4.6 Complex Orthogonal Designs

So far we have studied real STBCs that provide full diversity and simple ML decoding for any number of transmit antennas. The proposed structures only work for real constellations. Naturally, it is interesting to extend these schemes to complex signal constellations.

Definition 4.6.1

A complex orthogonal design of size N is an N N orthogonal matrix _N with complex entries x ₁, ? x ₁, x ₂, ? x ₂, , x _N, ? x _N, their conjugates , ? , , ? , , , ? , and multiples of these indeterminate variables by or ?j such that

(4.72)

In this section, we study the existence of complex orthogonal designs. The following construction shows the relationship between real and complex orthogonal designs.

Construction: Given a complex orthogonal design of size N, we replace each complex variable x _n = { x _n} + { x _n}j, 1 ? n ? N by the 2 2 real matrix

(4.73)

The resulting 2 N 2 N matrix is a real orthogonal design of size 2 N.

Note that the representation in (4.73) results in representing by

(4.74)

Since real orthogonal designs only exist for N = 2, 4, 8, the above construction shows that a complex orthogonal design can only exist for N = 2 or N = 4. For N