Z over and call it Zj,k It is a random variable, due to the presence of the noise, and it can be defined as the following probability (9.51) where Cj,k is a threshold fixed on IIIIIk Zi,k is the curve coordinate along p(Zj,k/II) is the probability density function of the projection Zj,k Let us optimize the threshold Cj,k to minimize the probability P(j,k) that is given by P(j,k) = (9.52) Once optimized the threshold, let us consider, for any k=1,2,...,M and different from j, the plane orthogonal to the segment and passing by Cj.k. In this way, M - 1 planes are indi- viduated. The optimum decision zone is the minimum volume individuated by such planes. As an example, the optimum decision zones for 4-QAM are the four quadrature plane quadrants, shown in Figure 9.38. Once the decision zones have been optimized, the decision procedure leads to the fol- lowing equation for the symbol error probability: (9.53) where Z indicates the points representing in the constellation space the received field Ck indicates the constellation kth point p(Z/Ck) is the probability density function that Z is received when Ck is transmitted Integral in dZ is a volume integral into the constellation space Telecommunication Networks 4-QAM If this signal is received, the estimate transmitted symbol is 11 If this signal is received, the estimate transmitted symbol is 00 FIGURE 9.38 Hard decision for 4-QAM. Several techniques have been devised to manage integrals like that in Equation 9.53, com- prising an n-dimensional extension of the Saddle point approximation [61]. However, since we will deal in any case with low symbol error probabilities, the easier way to go is to evaluate the integral (9.53) using the so-called union bound. The union bound is an asymptotically exact approximation of an integral in the form (9.53) that exploits the definition of the optimum decision zones to approximate the integral as follows (9.54) Equation 9.54 shows as the union bound reduces the decision process to a set of binary decisions along the direction conn