Define the boundary of the lattice

Question

Solve the following problem:

Let Z_n + ( ½ , ½ ,..., ½ ) denote the n-dimensional integer lattice shifted by ½, and let R be an n-dimensional hypercube centered at the origin with side length L which defines the boundary of this lattice. We further assume that n is even and L = 2^l is a power of 2; the number of bits per two dimensions is denoted by β, and we consider a constellation C based on the intersection of the shifted lattice Zn +(½ , ½ ,..., ½ ) and the boundary region R defined as an n-dimensional hypercube centered at the origin with side length L.

1. Show that β = 2^l + 2.

2. Show that for this constellation the figure of merit is approximated by

CFM(C)≈6/2^β

Note that this is equal to the CFM for a square QAM constellation.

3. Show the shaping gain of R is given by γs(R) = 1.