In this problem, revisit Example, which deals with coded binary antipodal signaling over an additive white Gaussian noise (AWGN) channel. Starting with (5.105) and the underlying theory, develop a software package for computing the minimum Eb/N0 required for a given bit error rate, where Eb is the signal energy per bit, and N0/2 is the noise spectral density. Hence, compute the results plotted in parts a and b of Figure 5.16.
As mentioned in Example 5.10, the computation of the mutual information between the channel input and channel output is well approximated using Monte Carlo integration. To explain how this method works, consider a function g(y) that is difficult to sample randomly, which is indeed the case for the problem at hand. (For this problem, the function g(y) represents the complicated integrand in the formula for the differential entropy of the channel output.) For the computation, proceed as follows:
• Find an area A that includes the region of interest and that is easily sampled.
• Choose N points, uniformly randomly inside the area A. Then the Monte Carlo integration theorem states that the integral of the function g(y) with respect to y is approximately equal to the area A multiplied by the fraction of points that reside below the curve of g, as illustrated in Figure P5.41. The accuracy of the approximation improves with increasing N.
