Calculation of shell frequency distributions obtained with shell-mapping schemes

In order to calculate the transmit power in shell-mapping-based transmission schemes, the frequencies of the shells are required. In this correspondence, a simple but general method for the calculation of these frequencies is derived. The method has approximately the same complexity as the shell-mapping encoder. As an example, the method is shown in detail for the shell-mapping scheme specified for the international telephone-line V.34 modem standard. Moreover, a very simple approximation is given which is tight for large constellations.