Coding gain: Difference between revisions

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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-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).

==Example==

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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]].

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==Bandwidth-limited regime==

In the ''bandwidth-limited regime'' (<math>\rho > 2b2~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>

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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.