Coding gain

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).

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

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

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

${\displaystyle P_{b}(E)\approx K_{b}(A)Q\left({\sqrt {\frac {2\gamma _{c}(A)E_{b}}{N_{0}}}}\right),}${\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 ${\displaystyle C}${\displaystyle C} with parameters ${\displaystyle (n,k,d)}${\displaystyle (n,k,d)}, the nominal spectral efficiency is ${\displaystyle \rho =2k/n}${\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 ${\displaystyle P_{b}(E)\approx 10^{-5}}${\displaystyle P_{b}(E)\approx 10^{-5}} for Reed–Muller codes of length ${\displaystyle n\leq 64}${\displaystyle n\leq 64}:

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

${\displaystyle \gamma _{c}(A)={(2^{\rho }-1)d_{\min }^{2}(A) \over 6E_{s}}.}${\displaystyle \gamma _{c}(A)={(2^{\rho }-1)d_{\min }^{2}(A) \over 6E_{s}}.}

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

${\displaystyle P_{s}(E)\approx K_{s}(A)Q{\sqrt {3\gamma _{c}(A)SNR_{\mathrm {norm} }}},}${\displaystyle P_{s}(E)\approx K_{s}(A)Q{\sqrt {3\gamma _{c}(A)SNR_{\mathrm {norm} }}},}

where ${\displaystyle K_{s}(A)}${\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

• Coding theory
• Error detection and correction

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