ley & Sons, 2004), Ch. 1; E. Desurvire, Erbium- Doped Fiber Amplifiers, Device and System Developments (New York: J. Wiley & Sons, 2002), Ch. 3. One can also define the received signal-to-noise ratio (SNR) as follows: SNR = Ps/PN, where Ps = (P1 + Po)/2 is the time-average signal power (assuming pseudo-random bit sequences), and PN is the total additive noise power. It can be shown (E. Desurvire, Erbium-Doped Fiber Amplifiers, Device and System Developments (New York: J. Wiley & Sons, 2002), Ch. 3.) that the SNR and the Q-factor are related through or, equivalently, 1++1+4(SNR) For high SNR, the second formula can be approximated by SNR~Q2/2. In optical communications, only half of the noise power is taken into account, because noise exists in two electromagnetic-field polarizations. In this case, the above definition reduces to SNR 2 Q2. 11.5 Corrected bit-error-rate Q2 (dB) Figure 11.5 Bit error rate (BER) as a function of the Q-factor expressed as Qab showing BER = 10-9 for Q2 = +15.5 dB. The probability of bit error (BER) is then given by the analytical expression and approx- imation: (11.27) where erfc(u) is the complementary error function, and where the approximation in the right-hand side is very accurate for Q > 2. It is easily checked that for Q = 6 we have BER = p(6)~~10-9, corresponding to one mistaken bit out of one billion received bits. One may also use the decibel definition for the Q-factor according to Q = 2010g10 Q, which gives Q2 = +15.5 dB for Q = 6, or BER = 10-9. Figure 11.5 shows a plot of BER = p(Q) = f(Q'p) = Long before the massive development and ubiquitous implementation of ECC in optical telecommunications systems, the values of BER = 10-9 or Q2 = +15.5 dB have been considered the de facto standard for "error-free" transmission. Having, thus, defined the bit error probability, p, we can now analyze to what extent a BER can be corrected, or reduced, by ECC, using a simple demonstration. 14 For simplicity, we shall assume that the communication channel is memoryless. This means that th