Coding gain: Article and Information about Coding gain

Coding gain - Fuhz Articles
Visually Impaired Edition.

Coding gain article, this Fuhz page will hopefully provide the answers to the who, what where and why on the Coding gain topic.
At the bottom of the page we often provide links to external documents relating to Coding gain which may also help your research. Every effort is made to ensure the content on this page is as accurate and error free as possible, however whenever researching information that requires the utmost accuracy such as a term paper it is always best to cross reference facts with numerous sources.

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).

1 Example
2 Power-limited regime
3 Example
4 Bandwidth-limited regime
5 See also
6 References

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 level of 2.5dB, then we say the coding gain = 4dB-2.5dB = 1.5dB, due to the code used (in this case BCH).

Power-limited regime

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

$\gamma_c(A) = {d^2_\min(A) \over 4E_b}.$

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

$P_b(E) \approx K_b(A)Q\sqrt(2\gamma_c(A)E_b/N_0),$

where $Q(\cdot)$ denotes the Gaussian probability of error function.

For the special case of a binary linear block code $C$ with parameters $(n, k, d)$ , the nominal spectral efficiency is $ρ = 2 k / n$ 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 $P_b(E) \approx 10^{-5}$ for Reed-Muller codes of length $n \le 64$ :

Code	$ρ$	$γ c$	$γ c$ (dB)	$K b$	$γ e f f$ (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 ( $ρ > 2 b / 2 D$ , i.e. the domain of non-binary signaling), the effective coding gain $γ e f f (A)$ of a signal set $A$ at a given target error rate $P s (E)$ is defined as the difference in dB between the $S N R n o r m$ required to achieve the target $P s (E)$ with $A$ and the $S N R n o r m$ required to achieve the target $P s (E)$ with M-PAM or (M×M)-QAM (i.e. no coding). The nominal coding gain $γ c (A)$ is defined as

$\gamma_c(A) = {(2^\rho - 1)d^2_\min (A) \over 6E_s}.$

This definition is normalized so that $γ c (A) = 1$ for M-PAM or (M×M)-QAM. The UBE becomes

$P_s(E) \approx K_s(A)Q\sqrt(3\gamma_c(A)SNR_{norm}),$

where $K s (A)$ is the average number of nearest neighbors per two dimensions.

References

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

Return to Fuhz Home - This article covering Coding gain is enhanced for the visually impaired.

Article from Wikipedia. All text is available under the terms of the GNU Free Documentation License - Our Privacy Policy - Thanks Auto Blog Commenter

Contents

Example

Power-limited regime

Example

Bandwidth-limited regime

See also

References