Z

Comparison with other polymerization methods

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2013-09-02T12:42:57Z

Clean up using AWB

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[[File:PSK BER curves.svg|thumb|right|280px|[[Bit-error rate]] (BER) vs ''E''b/''N''0 curves for different [[digital modulation]] methods is a common application example of ''E''b/''N''0. Here an [[AWGN]] channel is assumed.]]
'''''E''b/''N''0''' (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.

''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. When [[forward error correction]] (FEC) 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.<ref>
{{cite book
| title = Turbo coding
| edition =
| author = Chris Heegard and Stephen B. Wicker
| publisher = Kluwer
| year = 1999
| isbn = 978-0-7923-8378-9
| page = 3
| url = http://books.google.com/books?id=aTWkPD_COsoC&pg=PA3
}}</ref>

The [[noise spectral density]] ''N''0, usually expressed in units of [[watt]]s per [[hertz]], can also be seen as having dimensions of energy, or units of [[joule]]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 noise-limited rather than interference-limited communication, since additive white noise (with constant noise density ''N''0) is assumed.

==Relation to carrier-to-noise ratio==

''E''b/''N''0 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:

:<math>C/N=E_b/N_0\cdot\frac{f_b}{B}</math>

where
:''f''b is the channel data rate ([[net bitrate]]), and
:''B'' is the channel bandwidth

The equivalent expression in logarithmic form (dB):

:<math>\text{CNR}_{\text{dB}} = 10\log_{10}(E_b/N_0) + 10\log_{10}\left(\frac{f_b}{B}\right)</math>,

Caution: Sometimes, the noise power is denoted by <math>N_0/2</math> 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/''N''0==
''E''b/''N''0 can be seen as a normalized measure of the '''energy per symbol to noise power spectral density''' (''E''s/''N''0):

:<math>\frac{E_b}{N_0} =\frac{E_s}{\rho N_0}</math>,

where ''E''s is the energy per symbol in joules and <math>\rho</math> is the nominal [[spectral efficiency]] in (bit/s)/Hz.<ref>{{Cite web | last = Forney | first = David | title = MIT OpenCourseWare, 6.451 Principles of Digital Communication II, Lecture Notes section 4.2 | url=http://ocw.mit.edu/ | accessdate = 21 September 2010 }}</ref> ''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:

:<math>\frac{E_s}{N_0} =\frac{E_b}{N_0}\log_2 M </math>,

where ''M'' is the number of alternative modulation symbols.

Note that this is the energy per bit, not the energy per information bit.

''E''s/''N''0 can further be expressed as:

:<math>\frac{E_s}{N_0} = \frac{C}{N}\frac{B}{f_s}</math>,

where
:''C/N'' is the [[carrier-to-noise ratio]] or [[signal-to-noise ratio]].
:''B'' is the channel bandwidth in hertz.
:''f''s is the symbol rate in [[baud]] or symbols per second.

==Shannon limit==
{{main | Shannon–Hartley theorem}}

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:

:<math> I 

where
:''I'' is the [[information rate]] in [[bits per second]] excluding [[error-correcting code]]s;
:''B'' is the [[Bandwidth (signal processing)|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 gross bit rate ''R'' equal to the net bit rate ''I'' and therefore an average energy per bit of ''E''b = ''S/R'', with noise spectral density of ''N''0 = ''N/B''. For this calculation, it is conventional to define a normalized rate ''R''l = ''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:

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

Which can be solved to get the Shannon-limit bound on ''E''b/''N''0:

:<math>\frac{E_b}{N_0} > \frac{2^{2R_l}-1}{2R_l} </math>

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'',<ref>{{cite book | title = Algorithms for Communications Systems and Their Applications | author = Nevio Benvenuto and Giovanni Cherubini | year = 2002 | page = 508 | publisher = John Wiley & Sons | isbn = 0-470-84389-6}}</ref> is:

:<math>\frac{E_b}{N_0} > \ln(2) </math>

which corresponds to –1.59 dB because:
:<math>ln(2) = 0.693</math> and
:<math>10log(0.693) = -1.59 dB</math>

However, [[spread spectrum]] allows negative signal-to-noise ratio before despreading, since the bandwidth is much higher than the bit rate.

==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/''N''0 about 2 dB above the Shannon capacity limit. {{Citation needed|date=April 2009}}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]] or by the LDPC (Low Density Parity Check) codes.

==References==

{{reflist}}

==External links==
*[http://www.sss-mag.com/ebn0.html Eb/N0 Explained.] An introductory article on ''E''b/''N''0

{{Noise}}

{{DEFAULTSORT:Eb-N0}}
[[Category:Noise]]
[[Category:Signal processing]]
[[Category:Engineering ratios]]

Chain-growth polymerization - Revision history

137.122.205.68: /* Comparison with other polymerization methods */

en>Christian75: Clean up using AWB