I am trying to understand a text about beamforing :
Classic beamforming toy example. Single spatial stream, 2×2.
With one interferer, receiver alone can null it.
With two interferers, need to coordinate nulls in both receivers and transmitters. For example, the nulls for the transmitters to receivers are (1,2), (2,3), (3,1) and for the receivers to transmitters are (2,1), (3,2), and (1,3), thus using all six degrees of freedom to cancel the six interference paths.
With three interferers (four user pairs), we lack sufficient degrees of freedom to cancel interference. Thus, we resort to orthogonalization of channels.
What do we mean by nulling in this case? Are interference coming from the antennas inside the 2×2 system? is the case with two interferes a 3×3 system and the next one a 4×4 system? What is a user pair, is it an antenna in the MIMO setup? What is a degree of freedom in this case and why do we have only six of them?
Best Answer
Nulling is a RF technique for dealing with sources of interference. A boring example might be the noise coming off of a blender motor. When you null an interference source out, you identify the direction it comes from and then adjust the relative phases between the antennas such that signals from that particular direction interfere destructively, canceling out the noise.
The degrees of freedom are key because you only have so many knobs to twiddle. If you have a 2x2 system, you can adjust four phases, but what will matter for the purposes of nulling is the relationship between the phases: you have 3 relative phases that you can adjust. If you have one source of interference, its direction is well defined by an azimuth and elevation, so you have enough degrees of freedom to put a null at exactly that point.
In the case of two interference sources, you have to put 2 nulls into your directional antenna response. Each one has an azimuth and an elevation, so your problem has 4 degrees of freedom. Unfortunately, you only have 3 degrees of freedom in your antenna, so it is impossible for just the receiver to cancel out both signals. With a more advanced antenna pattern, you may be able to cancel out more.
I'm not familiar with cancellation where both transmitter and receiver cooperate to nullify multiple interference sources, so you'll need to look at the literature to better understand how that works. However, hopefully that is enough to get you started!