Coding gain

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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).

Example[]

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).

Power-limited regime[]

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

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

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

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

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

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

Example[]

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

Code	${\displaystyle \rho }$	${\displaystyle \gamma _{c}}$	${\displaystyle \gamma _{c}}$ (dB)	${\displaystyle K_{b}}$	${\displaystyle \gamma _{\mathrm {eff} }}$ (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

Bandwidth-limited regime[]

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

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

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

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

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

See also[]

• Channel capacity
• Eb/N0

References[]

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