ation 6.43 becomes (6.67) Transmission Systems Architectures and Performances Optical signal to noise ratio (dB, 10 log10(SNo)) Q factor (dB, 20 log 10 (Q)) FIGURE 6.21 Relationship between optical SN ratio and Q factor for 2. Let us assume that every in-line amplifier compensates exactly the loss of the span after it and that the booster compensates the first span while the preamplifier compensates the end terminal losses. For the sake of simplicity, in a first approximation, let us assume that the noise factors are all equal and independent from the gain of the amplifiers. Indicating with L the length of the vth span, Equation 6.67 can be rewritten as (6.68) If the span length is variable it is a relevant problem to find the optimum span distribution, that is, the span set that minimizes the overall ASE noise. This problem can be solved easily with the method of Lagrange multipliers, finding the minimum of (6.68) with the condition N + + L = L L (6.69) L being the overall link length. Calling y the Lagrange multiplier the Lagrangian function is (6.70) and the derivative, with respect to the unknown span lengths are (6.71) with the condition (6.69). Telecommunication Networks The only way in which all the N relations (6.71) can be equal to zero is that all the spans have the same length. Thus, as it is intuitive, the condition of equal length spans is an extreme of the ASE function. The fact that it is a minimum and not a maximum is readily demonstrated by applying to the equal span condition the smallest possible perturbation, that is, assuming that one span is AL longer than the others and, to fulfill condition (6.71), another is AL shorter. Calling SASE the noise with all the span equal and S'ASE the noise after the modification it is easily calculated (6.72) This quantity is always positive, demonstrating that the condition of equally long span is the optimum span distribution in the sense that it minimizes the noise power. The situation does not change if a dependence of the noise factor on the gain is cons