Summary

Chapter 17 Summary: MIMO III — Multi-User MIMO

Key Points

• 1.

  MAC capacity region and SIC. The $K$-user MIMO MAC capacity region is the polymatroid defined by $\sum_{k \in \mathcal{S}} R_k \leq \log_2\det(\mathbf{I} + \frac{1}{\sigma^2}\sum_{k \in \mathcal{S}}P_k\mathbf{h}_k\mathbf{h}_k^H)$ for all subsets $\mathcal{S}$. Every boundary point is achieved by successive interference cancellation (SIC) with Gaussian codebooks, and the sum rate is independent of the SIC decoding order.

• 2.

  BC-MAC duality and DPC. The MISO broadcast channel capacity region under a sum power constraint $P$ equals the union of dual MAC capacity regions: $\mathcal{C}_{\text{BC}}(P) = \bigcup_{P_1+\cdots+P_K = P} \mathcal{C}_{\text{MAC}}(P_1, \ldots, P_K)$. Dirty paper coding (DPC) achieves this region by pre-cancelling known interference, but its impractical encoding complexity motivates linear alternatives.

• 3.

  Linear precoding (ZF and RZF). Zero-forcing precoding $\mathbf{W}_{\text{ZF}} = \mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H})^{-1}$ eliminates inter-user interference at a power penalty of $[(\mathbf{H}\mathbf{H}^{H})^{-1}]_{kk}$. Regularised ZF (RZF) adds $\alpha\mathbf{I}$ to balance interference suppression and noise enhancement. Both achieve the same multiplexing gain as DPC ($\min(K, N_t)$ DoF), with a constant gap that vanishes as $N_t/K \to \infty$.

• 4.

  WMMSE algorithm. The weighted sum-rate maximisation problem is non-convex, but the WMMSE reformulation transforms it into block coordinate descent over convex subproblems (receive filters, MSE weights, precoders). The algorithm is monotonically non-decreasing in the WSR and converges to a KKT stationary point, bridging the gap between closed-form linear precoding and the DPC bound.

• 5.

  Interference alignment. The $K$-user interference channel has $K/2$ total DoF (Cadambe-Jafar, 2008): each user achieves $1/2$ DoF by aligning all interference into half the signal space. This dramatic result is asymptotic (requiring channel extensions and perfect global CSI) and serves as a theoretical benchmark rather than a practical scheme.

• 6.

  User scheduling and multi-user diversity. With $K$ users and $N_t$ BS antennas, scheduling the best near-orthogonal user subset provides a multi-user diversity gain of $\log_2 \ln K$ bits/s/Hz. The SUS algorithm achieves this scaling with $O(KN_t^2)$ complexity, and proportional fair scheduling balances throughput with fairness in practical systems.