Exercises

In the far-field, all active antennas see the RIS at nearly the same angle. The per-element phase pattern across the RIS is the same (up to a linear ramp) for all active antennas. The BS-RIS channel becomes a rank-1 outer product.

In the near-field, each active antenna sees the RIS at a slightly different geometry (different per-element distances and angles). This gives rise to distinct phase patterns per active antenna — rank $\approx \min(N_t, N_{\text{eff}})$.

Place the array close to the RIS ($d_{\text{AR}} \ll d_F$) to get high rank. This is the signature of the array-fed RIS architecture. $\blacksquare$

Up to $\min(N_t, N_{\text{eff}})$ streams. Under near- field design with $N_{\text{eff}} \gg N_t = 16$, the bottleneck is $N_t$: 16 streams.

The active array has only 16 RF chains. Each supports one baseband stream. The RIS can focus $\sim 16$ eigenmodes toward 16 users, but can't produce more independent streams than the array feeds.

To serve more users, either: (a) increase $N_t$, or (b) time-share $K > 16$ users across slots. $\blacksquare$

$\lambda = c/f = 3\cdot 10^8 / 28\cdot 10^9 = 1.07$ cm.

Largest: $D = 0.3$ m.

$d_F = 2 \cdot 0.09 / 0.0107 = 16.8$ m.

$d < 16.8$ m is near-field. Design target: $d_{\text{AR}} \lesssim d_F / 5 = 3.4$ m. Very generous for realistic array-RIS deployments (panels on adjacent building facades). $\blacksquare$

$O(n^3)$ for assignment over $n$ items. Here $n = K$.

$K^3$ for $K = 4$: 64 operations. Trivially fast. For $K = 16$: $4096$ operations. Still microseconds.

Scales well to moderate user counts ($K \leq 100$). For very large $K$, greedy heuristics are cheaper without losing much optimality. $\blacksquare$

Let $\boldsymbol{\phi}^{(k)}$ be the RIS phase designed to focus eigenmode $k$ toward user $k$. Then $\boldsymbol{\phi}^{(k)}_n = e^{-j\arg((\mathbf{u}_k^*)_n \mathbf{h}_{k,2}^*)}_n$ ... $\propto$ matched filter.

User $k$ receives signal through eigenmode $k$ AND leakage from eigenmode $j \neq k$. Leakage at user $k$ from eigenmode $j$: $(\mathbf{h}_{k,2})^H \text{diag}(\boldsymbol{\phi}^{(j)}) \mathbf{u}_j$. This is not zero in general; it depends on how well $\boldsymbol{\phi}^{(j)}$ aligns user $k$'s direction.

For users at widely-separated angles (typical in mmWave), $\boldsymbol{\phi}^{(j)}$ points away from user $k$, and the leakage is small. Orthogonality is an approximation: not zero but small. Typical leakage level: $-20\,\text{dB}$ to $-30\,\text{dB}$ below the desired signal in well-separated UEs. $\blacksquare$

$\mathbf{H}_1$ (BS-RIS): geometrically fixed, changes slowly (hours/days). $\mathbf{h}_{k,2}$ (RIS-UE): changes with UE mobility (ms-to-seconds).

Only $\mathbf{h}_{k,2}$ is re-estimated per coherence block. $K$ pilots (one per user) suffice, since $\mathbf{H}_1$ is factored out via SVD.

Conventional RIS: $N$ pilots per block. Array-fed RIS: $K$ pilots per block. For $K = 8, N = 256$: $32\times$ pilot savings. For large-$N$ deployments, this is the difference between feasible and infeasible. $\blacksquare$

$N_{\text{antennas}} = N_t + N$. Multi-user capacity with $K = N_t + N$ users: $\sum_k \log_2(1 + \text{SNR}_k)$ where each user gets aperture gain proportional to the array size. In favorable propagation: $(N_t + N) / K$ per user.

$N_t$ active + $N$ passive. $K = N_t$ users. Each user gets aperture $N$ via RIS focusing — not $N_t + N$. Cost: multiplexing capped at $N_t$.

Small $K$: array-fed wins (aperture gain $N$ focused on each user; no unused antennas). Large $K$: fully-digital wins (multiplexing gain up to total antenna count). For $K \leq N_t$, array-fed is competitive; for $K \gg N_t$, fully-digital is better.

Fully-digital cost scales as $N_t + N$ RF chains. Array-fed cost scales as $N_t$ RF chains + $N$ passive elements. For $N \gg N_t$, array-fed is much cheaper — the architectural case. $\blacksquare$

User $k$ → eigenmode $\pi(k)$. Let $\mathbf{v}_{\pi(k)}$ be the corresponding right singular vector of $\mathbf{H}_1$.

$\mathbf{v}_{k} = \sqrt{p_k} \mathbf{v}_{\pi(k)}$, where $p_k$ is the water-filling power to user $k$.

$\mathbf{W} = [\mathbf{v}_{1}, \ldots, \mathbf{v}_{K}] = \mathbf{V}_1 \mathbf{P}_\pi \text{diag}(\sqrt{p_k})$, where $\mathbf{P}_\pi$ is the permutation matrix mapping user index to assigned eigenmode. $\blacksquare$

Only $N_t$ eigenmodes exist. Cannot support more than $N_t$ independent streams simultaneously.

1. Time-share $K$ users across $\lceil K / N_t \rceil$ time slots, each serving $N_t$ users.
2. NOMA: superpose users on the same eigenmode with successive interference cancellation.
3. Increase $N_t$: add active antennas.

Option 1 (time-sharing) is simplest and most common. Per-user rate is divided by $\lceil K/N_t \rceil$, but aggregate throughput is similar. $\blacksquare$

Parallel channels with per-channel gain $\sigma_k^2 \cdot N / K$. Capacity per channel: $\log(1 + p_k \sigma_k^2 N / (K \sigma^2))$ with power $p_k \geq 0$, $\sum p_k = P$.

$p_k^\star = \max(0, \nu - \sigma^2 K / (\sigma_k^2 N))$, where $\nu$ (water level) is found from $\sum_k p_k^\star = P$.

Stronger eigenmodes ($\sigma_k$ large) get more power; weakest modes may get 0 (below water level). The optimal water level $\nu$ grows with $P$; at high $P$, all eigenmodes are active with near-equal power. $\blacksquare$

$\boldsymbol{\Phi}$ designs a single coherent beam from BS to one user (or jointly balances multiple users with shared BS-side beamforming).

$\boldsymbol{\Phi}$ serves as a beam-mapping matrix: each eigenmode of $\mathbf{H}_1$ is focused toward a different user. Composite $\boldsymbol{\phi}$ superposes per-user focusing.

Conventional: passive beamforming for coherent combining. Array-fed: passive beam-steering for multi-stream multiplexing. The same mathematical object (diagonal matrix) plays different architectural roles. $\blacksquare$

Active RF chain cost: $\sim \$200/unit$ (ADC + DAC + mixer). 32-antenna fully-digital: $\sim \$6400$. Passive metasurface: $\sim \$100/m^2$. Array-fed (8 active + 256 passive): $\$1600 + \$500 \approx \$2100$. Ratio: $\sim 3\times$ cost advantage.

Active RF chain cost: $\sim \$2000/unit$ (extreme bandwidth + mixer complexity). 32-antenna fully-digital: $\sim \$64,000$. Passive metasurface: $\sim \$200/m^2$ (smaller elements, same area). Array-fed (8 active + 256 passive): $\$16,000 + \$1000 \approx \$17,000$. Ratio: $\sim 4\times$ cost advantage.

Cost advantage grows with frequency. At sub-THz ($> 200$ GHz), fully-digital becomes effectively infeasible; array-fed RIS is the only practical option. $\blacksquare$

BS and RIS are both stationary (mounted on buildings / poles). The distance between them, their relative orientations, and their antenna positions don't change with UE motion.

Very slow effects only: temperature changes (array element position drift), mechanical vibration (tiny), hardware aging (over months).

$\mathbf{H}_1$ need not be estimated per coherence block. Calibrate once per deployment with a high-accuracy pilot sweep; store the result; re-calibrate only when drift is suspected (rare). This is the foundation of the two- timescale CSI approach. $\blacksquare$

$p_k = P/8$ for all $k$. Per-user SNR (linear) = $\sigma_k^2 N P / (8 \sigma^2) = \sigma_k^2 \cdot (512/8) \cdot 10 = 640\sigma_k^2$.

For $\sigma_k^2 = 2.25$: SNR = $1440$, $R = \log_2(1441) = 10.5$. For $\sigma_k = 1.2^2 = 1.44$: SNR = $922$, $R = 9.8$. Down to $\sigma_8 = 0.09$: SNR = $58$, $R = 5.9$. Sum roughly 8 × 8.5 ≈ 68 bits/s/Hz (taking geometric mean).

Water-filling would give more power to stronger eigenmodes, improving the stronger users slightly at the cost of weaker. Sum rate improvement: $\sim 1$-$3$ bits/s/Hz. For fairness, equal power is sometimes preferred. $\blacksquare$

$K = 32$ requires at least 32 eigenmodes. Single-panel array- fed RIS with $N_t = 32$ is expensive (32 active antennas at 140 GHz = very high cost).

Use 4 array-fed panels, each $N_t = 8, N = 256$. Split users into 4 groups of 8, one per panel. Each panel serves its group independently. See Chapter 12 for multi- panel coordination.

Alternatively: 1 panel with $N_t = 8$, but time-share across user groups. Each group gets $\sim 25\%$ coherence time. Lose in per-user rate but gain in hardware cost.

Multi-panel: higher cost but full-time serving. Time-share: lower cost but reduced per-user rate. At 6G deployment density (many hotspots), multi-panel wins. At single-site hotspot, time-share may be preferred. $\blacksquare$