Equations related to Spectral Efficiency in Digital Beamforming

 

 

 

Fig 1: Digital Beamforming 

 

The main working principle between the beamforming (or analog beamforming) is to maximize the signal strength in a particular direction towards the receiver. For example, you can steer the antenna manually towards the transmitter to maximize the signal strength, like dish antennas. However, the approach could be more practical for mobile communications. With the help of a phase shifter, we do it electronically. On the other hand, for example, a dish antenna has an aperture that adds some gain to the received signal.Similarly, placing many antennas at a particular space instant creates a beam in a specific direction, minimizing signal strength in the rest of the directions. Here, a combination of antennas creates virtual apertures. On the transmitter side, it transmits a more robust signal toward the receiver. Oppositely, it makes a virtual aperture at the receiver and captures the signal more efficiently.

The main advantage of beamforming is maximizing the signal strength by increasing the number of antenna elements, not the total power.

In the case of analog beamforming, there is only one transmitter and one receiver, and only a single data stream is possible between them.

In the case of digital beamforming, multiple data streams are possible between transmitter and receiver or receivers. The number of users may be one or many.

If there is only one number of users, then multiple data streams between the transmitter and receiver will increase data rates through spatial multiplexing. Similarly, if multiple users exist, each data stream may be allocated to a single user. The various users communicate with the transmitter by deploying spatial multiplexing.

 

For Single User Digital Beamforming

The received signal vector y at receiver side,

                                                                                 y= √ρHDs + n

                                               Here, D = digital beamforming matrix

                                                         ρ = average received power

                                                         H =channel matrix ( N_r X N_t)

                                                         n = additive white Gaussian noise vector

                                                     N_r and Nt are the number of antenna elements at the receiver and  transmitter side respectively.

 You may think this beamforming equation is the same as analog beamforming, but this is not. Here, the number of independent data streams between transmitter and receiver is min(Nr, Nt). If there are two transmitting antennas and three receiver antennas, then two independent data streams are possible between transmitter and receiver to communicate simultaneously.

 

For Multi-user Digital Beamforming

D= [D₁, D₂, …, D_U], & D_u denotes the u th user, a digital precoder (size of N_BS X N_MS)

N_BS and N_MS are the number of antenna elements at the transmitter and receiver side respectively 

Now cancel interference at u^th user due to other users, we need to design the baseband precoder in such a way that H_uD_n for nǂ u should be zero at the u th MS. Therefore, H_uD_n =0 cancels interferences at uth MS.