Spatially sparse hybrid precoding

 

In the case of digital beamforming, each antenna element in a completely connected architecture must have its own radio frequency (RF) series. The analog phase shifter (PSs) applies the analog beamformer; all of its elements have the same amplitude but different phases. The goal is to optimise the overall throughput or sum rate R (A, D) obtained over Gaussian signalling on MMwave channels by designing (A, D)

 

 

The related sum rate optimization problem looks like this:

 

 

Here, set F  consists all possible analog beamformers (size of N_t X N_RF^t matrices) with constant-magnitude entries.

Now for hybrid architecture we will find singular value decomposition (SVD) of channel matrix to find the stronger eigen values and we will only allocate power accordingly to these paths to achieve low overhead in hybrid architecture (Appendix B). For example we will further divide the eigen value matrix (Σ)  into two parts (Eqn 6.5) where Σ₁ is a N_s X N_s matrix; where N_s indicates rank of matrix or how many simultaneous data streams are available between BS and MS

 

 

On the other hand optimal unconstrained precoding is called fully digital precoding which is based on the singularvalue decomposition (SVD) of the channel matrix where we allocate power to each eigen path. Hybrid beamforming employs less number of RF chain in case of multiple-stream transmission as well as concept of OMP (orthogonal matching pursuit) is used.