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## MAC layer design for mmWave massive MIMO

G. Lee, Y. Sung, inmmWave Massive MIMO, 2017

### 10.4User Scheduling in Massive MIMO

As seen in the previous section, the sum rate of RBF and SUS scales with the same rate as the capacity with scaling $O(MloglogK)$ in MU-MISO downlink networks for fixed *M* under the i.i.d. Rayleigh fading assumption, as *K* increases [18, 19]. The considered situation with fixed *M* and large *K* models MU-MISO networks with small-scale BS antenna arrays of four or eight transmit antennas at the BS and several hundred or thousand active users in a cell, as in a 4G network. In massive MIMO, the number *M* of transmit antennas at the BS is very large, in the order of hundreds. To analyze the performance of user scheduling in massive MIMO, it is necessary to consider the scenario of large *M* and the performance behavior as *M* increases. One of the important findings in this regard is that the scheduling gain obtained from smart user selection is insignificant under rich scattering environments, commonly modeled as i.i.d. Rayleigh fading channel elements (10.8), when *M* is very large [21–23]. This is mainly due to the “channel hardening effect” [44] and the difficulty of finding a subset of semiorthogonal users in the massive MIMO case [21, 30]:

*Channel hardening effect:* This refers to the phenomenon that in massive MIMO the norm of each channel vector from the same Rayleigh fading distribution converges to the same constant by the law of large numbers. Assume the channel vector of user *k*${\mathbf{h}}_{k}\stackrel{i.i.d.}{\sim}\mathcal{C}\mathcal{N}(0,\mathbf{I})$. Then, ∥**h**_{k}∥^{2} is a chi-square random variable with 2*M* degrees-of-freedom and one can show [21]:

*x*> 1, using large deviations principle. Then:

*x*> 1, the right-hand side (RHS) of (10.14) converges to:

*x*> 1, when $logK$ grows slower than

*M*such as

*K*=

*μM*,

*K*=

*M*

^{3}, and $K={e}^{{M}^{0.9}}$. It is obvious that $\mathbb{P}({max}_{k}\mathfrak{\parallel}{\mathbf{h}}_{k}{\mathfrak{\parallel}}^{2}\le x)\downarrow 0$ for all

*x*< 1 due to $\mathbb{E}[\mathfrak{\parallel}{\mathbf{h}}_{k}{\mathfrak{\parallel}}^{2}]=1$ by the law of large numbers. Consequently, ${max}_{k}\mathfrak{\parallel}{\mathbf{h}}_{k}{\mathfrak{\parallel}}^{2}$ converges to the constant 1 in probability. This observation shows that there is no user that has a significantly large channel vector magnitude when ${lim}_{M,K\to \mathfrak{\infty}}\frac{logK}{M}=0$, i.e., $K=o({e}^{{c}^{\mathfrak{\prime}}M})$ with some constant

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*c*′ > 0.

*The negligible probability of finding a subset of semiorthogonal users:* Let us consider the probability of finding a set of *M* roughly orthogonal users from *K* users with ${\mathbf{h}}_{k}\stackrel{i.i.d.}{\sim}\mathcal{C}\mathcal{N}(0,\mathbf{I})$, *k* = 1,2,…,*K*. This probability can be computed by considering a set of *M* double cones constructed around each axis of the *M*-dimensional space:

**e**

_{i}is the

*i*th column of the

*M*×

*M*identity matrix. Note that a set of

*M*channel vectors each of which is contained in one distinct cone of the cone set $\mathfrak{\{}{\mathcal{C}}_{1}^{\mathfrak{\prime}},{\mathcal{C}}_{2}^{\mathfrak{\prime}},\dots ,{\mathcal{C}}_{M}^{\mathfrak{\prime}}\mathfrak{\}}$ is a set of

*M*roughly orthogonal channel vectors. The probability that the channel vector

**h**

_{k}

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falls in the cone ${\mathcal{C}}_{i}^{\mathfrak{\prime}}$ is given by [30]:*M*by the law of large numbers. Therefore, the probability that the cone ${\mathcal{C}}_{i}^{\mathfrak{\prime}}$ contains at least one out of the

*K*

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channel vectors is given by:*M*users for very large

*M*if ${lim}_{M,K\to \mathfrak{\infty}}\frac{logK}{M}=0$, i.e., $K=o({e}^{{c}^{\mathfrak{\prime}}M})$ with some constant

*c*′ > 0.

The above two facts show that if *K* grows slower than the exponential rate with respect to *M*, there is neither significant MU diversity gain nor a user subset of size *M* with roughly orthogonal channel vectors. Hence, as the number *M* of transmit antennas at the BS increases, *K* should increase exponentially with respect to *M* to obtain a nontrivial scheduling gain. However, in massive MIMO with *M* in the order of hundreds, the required number *K* of active users in the cell is too large, thus the scheduling gain is trivial due to the insufficient number of active users in practical networks under rich scattering environments.

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The channel hardening effect leads to simplified resource allocation because fast-fading channel variation in the frequency domain also disappears due to the channel hardening effect. Thus, scheduling over the frequency domain may not be required and the whole spectrum can be assigned simultaneously to all users scheduled in the antenna domain.

Consistent with the above observations, the sum rate of RBF is shown to scale as [18]:

for $M=O(logK)$ where *x* ∼ *y* indicates ${lim}_{M,K\to \mathfrak{\infty}}x/y=1$ and *c* is a positive constant. Furthermore, when ${lim}_{M,K\to \mathfrak{\infty}}\frac{logK}{M}=0$, [18]:

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In other words, in order to achieve linear sum rate scaling by RBF with respect to the number *M* of antennas, the number *K* of active users in the cell should increase exponentially with respect to the number *M* of antennas. This result presents a pessimistic prospect as to the scheduling methods based on partial CSI feedback, and suggests that it is necessary to obtain accurate CSI in the massive MIMO regime, since the sum capacity of an arbitrary subset of users with cardinality *M* among *K* users scales with *M*. The observation agrees with the result in Ravindran and Jindral [45]: given a constraint on the total amount of feedback, it is preferable to acquire accurate CSI feedback from a small number of users than to get coarse channel feedback from a large number of users. From negligible scheduling gains in massive MIMO under rich scattering environments, efficient user scheduling is less important. Thus, in massive MIMO under rich scattering environments simple scheduling methods that preselect users according to a certain probability distribution and exploit the CSI of only the selected users were considered [22, 23].