Ferkans — Interactive Telecom Tutor

Many Users, Few Antennas

In Chapter 16, we characterized the capacity region of the MIMO BC assuming that all $K$ users are served simultaneously. In practice, the number of users $K$ in a cell can be much larger than the number of transmit antennas $n_t$ . Since DPC can effectively multiplex at most $n_t$ spatial streams, the base station must schedule which users to serve in each time-frequency resource. This creates a new dimension of the problem: user selection.

The remarkable insight is that with many users, multiuser diversity provides a significant gain. With high probability, at least $n_t$ users have nearly orthogonal channels, allowing the base station to exploit the spatial degrees of freedom without the penalty of serving users with correlated channels. As $K \to \infty$ , the sum capacity scales as $n_t \log\log K$ — a logarithmic gain in the number of users.

Definition:
User Scheduling in the MIMO BC

Given $K$ users with channel matrices $\{\mathbf{H}_{k}\}_{k=1}^K$ and $n_t$ transmit antennas, the base station selects a subset $\mathcal{S} \subseteq [K]$ with $|\mathcal{S}| \leq n_t$ to serve in each time slot. The scheduled users are served via DPC (or linear precoding). The sum throughput is: $R_{\text{sum}}(\mathcal{S}) = \max_{\{\mathbf{K}_k\}_{k \in \mathcal{S}}} \sum_{k \in \mathcal{S}} R_{k}(\mathbf{K}_k)$ subject to $\sum_{k \in \mathcal{S}} \text{tr}(\mathbf{K}_k) \leq P$ .

The optimal scheduling problem is to select $\mathcal{S}^*$ that maximizes $R_{\text{sum}}(\mathcal{S})$ — a combinatorial optimization over $\binom{K}{|\mathcal{S}|}$ possibilities.

Theorem: Multiuser Diversity Gain

For the $K$ -user MISO BC with i.i.d. Rayleigh fading channels $\mathbf{h}_k \sim \mathcal{CN}(\mathbf{0}, \mathbf{I}_{n_t})$ , the sum capacity with optimal user scheduling scales as: $C_{\text{sum}} = n_t \log\left(1 + \frac{P}{n_t} \log K\right) + o(\log\log K)$ as $K \to \infty$ with fixed $n_t$ and $P$ .

That is, the sum capacity grows as $n_t \log\log K$ — each antenna contributes one spatial stream, and the multiuser diversity gain provides $\log K$ effective SNR enhancement per stream.

Among $K$ i.i.d. users, the maximum channel gain $\max_k \|\mathbf{h}_k\|^2$ grows as $\log K$ (extreme-value theory for chi-squared random variables). With $n_t$ antennas, we can find $n_t$ users whose channels are approximately orthogonal and each has gain approximately $\log K$ . Serving these users is equivalent to a parallel channel with $n_t$ streams, each at SNR $\approx (P/n_t) \log K$ .

Proof

Upper bound

The sum capacity is bounded by the capacity with full cooperation (all users' channels known, joint processing): $C_{\text{sum}} \leq \log\det\left(\mathbf{I} + \frac{P}{n_t} \sum_{k=1}^K \mathbf{h}_k \mathbf{h}_k^H\right)$ By the law of large numbers, $\frac{1}{K}\sum_k \mathbf{h}_k \mathbf{h}_k^H \to \mathbf{I}_{n_t}$ , so the upper bound grows as $n_t \log(1 + KP/n_t)$ . But scheduling only $n_t$ users gives a tighter analysis.

Achievability via semi-orthogonal user selection

The Yoo-Goldsmith algorithm selects users greedily: pick the user with the largest channel norm, then successively pick users whose channels are nearly orthogonal to the already selected channels. With $K$ users, this finds $n_t$ nearly orthogonal users with channel norms $\sim \log K$ , achieving the stated scaling.

The $\log\log K$ scaling

The sum rate grows as $n_t \log\log K$ because the effective SNR per stream is $(P/n_t)\log K$ , and $\log((P/n_t)\log K) \sim \log\log K$ for large $K$ . This is a diminishing-returns gain: doubling the number of users adds only a constant to the sum rate.

,

Multiuser Diversity and Scheduling

Animation showing a base station selecting the best users from many candidates. Channel gains fluctuate across users, and the scheduler picks users with the strongest channels. The multiuser diversity gain scales as

\log K

.

Example: User Scheduling Gain

A base station with $n_t = 4$ antennas serves a cell with $K$ users, each with i.i.d. Rayleigh fading. Compare the sum rate with $K = 4$ users (no scheduling choice) and $K = 100$ users (optimal scheduling) at $\text{SNR} = 20$ dB.

Solution

$K = 4$ users

All users are served. With ZF precoding (near-optimal for equal-power i.i.d. channels): $R_{\text{sum}} \approx n_t \cdot \log\left(1 + \frac{P}{n_t}\right) = 4 \cdot \log\left(1 + 25\right) \approx 4 \times 4.70 = 18.8 \text{ bits/use}$

$K = 100$ users

With optimal scheduling, the effective per-stream SNR is enhanced by the multiuser diversity factor $\approx \log K = \log 100 \approx 4.6$ : $R_{\text{sum}} \approx 4 \cdot \log\left(1 + 25 \times 4.6\right) = 4 \cdot \log(116) \approx 4 \times 6.86 = 27.4 \text{ bits/use}$ A gain of approximately 8.6 bits (46%) from user scheduling alone.

Multiuser Diversity: Sum Rate vs. Number of Users

Visualize how the sum rate of the MISO BC scales with the number of users $K$ for different numbers of antennas. The $\log\log K$ growth is evident.

Parameters

Number of Tx antennas (

n_t

)4

SNR (dB)20

Maximum number of users (

K

)200

🔧Engineering Note

Scheduling in 5G NR

5G NR implements multiuser scheduling through the proportional fair (PF) scheduler, which balances throughput and fairness. The PF scheduler serves user $k^* = \arg\max_k R_{k}(t) / \bar{R}_k(t)$ , where $\bar{R}_k(t)$ is user $k$ 's average throughput.

With massive MIMO ( $n_t = 64$ or more), the multiuser diversity gain is amplified: the base station can find $n_t$ nearly orthogonal users even with moderate $K$ , and the ZF precoding loss vanishes. This is why 5G massive MIMO cells can support 10-20 simultaneous spatial streams with high spectral efficiency.

Practical Constraints

•
5G NR scheduling granularity: 1 slot = 0.5 ms
•
CSI feedback delay: 4-8 ms in FDD, reduces multiuser diversity gains
•
Proportional fair scheduler is standard; round-robin for fairness-critical services
•
Typical: $n_t = 64$ , $K = 16-32$ simultaneous users in massive MIMO

Common Mistake: Scheduling More Users Than Antennas

Mistake:

Attempting to serve $K > n_t$ users simultaneously in the MIMO BC, believing that DPC can handle arbitrary numbers of users.

Correction:

DPC can encode for $K > n_t$ users, but the capacity region degenerates: the sum rate is limited by $n_t$ spatial degrees of freedom, so the per-user rate decreases as $K$ grows beyond $n_t$ . The optimal strategy is to select $\leq n_t$ users per time slot and exploit multiuser diversity across slots.

Multiuser diversity

The gain obtained by scheduling transmission to users with favorable channel conditions. In a system with $K$ users and i.i.d. fading, the best user's channel gain scales as $\log K$ , providing an SNR enhancement that grows logarithmically with the number of users.

Quick Check

The multiuser diversity gain scales as $\log\log K$ . Why is this a diminishing-returns effect?

Because $\log\log K$ grows extremely slowly — doubling $K$ from 100 to 200 adds less than 0.1 bits per stream

Because the channels become more correlated with more users

Because the base station runs out of power

Correction:

Because

\log\log K

grows extremely slowly — doubling

K

from 100 to 200 adds less than 0.1 bits per stream

$\\log\\log 100 \\approx \\log 4.6 \\approx 2.2$ , $\\log\\log 200 \\approx \\log 5.3 \\approx 2.4$ . The gain from adding 100 more users is only 0.2 bits per stream. The first users matter most; after $K \\approx 10 n_t$ , additional users provide negligible gains.

Key Takeaway

User scheduling transforms the MIMO BC from a fixed $K$ -user problem to an opportunistic selection problem. With $K \gg n_t$ users, multiuser diversity provides a $\log K$ SNR enhancement, but the sum-rate gain scales only as $\log\log K$ — a diminishing-returns effect. In practice, the combination of multiuser scheduling and massive MIMO (large $n_t$ ) is the dominant factor in 5G NR spectral efficiency.

The KKK-User MIMO Broadcast Channel