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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

• Power-limited regime

• Example

• Bandwidth-limited regime

• See also

• References

If the uncoded BPSK system in AWGN environment has a bit error rate (BER) of 10−2 at the SNR level 4 dB, and the corresponding coded (e.g., BCH) system has the same BER at an SNR of 2.5 dB, then we say the coding gain = 4 dB − 2.5 dB = 1.5 dB, due to the code used (in this case BCH).

In the power-limited regime (where the nominal spectral efficiency ρ ρ ≤ ≤ 2 {\displaystyle \rho \leq 2} [b/2D or b/s/Hz], i.e. the domain of binary signaling), the effective coding gain γ γ e f f ( A ) {\displaystyle \gamma _{\mathrm {eff} }(A)} of a signal set A {\displaystyle A} at a given target error probability per bit P b ( E ) {\displaystyle P_{b}(E)} is defined as the difference in dB between the E b / N 0 {\displaystyle E_{b}/N_{0}} required to achieve the target P b ( E ) {\displaystyle P_{b}(E)} with A {\displaystyle A} and the E b / N 0 {\displaystyle E_{b}/N_{0}} required to achieve the target P b ( E ) {\displaystyle P_{b}(E)} with 2-PAM or (2×2)-QAM (i.e. no coding). The nominal coding gain γ γ c ( A ) {\displaystyle \gamma _{c}(A)} is defined as

γ γ c ( A ) = d min 2 ( A ) 4 E b . {\displaystyle \gamma _{c}(A)={\frac {d_{\min }^{2}(A)}{4E_{b}}}.}

This definition is normalized so that γ γ c ( A ) = 1 {\displaystyle \gamma _{c}(A)=1} for 2-PAM or (2×2)-QAM. If the average number of nearest neighbors per transmitted bit K b ( A ) {\displaystyle K_{b}(A)} is equal to one, the effective coding gain γ γ e f f ( A ) {\displaystyle \gamma _{\mathrm {eff} }(A)} is approximately equal to the nominal coding gain γ γ c ( A ) {\displaystyle \gamma _{c}(A)} . However, if K b ( A ) > 1 {\displaystyle K_{b}(A)>1} , the effective coding gain γ γ e f f ( A ) {\displaystyle \gamma _{\mathrm {eff} }(A)} is less than the nominal coding gain γ γ c ( A ) {\displaystyle \gamma _{c}(A)} by an amount which depends on the steepness of the P b ( E ) {\displaystyle P_{b}(E)} vs. E b / N 0 {\displaystyle E_{b}/N_{0}} curve at the target P b ( E ) {\displaystyle P_{b}(E)} . This curve can be plotted using the union bound estimate (UBE)

P b ( E ) ≈ ≈ K b ( A ) Q ( 2 γ γ c ( A ) E b N 0 ) , {\displaystyle P_{b}(E)\approx K_{b}(A)Q\left({\sqrt {\frac {2\gamma _{c}(A)E_{b}}{N_{0}}}}\right),}

where Q is the Gaussian probability-of-error function.

For the special case of a binary linear block code C {\displaystyle C} with parameters ( n , k , d ) {\displaystyle (n,k,d)} , the nominal spectral efficiency is ρ ρ = 2 k / n {\displaystyle \rho =2k/n} and the nominal coding gain is kd/n.

The table below lists the nominal spectral efficiency, nominal coding gain and effective coding gain at P b ( E ) ≈ ≈ 10 − − 5 {\displaystyle P_{b}(E)\approx 10^{-5}} for Reed–Muller codes of length n ≤ ≤ 64 {\displaystyle n\leq 64} :

| Code | ρ {\displaystyle \rho } | γ c {\displaystyle \gamma _{c}} | γ c {\displaystyle \gamma _{c}} (dB) | K b {\displaystyle K_{b}} | γ e f f {\displaystyle \gamma _{\mathrm {eff} }} (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 |

In the bandwidth-limited regime ( ρ ρ > 2 b / 2 D {\displaystyle \rho >2~b/2D} , i.e. the domain of non-binary signaling), the effective coding gain γ γ e f f ( A ) {\displaystyle \gamma _{\mathrm {eff} }(A)} of a signal set A {\displaystyle A} at a given target error rate P s ( E ) {\displaystyle P_{s}(E)} is defined as the difference in dB between the S N R n o r m {\displaystyle SNR_{\mathrm {norm} }} required to achieve the target P s ( E ) {\displaystyle P_{s}(E)} with A {\displaystyle A} and the S N R n o r m {\displaystyle SNR_{\mathrm {norm} }} required to achieve the target P s ( E ) {\displaystyle P_{s}(E)} with M-PAM or (M×M)-QAM (i.e. no coding). The nominal coding gain γ γ c ( A ) {\displaystyle \gamma _{c}(A)} is defined as

γ γ c ( A ) = ( 2 ρ ρ − − 1 ) d min 2 ( A ) 6 E s . {\displaystyle \gamma _{c}(A)={(2^{\rho }-1)d_{\min }^{2}(A) \over 6E_{s}}.}

This definition is normalized so that γ γ c ( A ) = 1 {\displaystyle \gamma _{c}(A)=1} for M-PAM or (M×M)-QAM. The UBE becomes

P s ( E ) ≈ ≈ K s ( A ) Q 3 γ γ c ( A ) S N R n o r m , {\displaystyle P_{s}(E)\approx K_{s}(A)Q{\sqrt {3\gamma _{c}(A)SNR_{\mathrm {norm} }}},}

where K s ( A ) {\displaystyle K_{s}(A)} is the average number of nearest neighbors per two dimensions.

• Channel capacity

• Eb/N0

MIT OpenCourseWare, 6.451 Principles of Digital Communication II, Lecture Notes sections 5.3, 5.5, 6.3, 6.4