There will be some hardening, but much less than with co-located arrays. This is something that we analyzed in the following paper:

Zheng Chen, Emil Björnson, “Channel Hardening and Favorable Propagation in Cell-Free Massive MIMO with Stochastic Geometry,” IEEE Transactions on Communications, To appear. (https://arxiv.org/pdf/1710.00395)

]]>Hi,

I had the same observation during my MS research as well. I simulated a massive MIMO system with linear receiver in presence of an LMMSE estimator assuming i.i.d Rayleigh fading channels. What I observed was significantly high MSE with floor. The floor elevated with increase in the number of cells which is obviously due to increase in resulted pilot contamination. Despite this undesirable MSE, the linear receiver represented very good BER behavior thanks to the large M. I repeated the simulation with small M and observed that the BER curve encountered error floor. What I realized from these results was that despite the undesirable MSE behavior of LMMSE estimation, linear receiver still can operate efficiently under asymptotically favorable propagation condition due to large number of antennas. in other words, massive MIMO brings robustness. You can take advantage of linearity of estimator and receiver while guaranteeing robustness of the overall system.

These are good questions that I receive quite often. I therefore decided to reply to you by writing a new blog post:

http://ma-mimo.ellintech.se/2018/04/23/estimating-channels-under-channel-hardening/

]]>In the absence of favorable propagation, it might be better to serve one user at a time (OFDMA/FDMA/TDMA) than to spatially multiplex them.

]]>According to my understanding, in small cells mm-wave massive MIMO, since channels are mostly deterministic line-of-sight from the beginning, the channels can be thought of as already ‘hardened’ without the need for large M. The variance of ||h||^2/E[||h||^2] is almost 0 even with small number of BS antennas.

However, it seems that favourable propagation condition might not be met in this case. (h^H)*(h)/M would not go to 0 as M goes to infinity.

So what would be the implications on sum throughput/capacity of such systems where the channel hardening condition is already satisfied but favourable propagation condition is not met?

I read the paper, “No downlink pilots are needed in TDD Massive MIMO” and it explains what would happen in keyhole channels (without favourable propagation and without channel hardening), but I want to know what would happen in systems without favourable propagation but with channels already “hardened”?

I am asking this question in context of low mobility users since for high mobility users, it might be a different story.

]]>Yes! You will lose a bit in performance due to the imperfect channel estimation, but nothing fundamentally different will happen.

]]>If we talk about mm-wave Massive MIMO, where paths are mostly line-of-sight and assume that channel will be estimated by uplink pilots only (TDD mode and reciprocity), will Massive MIMO work in such conditions too? ]]>

If there is no fading, the channel is deterministic from the beginning and there is no need for channel hardening.

When you talk about stationarity, I think what you refer to are users that move around. The hardening only applies to small-scale fading effects and not variations in the large-scale fading effects (path-loss, shadowing).

]]>Each terminal is assigned an individual beamforming vector that is tailored to its channel, which can be viewed as directing a beam towards that terminal. The base station sends a superposition of the beamformed signals.

The better you know the channel vector to a terminal, the better you can direct a beam towards that terminal. This leads to a power gain. The more antennas you have, the more directive the beams is, which also leads to a power gain.

]]>Thank you for the clarification. I have another question about beamforming vector dimension. For each terminal, I have to assign a beamforming vector for each terminal or only one beamforming vector across all terminals. May I expect any kind of power gain or power control benefit by modelling channel?

]]>The terminology “channel after beamforming” was not used in this blog post, but it usually refers to the scalar channel that you get by taking the inner product between the channel vector and the beamforming vector.

When we talk about channel hardening, we are only considering one user channel and the beamforming selected to communicate over that channel.

The interfering channels will typically not harden.

]]>The SNR can be expressed as SNR = p*b/sigma^2, where p is the transmit power, b is the pathloss coefficient, and sigma^2 is the noise power. The noise power is not 1, but can be computed based on the thermal noise power level.

As you say, it is common that people normalize the noise power to 1 to reduce the amount notation i their papers. This means that you include the noise variance in the transmit power term or pathloss coefficient instead. For example, one can define the SNR as p*c/1=p*c, where c=b/sigma^2 is the normalized pathloss coefficient, and 1 is the normalized noise variance.

]]>I have another question regarding the noise power to calculate SNR of massive MIMO in LoS conditions.

Most current works on Massive MIMO I have come across model noise as a complex gaussian random variable with 0 mean and unit variance (in linear scale).

My question is why is noise power considered to be 1 in most works?

]]>Yes, Massive MIMO works well in both LoS and non-LoS.

]]>* Yes.

* Yes, if you with “transpose” mean the “conjugate transpose” (also known as “Hermitian transpose”). The fact that the diagonal elements converge to deterministic constants proves the channel hardening. The fact that the off-diagonal elements converges to zero proves the favorable propagation.

* The proof is based on that M->infinity and K is constant. A consequence from this is that M>>K (infinity is much larger than any constant…).

]]>* Another question is: with an uncorrelated Rayleigh fading channel matrix H (MxK) (with variance 1): transpose(H)*H/M = Identity matrix (KxK) if M goes to infinity. Does it means this channel is both favorable and hardens?

* I have seen in a lot of massive MIMO paper, they claim that transpose(H)*H/M = Identity matrix (KxK) if M goes to infinity and M>>K. But why do we need M>>K, because I saw in the paper “Energy and spectral efficiency of very large multiuser MIMO system”, they claim that a very large M is enough to get: transpose(H)*H/M = Identity matrix (KxK).

]]>I assumed uncorrelated channel coefficients to get a simple description of the channel hardening concept. For correlated fading, one can still analyze the variance of ||h||^2/E{||h||^2} and see how close to zero it is.

Correlated channels harden more slowly, due to the eigenvalue variations in the covariance matrix. But if the two conditions in Assumption 1 in https://arxiv.org/pdf/1705.00538.pdf hold, we will get asymptotic channel hardening anyway.

In practice, I think you will have a finite-sized area where you can deploy antenna. I would then deploy as many antennas as I can with half-wavelength spacing. Adjacent antennas will have correlated channel coefficients, but it is better to have these antennas than to deploy fewer antennas with a larger antenna spacing.

]]>While reading the piece, I was wondering whether there are any requirements on the spacing of the antenna array elements wrt the carrier wave length? Being too close could mean higher correlation which would then break the assumption behind the probability calculation you made.

]]>What you observe is correct for independent Rayleigh fading channels. The channel to each BS antenna is estimated independently when there is no correlation between the channels, thus the MSE is not affected by the number of antennas.

If you would instead consider correlated Rayleigh fading, then the MSE per antenna will reduce as you add more antennas.

]]>Does the channel hardening of massive MIMO improves the error mean of MMSE estimator?

]]>