Coding gain
Encyclopedia
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).
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).
[b/2D or b/s/Hz], i.e. the domain of binary signaling), the effective coding gain 
of a signal set 
at a given target error probability per bit 
is defined as the difference in dB between the 
required to achieve the target 
with 
and the 
required to achieve the target 
with 2-PAM
or (2×2)-QAM
(i.e. no coding). The nominal coding gain
is defined as
This definition is normalized so that
for 2-PAM or (2×2)-QAM. If the average number of nearest neighbors per transmitted bit 
is equal to one, the effective coding gain 
is approximately equal to the nominal coding gain 
. However, if 
, the effective coding gain 
is less than the nominal coding gain 
by an amount which depends on the steepness of the 
vs. 
curve at the target 
. This curve can be plotted using the union bound estimate (UBE)
where
denotes the Gaussian probability of error function
.
For the special case of a binary linear block code
with parameters 
, the nominal spectral efficiency is 
and the nominal coding gain is kd/n.
for Reed-Muller codes of length 
:
! Code !! 
!! 
!! 
(dB) !! 
!! 
(dB)
|-
| [8,7,2] > 1.75
7/4
2.43
4
| [8,4,4] 1.0
2
3.01
4
| [16,15,2] 1.88
15/8
2.73
8
| [16,11,4] 1.38
11/4
4.39
13
| [16,5,8] 0.63
5/2
3.98
6
| [32,31,2] 1.94
31/16
2.87
16
| [32,26,4] 1.63
13/4
5.12
48
| [32,16,8] 1.00
4
6.02
39
| [32,6,16] 0.37
3
4.77
10
| [64,63,2] 1.97
63/32
2.94
32
| [64,57,4] 1.78
57/16
5.52
183
| [64,42,8] 1.31
21/4
7.20
266
| [64,22,16] 0.69
11/2
7.40
118
| [64,7,32] 0.22
7/2
5.44
18
, i.e. the domain of non-binary signaling), the effective coding gain 
of a signal set 
at a given target error rate 
is defined as the difference in dB between the 
required to achieve the target 
with 
and the 
required to achieve the target 
with M-PAM
or (M×M)-QAM
(i.e. no coding). The nominal coding gain
is defined as
This definition is normalized so that
for M-PAM or (M×M)-QAM. The UBE becomes
where
is the average number of nearest neighbors per two dimensions.
Coding theory
Coding theory is the study of the properties of codes and their fitness for a specific application. Codes are used for data compression, cryptography, error-correction and more recently also for network coding...
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 at the SNR level 4dBDecibel
The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...
, and the corresponding coded (e.g., BCH
BCH code
In coding theory the BCH codes form a class of parameterised error-correcting codes which have been the subject of much academic attention in the last fifty years. BCH codes were invented in 1959 by Hocquenghem, and independently in 1960 by Bose and Ray-Chaudhuri...
) 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 efficiencySpectral efficiency
Spectral efficiency, spectrum efficiency or bandwidth efficiency refers to the information rate that can be transmitted over a given bandwidth in a specific communication system...
[b/2D or b/s/Hz], i.e. the domain of binary signaling), the effective coding gain of a signal set at a given target error probability per bit is defined as the difference in dB between the required to achieve the target with and the required to achieve the target with 2-PAMPulse-amplitude modulation
Pulse-amplitude modulation, acronym PAM, is a form of signal modulation where the message information is encoded in the amplitude of a series of signal pulses....
or (2×2)-QAM
Quadrature amplitude modulation
Quadrature amplitude modulation is both an analog and a digital modulation scheme. It conveys two analog message signals, or two digital bit streams, by changing the amplitudes of two carrier waves, using the amplitude-shift keying digital modulation scheme or amplitude modulation analog...
(i.e. no coding). The nominal coding gain
is defined as
This definition is normalized so that
for 2-PAM or (2×2)-QAM. If the average number of nearest neighbors per transmitted bit is equal to one, the effective coding gain is approximately equal to the nominal coding gain . However, if , the effective coding gain is less than the nominal coding gain by an amount which depends on the steepness of the vs. curve at the target . This curve can be plotted using the union bound estimate (UBE)
where
denotes the Gaussian probability of error functionError function
In mathematics, the error function is a special function of sigmoid shape which occurs in probability, statistics and partial differential equations...
.
For the special case of a binary linear block code
with parameters , the nominal spectral efficiency is 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 for Reed-Muller codes of length : !! !! (dB) !! !! (dB)|-
| [8,7,2] >
| [8,4,4]
| [16,15,2]
| [16,11,4]
| [16,5,8]
| [32,31,2]
| [32,26,4]
| [32,16,8]
| [32,6,16]
| [64,63,2]
| [64,57,4]
| [64,42,8]
| [64,22,16]
| [64,7,32]
Bandwidth-limited regime
In the bandwidth-limited regime (, i.e. the domain of non-binary signaling), the effective coding gain of a signal set at a given target error rate is defined as the difference in dB between the required to achieve the target with and the required to achieve the target with M-PAMPulse-amplitude modulation
Pulse-amplitude modulation, acronym PAM, is a form of signal modulation where the message information is encoded in the amplitude of a series of signal pulses....
or (M×M)-QAM
Quadrature amplitude modulation
Quadrature amplitude modulation is both an analog and a digital modulation scheme. It conveys two analog message signals, or two digital bit streams, by changing the amplitudes of two carrier waves, using the amplitude-shift keying digital modulation scheme or amplitude modulation analog...
(i.e. no coding). The nominal coding gain
is defined as
This definition is normalized so that
for M-PAM or (M×M)-QAM. The UBE becomes
where
is the average number of nearest neighbors per two dimensions.