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

Power-limited regime

In the power-limited regime (where the nominal spectral efficiency [b/2D or b/s/Hz], i.e. with 2-PAM or (22)-QAM (i.e. is defined as

for 2-PAM or (2 vs.. This curve can be plotted using the union bound estimate (UBE)

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

For the special case of a binary linear block code and the nominal coding gain iskd/n.

Example

for Reed–Muller codes:

Bandwidth-limited regime

In the bandwidth-limited regime, i.e. with M-PAM or (MM)-QAM (i.e. is defined as

for M-PAM or (MM)-QAM. The UBE becomes

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