Chapter Summary

Chapter 20 Summary

Key Points

• 1.

  At mmWave and sub-THz, fully digital beamforming is infeasible because each RF chain (ADC + DAC + LO + mixer + LNA) consumes roughly 1 W at GSps sampling rates. The hybrid architecture factors the precoder as $\mathbf{W} = \mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}}$ with $\mathbf{F}_{\text{RF}}$ constant-modulus (phase shifters only) and $\mathbf{F}_{\text{BB}}$ unconstrained digital, using $N_{\text{RF}}$ RF chains with $K \leq N_{\text{RF}} \leq N_t$.

• 2.

  When $N_{\text{RF}} \geq 2K$, the hybrid architecture can realize any fully-digital precoder exactly via the amplitude-phase decomposition of Theorem TWhen Hybrid Matches Fully Digital. This architectural result means the cost of going hybrid is hardware (phase shifters) rather than spectral efficiency.

• 3.

  Fully-connected (FC) topologies provide programmable entries at $N_{\text{RF}} \cdot N_t$ phase shifters but pay $10\log_{10}(N_{\text{RF}})$ dB of combiner insertion loss. Subarray (SA) topologies use only $N_t$ phase shifters with no combiner, at a worst-case $10\log_{10}(N_{\text{RF}})$ dB per-beam array-gain penalty when streams target distinct directions. Deployed 5G mmWave base stations predominantly use subarray topologies.

• 4.

  Beam codebooks (DFT, oversampled DFT, 5G NR Type I/II) replace continuous beam optimization with finite-set search. Exhaustive beam sweeps cost $O(M_t M_r)$ pilot symbols; hierarchical codebooks reduce this to $O(\log M_t + \log M_r)$. The worst-case DFT quantization loss at the midpoint between two codewords is $\text{sinc}^2(N_t/M)$ - about 3.9 dB for $M = 2N_t$.

• 5.

  OMP-based spatially sparse precoding exploits the mmWave channel's $L$-path sparsity. When $N_{\text{RF}} \geq L$ and the dictionary contains the true steering vectors, OMP recovers the optimal hybrid precoder exactly. Alternating minimization works on arbitrary channels but converges only to local minima; the manifold-optimization variant (MO-AltMin) offers stronger guarantees.

• 6.

  Phase-shifter quantization to $b$ bits introduces a beamforming-gain loss $\eta(b) = \text{sinc}^2(1/2^b)$, which is 3.92 dB at $b = 1$, 0.91 dB at $b = 2$, 0.22 dB at $b = 3$, and under 0.1 dB beyond. Three-bit phase shifters are the practical sweet spot and the default in deployed 5G mmWave arrays.

• 7.

  Lens-based beamforming replaces the phase-shifter network with a passive quasi-optical structure (Butler matrix, Rotman lens, parabolic reflector). The architecture trades reconfigurability for zero power consumption, lower insertion loss, and wideband operation. At sub-THz 6G the CommIT array-fed multibeam reflector architecture scales to $10^4$-element-equivalent apertures using only $N_{\text{RF}}$ active chains - a key design point for 140-300 GHz systems.