ion threshold. They are obtained by summing (i.e., integrating) all of the values of the probability density function in the regions of mistaken interpretation (6.15) (6.16) The probability density functions are from equation 6.13. erfc is known as the "comple- mentary error function," which is tabulated in Abramowitz and Stegan (1972) together with a comprehensive list of its properties. It results from the definite integral of a Gaussian function (6.17) By the use of equations 6.14, 6.15, and 6.16, we have (6.18) Chapter 6 Optical Fiber Transmission A key parameter in equation 6.18 is the decision threshold current, ID, because we are at liberty to set it to any value between I and I. The best procedure is to select a value that makes the BER as low as possible. In this case, the necessary value is found by differentiat- ing and equating to zero: d(BER)/d(ID) = 0. If we make use of the definition of erfc(u) in equation 6.17, we find that (6.19) In most circumstances, o and O1 are sufficiently close that the logarithmic term in equa- tion 6.19 can be ignored. Then we obtain 10-10-1-1-10=0 (6.20) equation 6.20 defines Q, which we identify as a quality factor. Large values of Q are obtained when the standard deviations of the signal currents, and O1, are small. The optimal value of the decision threshold, I, is found from equation 6.20 to be I = (o0 I 1 + I0) / + Equation 6.20 also leads to an expression for Q in terms of the means and standard devia- tions of the photocurrents: Q=(11-I0)/(00+01). (6.21) In the special case where there are equal noise currents on the 1s and Os, o = O1, and SO ID = (I1 + I0)/2. In other words, only in these particular circumstances is the best threshold current at the 50 percent level. It is straightforward to substitute into equation 6.18 to obtain an expression for the BER in terms of the Q value: (6.22) Sometimes the complementary error function is inconvenient to compute or we wish to use equation 6.18 to provide further analytical results. Fortunately, it can be approximated w