2. Yes, this is actually what multi-user MIMO (and Massive MIMO) is about.

I recommend you to read my book “Massive MIMO Networks” (you can download it from massivemimobook.com) for further details on how this is done.

]]>Thanks for your comprehensive explanation.

1- Assuming using digital beamforming, is it possible that an array antenna generates a single beam direction in different frequencies (Like different sub-carriers in OFDM) at the same time? if so, how ?

2- The same question but for multiple beams. I mean, is it possible that an array antenna generates multiple beams in different frequencies at the same time?

Regards,

Erfan

2. Constraint 1 means \sum_i Q_ii <= constant so you can assign the power in any way you like between the diagonal elements in Q. In contrast, Constraint 3 means Q_11 <= constant, Q_22 <= constant, ..., so there is a strict upper limit for each of the diagonal values. One cannot "borrow" power from one antenna and give it to another.

]]>1. I still wonder in some papers, they said 2 is for independent users (no cooperation, no sharing information) but in 3, they can “cooperate with each other in terms of signaling”?

2. Moreover, can you explain in more detail mathematically? why in 1 they can share between each other but not in 2, 3? ]]>

1: The total power may not be larger than P, but it can be distributed arbitrarily between the antennas. Note that trace(Q) = \sum_i Q_ii.

2 and 3: These are essentially the same thing and put a constraint on the maximum power for each individual antenna.

]]>1. trace (Q) <= P.

2. Q <= P (semidefinite matrix inequality, P is diagonal matrix with diagonal elements i_th are power buget for i_th user).

3. Q_ii <= P_i.

where P is the total power budget of all users, P_i is the maximum budget for i_th user and Q is the correlation matrix between all users. Q = E[x*x^H]. (x is user signal vector). ]]>

Thanks for the detailed explanation.

Regarding the second answer ‘precoding is equivalent to digital beamforming’, I think it’s not accurate in some cases. I would say from standard perspective, precoding describes the mapping between data layers and ‘logical’ antenna port. When the number of ‘logical’ antenna port is smaller than the number of TXRUs (antenna port to TXRU is not 1-to-1 mapping), precoding is not equivalent to digital beamforming. There is another mapping between the ‘logical’ antenna port and the TXRUs. (I am not sure what to call this mapping relationship) What’s your thinking?

2. The digital part in hybrid beamforming can be used to get different beams in different parts of the band, as long as the beams are formed as linear combinations of the beams created in the analog domain.

3. Correct.

4. The terminology can be confusing since “precoding” and “spatial multiplexing” are terms being used in both single-user and multi-user communications. TM6 uses precoding/beamforming to serve a single user with a single data stream.

]]>First off: many thanks for the clarity with which you explain these issues! If you allow, I have a few questions/double-checks to further enhance my understanding:

1. You write ‘what I advocate is a fully digital implementation’. Despite the optimality in terms of achievable performacnce, from other sources I learned that pure digital beamforming would be too expensive in terms of required components and also be too power-hungry, which is why hybrid digital/analog solutions are recommended elsewhere. I’d appreciate you comments on this view.

2. Does such a hybrid solution still suffer from the undesirable characteristic of analog beamforming that you cannot direct beams differently in different parts of the frequency band/carrier? Would the key advantage then be that the hybrid approach supports single-user spatial multiplexing, while pure analog beamforming does not?

3. Regarding the comment that was made by Dr. Shafin (‘that precoding can be viewed as beamforming in the eigendirection of the channel rather than in any physical angular direction’), just to double-check: in case of a perfect LOS channel, the mentioned eigendirection is (I hope) actually identical to the physical angular direction towards the receiver location, right?

4. You state ‘precoding refers to the superposition of multiple beams for spatial multiplexing of several data streams’. In my understanding, transmission mode TM6 in LTE uses precoding based on PMI feedback for single-codeword/data stream transmission, so I suppose precoding is then not equivalent to spatial multiplexing, or am I misunderstanding?

Thanks again, your explanations are much appreciated.

Regards,

Remco

]]>However, the typical digital beamforming in LTE consists of a codebook of precoding vectors, which I believe corresponds to different angular directions. The PMI is then used to select the most suitable precoding vector in the codebook. This approach is ok for single-user precoding, but not efficient for multi-user precoding, you need to fine-tune the precoding to reduce interference. In canonical Massive MIMO, this is achieved by sending uplink pilots to estimate the full channels, so that you can do zero-forcing or similar methods.

]]>I’m not sure what you mean with “eigendirections” since this word has many different meanings in the literature, but it is not important here.

For more details, you can check out my article:

Emil Björnson, Mats Bengtsson, Björn Ottersten, “Optimal Multiuser Transmit Beamforming: A Difficult Problem with a Simple Solution Structure,” IEEE Signal Processing Magazine, vol. 31, no. 4, pp. 142-148, July 2014.

]]>Please, explain in more detail. ]]>

With digital beamforming you can create antenna patterns that are not achievable with analog beamforming. It will basically be a superposition of any number of analog beams, resulting in an antenna pattern that might not have a distinct direction. For example, it can be tailored to a user channel to provide a strong signal in a non-line-of-sight situation (with many multipath components in different directions). In addition to that, you can assign different antenna patterns to different parts of the frequency spectrum, while analog beamforming must assign the same phases to the entire frequency band.

]]>If there is only a small number of resolvable paths, I think that the optimal precoder will be a linear combination of the steering vectors for the angles of arrivals of these paths. If the angles are all similar, the resulting beam will be almost as the steering vector of one of the angles. If the angles are very different, none of the steering vectors will be a good match.

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