Thank you for the interesting discussion! It really provides clear explanations on some otherwise confusing explanations.

]]>Actually, I think there is no “concrete and official” way to calculate SINR in case of imperfect CSI. Instead, we just rely on some reasonable assumptions (like the assumption in Dr. Ngo’s paper or your paper). ]]>

http://ma-mimo.ellintech.se/2017/10/03/what-is-the-difference-between-beamforming-and-precoding/

If you have further questions related to this, I suggest that you submit a question to that blog post.

]]>This is yet another paper that should have been rejected due to using incorrect rate expressions.

]]>The above questions really did make me confused, now I understand clearly these two assumptions. But there is an other way of calculating the SINR which is used in many papers, such as [1]. In [1], the formula (5) is data detection using estimated channel but after that, the authors of [1] calculated the SINR as in (6), this is very weird to me because I think (6) is only for perfect CSI. Could you please explain that?

[1] “Soft Pilot Reuse and Multicell Block Diagonalization Precoding for Massive MIMO systems” – Xudong Zhu, Zhaocheng Wang.

]]>The capacity bound used in Ngo’s paper assumes that the MMSE channel estimates are used for both selecting the linear receiver and for data detection. This makes practical sense! In contrast, Alkhaled’s paper assumes that the estimates are used for selecting the linear receiver, but then in the data detection, the receiver suddenly has perfect CSI (otherwise (10) in Alkhaled’s paper will not be an achievable spectral efficiency). This weird assumption is probably an unintentional mistake, caused by guessing what the SINR would look like instead of using a rigorous capacity bound. I would have rejected this paper if I was the reviewer.

In summary, you should use the bound from Ngo’s paper.

]]>1. In Dr. Ngo’s paper, he claims ”BS treats estimated channel as true channel” and assumes that the desired signal is v_jk^H*g_jk*x_jk (g is estimated channel, v is a linear receiver derived from g, h is the true channel, x is the data symbol), and the difference between estimated channel and true channel (e_jk = g_jk – h_jjk) is treated as noise: e_jk*x_jk. The instantaneous SINR (SINR in each small-scale channel realization) is calculated as (36) in Dr. Ngo’s paper.

2. In some other papers, I saw they consider the desired signal is v_jk^H*h_jjk*x_jk. Instantaneous SINR is calculated as formula (9) in paper ”Adaptive pilot allocation algogrithm for Pilot Contamination Mitigation in TDD massive MIMO system” by Makram Alkhaled, Emad Alsusa.

Are they both legitimate assumptions and I can use any of them in my simulation?

]]>In the simulation, how can I compare these bounds with the true value of SINR? For example: I generate 10000 realizations of the small scale matrix (suppose that users’ positions do not change so large-scale coefficients do not change). I can calculate the lower bound of SINR user as your formula (12), but in each realization, what is the correct formula to calculate the true value of SINR (not lower bound) so that I can compare the lower bound and the true SINR (and check if the lower is tigh or loose). ]]>

You can compare this with Theorem 4.1 in my new book, which is a tighter bound on the capacity than Theorem 4.4 in my book. However, the benefit with Theorem 4.4 and (12) in my paper is that you can compute the SINR expression in closed form when using MR combining.

2. You can use any of the two capacity bounds, but not mix them. If you have good channel hardening in your system, the difference between the two bounds will be small. But if the hardening is weak, then the difference can be large. This is exemplified in Figure 4.11 of my new book.

]]>– paper “Energy and spectral efficiency of very large multi-user MIMO systems-N.Q. Ngo” they consider the estimated channel is true channel and they give formula (36).

– paper “Massive MIMO for Maximal Spectral Efficiency: How Many Users and Pilots Should Be Allocated?” is your paper, you consider noise as the worst-case Gaussian distributed in the decoding and give the formula (12).

So I have some question:

1. Are they equivalent?

2. In simulation, can I calculate the effective SINR of a user similar with the formula (12) in your paper but the Expectation respect to channel realization is outside the whole formula (12) as: SINR = E {(|g_jk^H*h_jjk|^2)/(∑_(l,m) g_jk^H*h_jlm -|g_jk^H*h_jjk|^2 + |g_jk|^2)}?