Coding gain: Difference between revisions

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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 -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==

If the uncoded [[BPSK]] system in [[AWGN]] environment has a [[Bitbit error rate]] (BER) of 10<mathsup>10^{-2}−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 level of 2.5dB5&nbsp;dB, then we say the ''coding gain'' = 4dB-{{nowrap|1=4 dB − 2.5dB5 dB = 1.5dB5 dB}}, due to the code used (in this case BCH).

==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) \over }{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\sqrtleft(\sqrt{\frac{2\gamma_c(A)E_b/}{N_0}}\right),</math>

where <math>''Q(\cdot)</math>'' denotesis 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-MullerReed–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

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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>

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.