I wonder how 2x4 (2 Rx, 4 Tx antennas) MIMO (TM 3 and 4) may actually work:
Generally the transmission - reception chain can be modelled with the following matrix multiplications:
r = W+ E H W s
s ... transmit symbol vector
W ... precoding matrix
H ... channel matrix
E ... equalizer matrix
W+ ... pseudoinverse of W
r ... received symbol vector
The equalizer matrix E suffers from noise and interference, but, in the ideal case, equals the pseudo-inverse of H, i.e. E = H+.
In the case of 2x2, 4x2 or 4x4 MIMO the product H+ H is (ideally) very close to identity, so things work fine.
But now consider 2x4: H is a 2x4 matrix, H+ is 4x2, so H+ H is 4x4. And it is very very far from identity. The simple reason being that calculating the pseudo-inverse of a 2x4 matrix is an under-determined problem. It is impossible to make H+ H close to identity for an arbitrary H in that case.
So based on the above model, it is mathematically impossible to do proper channel equalization for 2x4 MIMO.
How is it solved technically in real life then? Or does 2x4 simply not exist and is handled as 2x2? But what do I do with the remaining two Tx antennas then, how do I shrink H from 2x4 to 2x2 and what do I do with W (being 4x2)? And how would I model the potential Tx diversity gain? Or is it the wrong model?
Any help is very much appreciated!
In LTE, all of the MIMO options are required to have the same number of transmit and receive antennas; 2x4 is not a supported option.
4 Tx and 2 Rx mode : This is kind of replicates of 2Tx ,1Rx mode ( Transmit diversity) two times. Basically the advantage might be both increase in data rate ( equal to 2Tx,2Rx- Spatial Multiplexing ) and better SINR performance ( equal to 2Tx,1Rx -SFBC).
After consulting someone who actually develops LTE equipment we found that the above modelling is slightly wrong:
The equalizer does not try to invert H (which is intractable in case of H being 2x4), instead it inverts H W (which is 2x2 in this case). For data reception the UE does not care about single antenna links, all what matters are the layers, so the "precoded channel matrix" is what needs to be worked on.
So what happens at the UE is that it estimates H (using the Reference signals) and multiplies it with W (which is known by its PMI). This product is fed into the equalizer. The resulting equalizer matrix E is then used to "decode" the data (which was precoded at Tx side).