“Eb/N0”的意思、由来-开放百科全书

E_b/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.

As the description implies, E_b is the signal energy associated with each user data bit; it is equal to the signal power divided by the user bit rate (not the channel symbol rate). If signal power is in watts and bit rate is in bits per second, E_b is in units of joules (watt-seconds). N₀ is the noise spectral density, the noise power in a 1 Hz bandwidth, measured in watts per hertz or joules.

These are the same units as E_b so the ratio E_b/N₀ is dimensionless; it is frequently expressed in decibels. E_b/N₀ directly indicates the power efficiency of the system without regard to modulation type, error correction coding or signal bandwidth (including any use of spread spectrum). This also avoids any confusion as to which of several definitions of "bandwidth" to apply to the signal.

But when the signal bandwidth is well defined, E_b/N₀ is also equal to the signal-to-noise ratio (SNR) in that bandwidth divided by the "gross" link spectral efficiency in bit/s⋅Hz, where the bits in this context again refer to user data bits, irrespective of error correction information and modulation type.^[1]

E_b/N₀ must be used with care on interference-limited channels since additive white noise (with constant noise density N₀) is assumed, and interference is not always noise-like. In spread spectrum systems (e.g., CDMA), the interference is sufficiently noise-like that it can be represented as I₀ and added to the thermal noise N₀ to produce the overall ratio {{nowrap|E_b/(N₀ + I₀)}}.

Relation to carrier-to-noise ratio

E_b/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:

Caution: Sometimes, the noise power is denoted by N₀/2 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/N₀

E_b/N₀ can be seen as a normalized measure of the energy per symbol to noise power spectral density (E_s/N₀):

where E_s is the energy per symbol in joules and ρ is the nominal spectral efficiency in bit/s⋅Hz.^[2] E_s/N₀ 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.g. M = 4 for QPSK and M = 8 for 8PSK.

Shannon limit

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:

This equation can be used to establish a bound on E_b/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 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:

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,^[3] is:

Note that this often-quoted limit of −1.59 dB applies only to the theoretical case of infinite bandwidth. The Shannon limit for finite-bandwidth signals is always higher.

Cutoff rate

For any given system of coding and decoding, there exists what is known as a cutoff rate R₀, typically corresponding to an E_b/N₀ 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 and low-density parity-check (LDPC) codes.

References

1. ^{{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 = https://books.google.com/books?id=aTWkPD_COsoC&pg=PA3 }}
2. ^{{Cite web | last = Forney | first = David | title = MIT OpenCourseWare, 6.451 Principles of Digital Communication II, Lecture Notes section 4.2 | url=https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-451-principles-of-digital-communication-ii-spring-2005/readings-and-lecture-notes/MIT6_451S05_FullLecNotes.pdf#page=35 | accessdate = 8 November 2017 }}
3. ^{{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}}

Relation to carrier-to-noise ratio

Relation to Es/N0

Shannon limit

Cutoff rate

References

External links

Relation to E_s/N₀