ns: (11.31) From the above evaluations, it is seen that the series BER, is very rapidly converging, and that only the first contribution BER5 is actually significant. This means that for the corrected code, the primary source of errors is the "extra" bit error out of five error events, which cannot be corrected by the code. It is seen, however, that the ECC has reduced the BER from p = 10-4 to p = 2.5 X 10-12, which represents quite a substantial improvement! Figure 11.6 shows plots of the uncorrected and corrected BER, using in the latter case the definition of BER5 in Eq. (11.31). which illustrates the BER improvement due to ECC. Considering ON-OFF keying, and the above example, the uncorrected and corrected BER correspond to Qunc 2 3.73 (Qunc 11.4 dB) and Qcorr 2 6.91 (Qcorr = 16.8 dB). One can then define the coding gain as the decibel ratio y = 20 or the decibel difference y = Qcorr - Qunc, which, in this example, yields y = 20 log10[6.91/3.73] = 16.4 - 11.8 = 5.4 dB. This coding gain is indicated in Fig. 11.6 through the horizontal arrow. As a matter of fact, a 5.4 dB coding gain means that an identical BER can be achieved through ECC when the signal-to-noise ratio or SNR is decreased by 5.4 dB, as the figure illustrates. As mentioned in the previous section, the 11.5 Corrected bit-error-rate improvement Coding gain Q2 (dB) Figure 11.6 Uncorrected BER (open circles) and corrected BER (closed circles) as a function of Q2 , showing BER improvement from 10-4 to 2.5 X 10-12, , and corresponding coding gain of 5.4 dB. Uncorrected RS(255,239) RS(255,239) + RS(255,239) RS(255,239) + RS(255,223) Figure 11.7 Measured experimental data for uncorrected BER (open circle) and corrected BERs using RS(255, 239) Reed-Solomon ECC (dashed line), and two concatenations RS(255, 239) + RS(255,239) and RS(255,239) + RS(255,223). coding gain can be increased by the concatenation of two ECC codes, one serving as a first outer code, and the other as a second inner code. This feature is illustrated by the experimental data¹5 shown i