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In [[coding theory]] and related engineering problems, '''coding gain''' 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 [[error correcting code]] (ECC).▼
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▲In [[coding theory]], [[telecommunications engineering]] and other related engineering problems, '''coding gain''' is the measure in the difference between the [[signal
==Example==
If the uncoded [[BPSK]] system in [[AWGN]] environment has a [[
==Power-limited regime==
In the ''power-limited regime'' (where the nominal [[spectral efficiency]] <math>\rho \le
: <math>\gamma_c(A) = \frac{d^2_{\min}(A)
This definition is normalized so that <math>\gamma_c(A) = 1</math> for 2-PAM or (2×2)-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>
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▼
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>
==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 >
: <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''
: <math>P_s(E) \approx K_s(A)Q\sqrt{3\gamma_c(A)SNR_\mathrm{norm}},</math>
==References==▼
where <math>K_s(A)</math> is the average number of nearest neighbors per two dimensions.
: MIT OpenCourseWare (http://ocw.mit.edu), 6.451 Principles of Digital Communication II, Lecture Notes sections 5.3, 5.5, 6.3▼
==See also==
[[Category: Coding theory]]▼
*[[Channel capacity]]
[[Category: Error detection and correction]]▼
*[[Eb/N0]]
▲==References==
▲
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