Hybrid Beamforming Architectures

The mmWave MIMO channel between a transmitter with $N_t$ antennas and a receiver with $N_r$ antennas is well modelled by the Saleh-Valenzuela geometric channel:

$$\mathbf{H} = \sqrt{\frac{N_t N_r}{L}} \sum_{\ell=1}^{L} \alpha_\ell\, \mathbf{a}_r(\phi_\ell^r, \theta_\ell^r)\, \mathbf{a}_t^H(\phi_\ell^t, \theta_\ell^t)$$

where $L$ is the number of scattering paths (clusters), $\alpha_\ell$ is the complex gain of the $\ell$-th path (incorporating path loss and phase), and $\mathbf{a}_t(\phi, \theta)$, $\mathbf{a}_r(\phi, \theta)$ are the transmit and receive array response vectors at azimuth $\phi$ and elevation $\theta$.

For a uniform planar array (UPA) with $W$ elements horizontally and $H$ elements vertically ($N = WH$), the array response vector is:

$$\mathbf{a}(\phi, \theta) = \frac{1}{\sqrt{N}} \left[1, e^{j\pi\sin\phi\sin\theta}, \ldots, e^{j\pi(W-1)\sin\phi\sin\theta}\right]^T \otimes \left[1, e^{j\pi\cos\theta}, \ldots, e^{j\pi(H-1)\cos\theta}\right]^T$$

At mmWave, $L$ is typically 2–5 in NLOS (and 1 dominant in LOS), making the channel matrix low-rank and spatially sparse — the key property exploited by hybrid beamforming.

The OMP algorithm greedily approximates the optimal unconstrained precoder $\mathbf{F}_\text{opt}$ by selecting $N_\text{RF}$ columns from a dictionary of array response vectors. Key properties:

• Near-optimal for sparse channels: When $L \leq N_\text{RF}$ (number of paths $\leq$ number of RF chains), OMP can recover the dominant paths exactly, achieving spectral efficiency within 1–2 dB of the fully digital baseline.
• Graceful degradation: As $N_\text{RF}$ decreases below $2N_s$, performance degrades smoothly. The rule of thumb $N_\text{RF} \geq 2N_s$ ensures that both the real and imaginary parts of each stream can be independently controlled.
• Dictionary design matters: The standard DFT codebook works well for ULAs, but UPAs benefit from 2D oversampled DFT dictionaries to capture elevation angles.

Hardware constraints beyond unit-modulus also affect practical implementations:

• Finite-resolution phase shifters: Commercial phase shifters typically have 4–6 bits of resolution (16–64 discrete phases). Quantising the OMP solution to the nearest discrete phase incurs $\sim$0.5–1 dB loss at 4 bits and $\sim$0.1 dB at 6 bits.
• Insertion loss: Each phase shifter introduces 3–6 dB insertion loss at mmWave, which must be compensated by the PA or accounted for in the link budget.
• Switching networks: Architectures using switches instead of phase shifters (selection-based hybrid BF) further reduce power but sacrifice beamforming resolution.

The achievable spectral efficiency with hybrid precoding $\mathbf{F} = \mathbf{F}_\text{RF}\mathbf{F}_\text{BB}$ and hybrid combining $\mathbf{W} = \mathbf{W}_\text{RF}\mathbf{W}_\text{BB}$ is:

$$R_\text{hybrid} = \log_2\!\det\!\left(\mathbf{I} + \frac{P}{N_s \sigma_n^2} \mathbf{R}_n^{-1}\,\widetilde{\mathbf{H}}\,\widetilde{\mathbf{H}}^H\right)$$

where $\widetilde{\mathbf{H}} = \mathbf{W}_\text{BB}^H \mathbf{W}_\text{RF}^H \mathbf{H} \mathbf{F}_\text{RF}\mathbf{F}_\text{BB}$ is the effective channel after hybrid precoding and combining, and $\mathbf{R}_n = \sigma_n^2 \mathbf{W}_\text{BB}^H \mathbf{W}_\text{RF}^H \mathbf{W}_\text{RF}\mathbf{W}_\text{BB}$ is the effective noise covariance.

The fully digital upper bound is:

$$R_\text{digital} = \sum_{i=1}^{N_s} \log_2\!\left(1 + \frac{P}{N_s \sigma_n^2}\,\sigma_i^2(\mathbf{H})\right)$$

where $\sigma_i(\mathbf{H})$ are the singular values of $\mathbf{H}$.

Simulation studies (Heath et al., 2016) consistently show that with $N_\text{RF} \geq 2N_s$, the fully connected hybrid architecture achieves within 1–3 dB of the fully digital baseline. The sub-connected architecture incurs an additional 1–2 dB penalty. As $N_\text{RF}/N_t \to 1$, hybrid and digital performance converge identically.