Hi,
I have studied that in massive MIMO systems as the number of BS antennas tends to infinity the channel hardens and also the channel vector of the users get orthogonal. Is this also true for the case were the number of BS antennas and the number of users increase simultaneously but there ratio is constant?
Does the channel between each transmit and receive antenna has to have a certain condition in order to the channel hardening and the orthogonality to hold? I mean for example, does the channel of each antenna has to be independent or can they be correlated? Does the channel of each user has to be independent? Does the channel has to have a Rayleigh distribution?
Thanks.
My answers:
1. Is this also true for the case were the number of BS antennas and the number of users increase simultaneously but there ratio is constant?
Yes. The generally important point is to increase the number of propagation paths that are highly uncorrelated.
2. Does the channel between each transmit and receive antenna has to have a certain condition in order to the channel hardening and the orthogonality to hold?
The fundamental concept here is that if you have enough antennas, you will be able to create many highly uncorrelated propagation paths. So, if you experience fading in some of the paths, chances are good that with many dozens of antennas, you will find a propagation path to each UE, which experiences minimal fading.
3. Does the channel of each user has to be independent?
Usually, when "massive MIMO" is discussed, we are talking about simple UE, with a single antenna. So if this is what you are asking, the answer is yes. If, instead, you are asking about UE with multiple antennas, then you would be looking for multiple highly uncorrelated propagation paths, to this UE.
4. Does the channel has to have a Rayleigh distribution?
Rayleigh fading is a typical and worst case example. But the answer to your specific question is that if you experience channel hardening, it means you have found paths with minimal Rayleigh fading. If you can't find such paths, then you may not be seeing channel hardening.
Channel hardening means that channels now appear deterministic.
http://ma-mimo.ellintech.se/2017/01/25/channel-hardening-makes-fading-channels-behave-as-deterministic/
This paper has enough math to satisfy the most hard core, as it looks to create a more optimal channel estimator.
https://arxiv.org/pdf/1312.1004.pdf
Thanks for the answers. One final question, does the channel of each BS antenna has to be independent in order to the channel hardening and the orthogonality hold? or can the antennas at the BS be correlated?
The answer to that appears to be more nuanced. The easier answer is yes, that better results are achieved with low spatial correlation among the BS antennas:
http://www.radio-electronics.com/info/antennas/mimo/large-massive-mimo-systems.php
But then some have suggested that the answer depends on your channel estimation strategy:
https://arxiv.org/pdf/1312.1004.pdf
The above argues that you can still get channel hardening, assuming certain estimation strategies are used, even if BS antenna spatial correlation is non-zero (Section II):
This theorem implies that although the nonzero spatial correlation does cause the increase of variance of N0, the channel hardening effect still exist and N0 is asymptotically diminishing provided that Assumption 1 holds.
The channels from the different antennas at the base station can be correlated. Channel hardening then occurs under two technical conditions on the channel covariance matrices. These are listed as Assumption 1 in the following paper (but can be found elsewhere):
https://arxiv.org/pdf/1611.09152.pdf
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