Chapter Summary

Chapter Summary

Key Points

• 1.

  Three linear receivers span the complexity–performance tradeoff. MRC ($\mathbf{G} = \mathbf{H}$) maximizes the single-user SNR at $\mathcal{O}(N_tK)$ cost; ZF ($\mathbf{G} = \mathbf{H}(\mathbf{H}^{H}\mathbf{H})^{-1}$) nulls interference at $\mathcal{O}(K^{3})$ cost but amplifies noise; MMSE ($\mathbf{G} = \mathbf{H}(\mathbf{H}^{H}\mathbf{H} + \frac{\sigma^2}{P}\mathbf{I})^{-1}$) optimally balances interference and noise at the same cubic cost as ZF.

• 2.

  In the massive MIMO limit $N_t \gg K$, all three receivers converge. The per-user SINR scales as $N_t P \beta_k / \sigma^2$, confirming that simple linear processing becomes near-optimal with a large antenna surplus — the foundational promise of massive MIMO.

• 3.

  MMSE is the optimal linear receiver for all operating regimes. It reduces to MRC at low SNR (noise-dominated) and to ZF at high SNR (interference-dominated), smoothly interpolating between the two extremes. The regularization strength $\sigma^2/P$ is not arbitrary but dictated by the Bayesian LMMSE framework.

• 4.

  MMSE-SIC achieves the MAC capacity region. By decoding users sequentially and cancelling each decoded signal, MMSE-SIC achieves the sum rate $\log_2 \det(\mathbf{I} + \frac{1}{\sigma^2}\mathbf{H}\mathbf{P}\mathbf{H}^{H})$ for any decoding order — the information-theoretic limit for the MIMO uplink.

• 5.

  Low-complexity alternatives avoid the cubic inversion cost. The Neumann series approximation with $L = 2$–$3$ terms achieves near-MMSE performance at $\mathcal{O}(L K^{2})$ cost when $N_t/K \geq 4$. Conjugate gradient methods offer similar benefits with adaptive convergence.

• 6.

  1-bit ADCs incur a fixed $2/\pi$ rate penalty, independent of $N_t$. The Bussgang decomposition linearizes the quantizer, enabling modified MMSE (box) detection with standard LMMSE machinery. The massive antenna array compensates for the coarse quantization, making 1-bit receivers viable for energy-constrained deployments at mmWave and sub-THz.