Coding gain: Difference between revisions

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''Coding Gain'', is the measure in the difference between the SNR levels between the uncoded system & coded system required to reach the same BER levels when used with the [[ECC]].

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In [[coding theory]], [[telecommunications engineering]] and other related engineering problems, ''Coding'coding Gaingain''', is the measure in the difference between the [[signal-to-noise ratio]] (SNR) levels between the uncoded system &and coded system required to reach the same [[bit error rate]] (BER) levels when used with the [[ECCerror correcting code]] (ECC).

==Example==

If the uncoded [[BPSK]] system in [[AWGN]] environment has a [[bit error rate]] (BER) of 10<mathsup>10^-3−2</mathsup> at the SNR level 4&nbsp;[[decibel|dB]], and the corresponding coded (e.g., [[BCH code|BCH]]) system has the same BER at an SNR of 2.5&nbsp;dB, then we say the ''coding gain'' = {{nowrap|1=4 dB − 2.5 dB = 1.5 dB}}, due to the code used (in this case BCH).

SNR level 3dB, and the corresponding Coded (eg BCH) systsem has the same BER at SNR level of 1dB,

then we say the ''coding gain'' = 3dB-1.5dB = 1.5dB, due to the code (BCH in this case) used.

==Power-limited regime==

In the ''power-limited regime'' (where the nominal [[spectral efficiency]] <math>\rho \le 2</math> [b/2D or b/s/Hz], ''i.e.'' the ___domain of binary signaling), the effective coding gain <math>\gamma_\mathrm{eff}(A)</math> of a signal set <math>A</math> at a given target error probability per bit <math>P_b(E)</math> is defined as the difference in dB between the <math>E_b/N_0</math> required to achieve the target <math>P_b(E)</math> with <math>A</math> and the <math>E_b/N_0</math> required to achieve the target <math>P_b(E)</math> with 2-[[Pulse-amplitude modulation|PAM]] or (2&times;2)-[[Quadrature amplitude modulation|QAM]] (''i.e.'' no coding). The nominal coding gain <math>\gamma_c(A)</math> is defined as

: <math>\gamma_c(A) = \frac{d^2_{\min}(A)}{4E_b}.</math>

This definition is normalized so that <math>\gamma_c(A) = 1</math> for 2-PAM or (2&times;2)-QAM. If the average number of nearest neighbors per transmitted bit <math>K_b(A)</math> is equal to one, the effective coding gain <math>\gamma_\mathrm{eff}(A)</math> is approximately equal to the nominal coding gain <math>\gamma_c(A)</math>. However, if <math>K_b(A)>1</math>, the effective coding gain <math>\gamma_\mathrm{eff}(A)</math> is less than the nominal coding gain <math>\gamma_c(A)</math> by an amount which depends on the steepness of the <math>P_b(E)</math> ''vs.'' <math>E_b/N_0</math> curve at the target <math>P_b(E)</math>. This curve can be plotted using the [[union bound]] estimate (UBE)

: <math>P_b(E) \approx K_b(A)Q\left(\sqrt{\frac{2\gamma_c(A)E_b}{N_0}}\right),</math>

where ''Q'' is the [[error function|Gaussian probability-of-error function]].

For the special case of a binary [[linear block code]] <math>C</math> with parameters <math>(n,k,d)</math>, the nominal spectral efficiency is <math>\rho = 2k/n </math> and the nominal coding gain is&nbsp;''kd''/''n''.

==Example==

The table below lists the nominal spectral efficiency, nominal coding gain and effective coding gain at <math>P_b(E) \approx 10^{-5}</math> for [[Reed–Muller code]]s of length <math>n \le 64</math>:

{| class="wikitable"

! Code !! <math>\rho</math> !! <math>\gamma_c</math> !! <math>\gamma_c</math> (dB) !! <math>K_b</math> !! <math>\gamma_\mathrm{eff}</math> (dB)

|-

| [8,7,2] || 1.75 || 7/4 || 2.43 || 4 || 2.0

|-

| [8,4,4] || 1.0 || 2 || 3.01 || 4 || 2.6

|-

| [16,15,2] || 1.88 || 15/8 || 2.73 || 8 || 2.1

|-

| [16,11,4] || 1.38 || 11/4 || 4.39 || 13 || 3.7

|-

| [16,5,8] || 0.63 || 5/2 || 3.98 || 6 || 3.5

|-

| [32,31,2] || 1.94 || 31/16 || 2.87 || 16 || 2.1

|-

| [32,26,4] || 1.63 || 13/4 || 5.12 || 48 || 4.0

|-

| [32,16,8] || 1.00 || 4 || 6.02 || 39 || 4.9

|-

| [32,6,16] || 0.37 || 3 || 4.77 || 10 || 4.2

|-

| [64,63,2] || 1.97 || 63/32 || 2.94 || 32 || 1.9

|-

| [64,57,4] || 1.78 || 57/16 || 5.52 || 183 || 4.0

|-

| [64,42,8] || 1.31 || 21/4 || 7.20 || 266 || 5.6

|-

| [64,22,16] || 0.69 || 11/2 || 7.40 || 118 || 6.0

|-

| [64,7,32] || 0.22 || 7/2 || 5.44 || 18 || 4.6

|-

|}

==Bandwidth-limited regime==

In the ''bandwidth-limited regime'' (<math>\rho > 2~b/2D</math>, ''i.e.'' the ___domain of non-binary signaling), the effective coding gain <math>\gamma_\mathrm{eff}(A)</math> of a signal set <math>A</math> at a given target error rate <math>P_s(E)</math> is defined as the difference in dB between the <math>SNR_\mathrm{norm}</math> required to achieve the target <math>P_s(E)</math> with <math>A</math> and the <math>SNR_\mathrm{norm}</math> required to achieve the target <math>P_s(E)</math> with M-[[Pulse-amplitude modulation|PAM]] or (M&times;M)-[[Quadrature amplitude modulation|QAM]] (''i.e.'' no coding). The nominal coding gain <math>\gamma_c(A)</math> is defined as

: <math>\gamma_c(A) = {(2^\rho - 1)d^2_{\min} (A) \over 6E_s}.</math>

This definition is normalized so that <math>\gamma_c(A) = 1</math> for M-PAM or (''M''&times;''M'')-QAM. The UBE becomes

: <math>P_s(E) \approx K_s(A)Q\sqrt{3\gamma_c(A)SNR_\mathrm{norm}},</math>

where <math>K_s(A)</math> is the average number of nearest neighbors per two dimensions.

==See also==

*[[Channel capacity]]

*[[Eb/N0]]

==References==

[http://ocw.mit.edu MIT OpenCourseWare], 6.451 Principles of Digital Communication II, Lecture Notes sections 5.3, 5.5, 6.3, 6.4

[[Category: Error detection and correction]]

[[Category:Coding theory]]

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[[Category: Error detection and correction]]