A universal lattice code decoder for fading channels

1639 the average frequency distribution H avg (s) := (1=N) N i=1 H(s; i): The behavior of the distributions for different values of K is discussed in Fig. 4. For M = 9 and K = 12; 16; 20; 24 the average frequency distribution Havg(s) (solid lines) and the approximation Happ(s) (dashed lines) are compared to the Maxwell–Boltzmann distribution H M-B (s) (dash-dotted lines). Here, the parameter is chosen so that the entropy is equal to K=8. Even for low K the approximation H app (s) is very close to the true average frequency distribution H avg (s). The approximation improves as K increases. Unfortunately, the Maxwell–Boltzmann distribution HM-B(s) does not provide a good estimate of H app (s). Shells with low index occur less often than expected from the optimal entropy–power tradeoff. Finally, in Table II the average energy 2 a of the signal points in V.34 are summarized. For a symbol rate of 3200 Hz, the true average energy 2 a (cf. (2)), the approximate energy 2 a; app based on H app (s), and the energy 2 a; M-B derived from the Maxwell–Boltzmann distribution are given for all possible data rates and associated mapping parameters K, M, and q [5, Table 10, expanded]. The underlying signal constellation is specified in [5, Fig. 5]. Again, the exact calculation and the approximation are very close. Obviously, the energies derived from the Maxwell–Boltzmann distribution underestimate the actual energies as they are lower bounds. The approximation (13) provides much better results. IV. CONCLUSIONS In this correspondence, a simple but general method for the calculation of the frequencies of the shells in shell-mapping schemes was derived. As an example, the method was shown in detail for the shell-mapping scheme specified for the international telephone-line modem standard ITU Recommendation V.34. The method starts with partial histograms that give the number of occurrences of shells within all possible combinations of n-tuples of shells with some fixed total cost. These histograms can be calculated easily using the generating functions that are needed in the encoder in any case. Then, the shell-mapping encoder is run with a specific input, namely, the maximum K-tuple. To each step of the encoding procedure a partial histogram can be assigned. Summing up these parts yields the final histograms. Thus the calculation has approximately the same complexity as the mapping encoder itself. With the knowledge of the frequencies of shells, the …