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Coding gain
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<h2 id="example">Example</h2> <p>If the uncoded <a href="/facts/BPSK/TFnpYe64">BPSK</a> system in <a href="/facts/AWGN/IE4Gof7T">AWGN</a> environment has a <a href="/facts/Bit_error_rate/ovAAPFjV">bit error rate</a> (BER) of 10−2 at the SNR level 4 <a href="/facts/Decibel/0LyxAPXh">dB</a>, and the corresponding coded (e.g., <a href="/facts/BCH_code/1KelJGxF">BCH</a>) system has the same BER at an SNR of 2.5 dB, then we say the <i>coding gain</i> = 4 dB − 2.5 dB = 1.5 dB, due to the code used (in this case BCH). </p> <h2 id="power-limited-regime">Power-limited regime</h2> <p>In the <i>power-limited regime</i> (where the nominal <a href="/facts/Spectral_efficiency/GaePAvv9">spectral efficiency</a> ρ ≤ 2 {\displaystyle \rho \leq 2} [b/2D or b/s/Hz], <i>i.e.</i> the domain of binary signaling), the effective coding gain γ e f f ( A ) {\displaystyle \gamma _{\mathrm {eff} }(A)} of a signal set A {\displaystyle A} at a given target error probability per bit P b ( E ) {\displaystyle P_{b}(E)} is defined as the difference in dB between the E b / N 0 {\displaystyle E_{b}/N_{0}} required to achieve the target P b ( E ) {\displaystyle P_{b}(E)} with A {\displaystyle A} and the E b / N 0 {\displaystyle E_{b}/N_{0}} required to achieve the target P b ( E ) {\displaystyle P_{b}(E)} with 2-<a href="/facts/Pulse-amplitude_modulation/Y30VmMJI">PAM</a> or (2×2)-<a href="/facts/Quadrature_amplitude_modulation/02j52SWG">QAM</a> (<i>i.e.</i> no coding). The nominal coding gain γ c ( A ) {\displaystyle \gamma _{c}(A)} is defined as </p> γ c ( A ) = d min 2 ( A ) 4 E b . {\displaystyle \gamma _{c}(A)={\frac {d_{\min }^{2}(A)}{4E_{b}}}.} <p>This definition is normalized so that γ 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 K b ( A ) {\displaystyle K_{b}(A)} is equal to one, the effective coding gain γ e f f ( A ) {\displaystyle \gamma _{\mathrm {eff} }(A)} is approximately equal to the nominal coding gain γ c ( A ) {\displaystyle \gamma _{c}(A)} . However, if K b ( A ) > 1 {\displaystyle K_{b}(A)>1} , the effective coding gain γ e f f ( A ) {\displaystyle \gamma _{\mathrm {eff} }(A)} is less than the nominal coding gain γ c ( A ) {\displaystyle \gamma _{c}(A)} by an amount which depends on the steepness of the P b ( E ) {\displaystyle P_{b}(E)} <i>vs.</i> E b / N 0 {\displaystyle E_{b}/N_{0}} curve at the target P b ( E ) {\displaystyle P_{b}(E)} . This curve can be plotted using the <a href="/facts/Union_bound/HDRr2yUH">union bound</a> estimate (UBE) </p> P b ( E ) ≈ K b ( A ) Q ( 2 γ c ( A ) E b N 0 ) , {\displaystyle P_{b}(E)\approx K_{b}(A)Q\left({\sqrt {\frac {2\gamma _{c}(A)E_{b}}{N_{0}}}}\right),} <p>where <i>Q</i> is the <a href="/facts/Error_function/Ca2H7pGr">Gaussian probability-of-error function</a>. </p><p>For the special case of a binary <a href="/facts/Linear_block_code/QVuqpGga">linear block code</a> C {\displaystyle C} with parameters ( n , k , d ) {\displaystyle (n,k,d)} , the nominal spectral efficiency is ρ = 2 k / n {\displaystyle \rho =2k/n} and the nominal coding gain is <i>kd</i>/<i>n</i>. </p> <h2 id="example">Example</h2> <p>The table below lists the nominal spectral efficiency, nominal coding gain and effective coding gain at P b ( E ) ≈ 10 − 5 {\displaystyle P_{b}(E)\approx 10^{-5}} for <a href="/facts/Reed%E2%80%93Muller_code/HUKw1xrV">Reed–Muller codes</a> of length n ≤ 64 {\displaystyle n\leq 64} : </p> <table><tbody><tr><th>Code</th><th> ρ {\displaystyle \rho } </th><th> γ c {\displaystyle \gamma _{c}} </th><th> γ c {\displaystyle \gamma _{c}} (dB)</th><th> K b {\displaystyle K_{b}} </th><th> γ e f f {\displaystyle \gamma _{\mathrm {eff} }} (dB)</th></tr><tr><td>[8,7,2]</td><td>1.75</td><td>7/4</td><td>2.43</td><td>4</td><td>2.0</td></tr><tr><td>[8,4,4]</td><td>1.0</td><td>2</td><td>3.01</td><td>4</td><td>2.6</td></tr><tr><td>[16,15,2]</td><td>1.88</td><td>15/8</td><td>2.73</td><td>8</td><td>2.1</td></tr><tr><td>[16,11,4]</td><td>1.38</td><td>11/4</td><td>4.39</td><td>13</td><td>3.7</td></tr><tr><td>[16,5,8]</td><td>0.63</td><td>5/2</td><td>3.98</td><td>6</td><td>3.5</td></tr><tr><td>[32,31,2]</td><td>1.94</td><td>31/16</td><td>2.87</td><td>16</td><td>2.1</td></tr><tr><td>[32,26,4]</td><td>1.63</td><td>13/4</td><td>5.12</td><td>48</td><td>4.0</td></tr><tr><td>[32,16,8]</td><td>1.00</td><td>4</td><td>6.02</td><td>39</td><td>4.9</td></tr><tr><td>[32,6,16]</td><td>0.37</td><td>3</td><td>4.77</td><td>10</td><td>4.2</td></tr><tr><td>[64,63,2]</td><td>1.97</td><td>63/32</td><td>2.94</td><td>32</td><td>1.9</td></tr><tr><td>[64,57,4]</td><td>1.78</td><td>57/16</td><td>5.52</td><td>183</td><td>4.0</td></tr><tr><td>[64,42,8]</td><td>1.31</td><td>21/4</td><td>7.20</td><td>266</td><td>5.6</td></tr><tr><td>[64,22,16]</td><td>0.69</td><td>11/2</td><td>7.40</td><td>118</td><td>6.0</td></tr><tr><td>[64,7,32]</td><td>0.22</td><td>7/2</td><td>5.44</td><td>18</td><td>4.6</td></tr></tbody></table> <h2 id="bandwidth-limited-regime">Bandwidth-limited regime</h2> <p>In the <i>bandwidth-limited regime</i> ( ρ > 2 b / 2 D {\displaystyle \rho >2~b/2D} , <i>i.e.</i> the domain of non-binary signaling), the effective coding gain γ e f f ( A ) {\displaystyle \gamma _{\mathrm {eff} }(A)} of a signal set A {\displaystyle A} at a given target error rate P s ( E ) {\displaystyle P_{s}(E)} is defined as the difference in dB between the S N R n o r m {\displaystyle SNR_{\mathrm {norm} }} required to achieve the target P s ( E ) {\displaystyle P_{s}(E)} with A {\displaystyle A} and the S N R n o r m {\displaystyle SNR_{\mathrm {norm} }} required to achieve the target P s ( E ) {\displaystyle P_{s}(E)} with M-<a href="/facts/Pulse-amplitude_modulation/Y30VmMJI">PAM</a> or (M×M)-<a href="/facts/Quadrature_amplitude_modulation/02j52SWG">QAM</a> (<i>i.e.</i> no coding). The nominal coding gain γ c ( A ) {\displaystyle \gamma _{c}(A)} is defined as </p> γ c ( A ) = ( 2 ρ − 1 ) d min 2 ( A ) 6 E s . {\displaystyle \gamma _{c}(A)={(2^{\rho }-1)d_{\min }^{2}(A) \over 6E_{s}}.} <p>This definition is normalized so that γ c ( A ) = 1 {\displaystyle \gamma _{c}(A)=1} for M-PAM or (<i>M</i>×<i>M</i>)-QAM. The UBE becomes </p> P s ( E ) ≈ K s ( A ) Q 3 γ c ( A ) S N R n o r m , {\displaystyle P_{s}(E)\approx K_{s}(A)Q{\sqrt {3\gamma _{c}(A)SNR_{\mathrm {norm} }}},} <p>where K s ( A ) {\displaystyle K_{s}(A)} is the average number of nearest neighbors per two dimensions. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Channel_capacity/55wJYsS6">Channel capacity</a></li> <li><a href="/facts/Eb/N0/DU5sMhd6">Eb/N0</a></li></ul> <p><a href="http://ocw.mit.edu">MIT OpenCourseWare</a>, 6.451 Principles of Digital Communication II, Lecture Notes sections 5.3, 5.5, 6.3, 6.4 </p>