Chapter Summary

Chapter Summary

Key Points

• 1.

  The MIMO detection problem is $\hat{\mathbf{x}} = \arg\min_{\mathbf{x} \in \mathcal{A}^{n_t}} \|\mathbf{y} - \mathbf{H}\mathbf{x}\|^2$, a bounded closest-vector problem that is $\mathcal{NP}$-hard in the worst case and combinatorial in size $|\mathcal{A}|^{n_t}$.

• 2.

  Linear detectors (ZF, MMSE) are convex-quadratic solutions: cheap, parallelizable, and always suboptimal. MMSE dominates ZF in per-stream SINR by the Wiener orthogonality principle, coinciding only in the noise-free limit.

• 3.

  Successive interference cancellation (MMSE-SIC, V-BLAST) achieves Gaussian MIMO capacity under genie-aided error-free cancellation — a direct operational realization of the chain rule of mutual information. Ordering by highest post-detection SINR mitigates error propagation.

• 4.

  Sphere decoding restricts the ML search to a Euclidean ball around the received vector. The Schnorr-Euchner enumeration on a QR-triangularized system finds ML exactly with expected complexity polynomial in $n_t$ at practical SNR — though provably exponential at any fixed SNR as $n_t \to \infty$.

• 5.

  Lattice reduction (LLL) re-parameterizes the channel with a well-conditioned basis. LLL-aided ZF/MMSE recovers full receive diversity $n_r$ at polynomial cost, closing most of the gap to ML with linear per-symbol complexity.