Unsymmetrical boundary shaping with application to spectral shaping

We discuss the selection of the boundary of a signal constellation which has nonequal values of average energy along different dimensions. This nonequal energy allocation, in conjunction with a nondiagonal modulating matrix, is used to shape the corresponding power spectrum. The objective is to optimize the "rate versus energy trade-off" subject to some constraints on the corresponding power spectrum. In a rectangular constellation, the nonequal energy allocation is achieved by using different number of points along different dimensions. It is shown that the optimization procedure reduces to maximizing the determinant of an autocorrelation matrix subject to some linear constraints on its elements. In a shaped constellation, the number of points along the one-D (one-dimensional) subspaces are the same and the nonequal energy allocation is achieved by selecting an appropriate boundary in a higher dimensional space.