Bit-Interleaved Coded Modulation: Fundamentals, Analysis and Design

6.1.3 Bounding Techniques

The approach we used to evaluate the TP in 2D constellations can be extended to the general -dimensional case. However, the geometric considerations get more tedious as the dimension of the integration space increases. Therefore, simple approximations are often used, which may also be sufficient for the purpose of the analysis and/or design. A popular approach relies on finding an upper bound on the TP , which we do in the following.

For a given , and for any with , we have

6.72

which follows from the fact that, by using one inequality, we relax the constraint imposed by inequalities defining . The TP is therefore upper bounded, for any , as

6.73

Setting we obtain ...