Exercises

Cell-free: $L$ distributed APs (each small, $N_a$ antennas) jointly serve $K$ UEs without cell boundaries. Controlled by a central processing unit via fronthaul.

UE sees many spatially-distributed paths — built-in redundancy.

Nearest AP is always close; smooth coverage.

No handovers — CPU dynamically reassigns APs as UEs move.

First performance evaluation of cell-free OTFS. Establishes the architecture, pilot design (embedded + superimposed), and conjugate beamforming in the DD domain.

~35% gain in 95%-likely per-user throughput vs cell-free OFDM at typical urban mobility (60-150 km/h).

DD-diversity × macro-diversity compounding + superimposed pilot efficiency + ICI robustness under Doppler.

$\mathrm{SNR}_l = \alpha_k |\mathbf{H}^{(l, k)}|^2 / \sigma_w^2$.

Conjugate BF: sum across APs adds coherently. $\sum_l |\mathbf{H}^{(l, k)}|^2 = L \bar\beta_k$ (LLN).

Leakage from $K-1$ other UEs. Scales as $K$ per user, averaged over spatial randomness.

$\mathrm{SINR}_k \to L \bar\beta_k / (\sigma_w^2 + K \bar\beta \kappa / L)$ where $\kappa$ is pilot contamination. Linear growth in $L$.

$4P = 32$ floats × 4 bytes = 128 bytes per UE per AP per frame.

$K = 200$ UEs: $200 \cdot 128 = 25.6$ kB per frame.

At 100 Hz frame rate: 2.56 MB/s per AP.

$L = 100$ APs: 256 MB/s total fronthaul. With user-centric clustering ($L_k = 8$): $256 \cdot 8/200 = 10$ MB/s. 25× reduction. Both fit in 1 GbE per AP.

MSE $\sim \sigma_w^2/(MN \rho_p) + (1-\rho_p)/(MN \rho_p) \cdot$ data interference. At optimal $\rho_p = 1/\sqrt{\text{SNR}}$: MSE $\propto 1/(MN \sqrt{\text{SNR}})$.

MSE $\sim \sigma_w^2/MN$. Independent of SNR (no data interference).

Superimposed / Embedded $= 1/\sqrt{\text{SNR}}$. At 20 dB SNR: $1/\sqrt{100} = 0.1$. 10× better.

At low SNR ($< 5$ dB), superimposed MSE can be worse due to pilot-data interference. Embedded pilots win. Deploy adaptive scheme: switch to superimposed only at high SNR.

Channels at two UEs $\mathbf{r}_1, \mathbf{r}_2$ correlate as $\rho(\mathbf{r}_1, \mathbf{r}_2) = \exp(-\|\mathbf{r}_1 - \mathbf{r}_2\|/d_c)$ where $d_c \sim \lambda \cdot \text{SNR}^{1/\alpha}$, path-loss exponent $\alpha$.

For independent pilot estimates: $\rho < 0.1$, i.e., $d \geq 2.3 d_c$.

At 28 GHz ($\lambda = 0.01$ m), $\text{SNR} = 15$ dB, $\alpha = 3$: $d_c \approx 0.01 \cdot 100^{1/3} \approx 5$ cm (small-scale). Adding large-scale fading: $d_c \sim 10$-$50$ m (geometric). Minimum UE spacing: 20-100 m for safe pilot reuse.

At typical urban UE density (100 UEs/km²): ~100 m spacing achievable. Pilot reuse is practical.

Per-UE average: $L \cdot \bar\beta \cdot \text{SNR} / K$. $64 / 50 \cdot 100 = 128 \to 21$ dB average.

95%-tile is 10-15 dB below average in cellular. In cell-free with uniformity 0.5: 95%-tile at 18-20 dB.

$R = \log_2(1 + 10^{18/10}) \approx 6.4$ bits/s/Hz.

Cellular OFDM at same SNR: 0.4 bits/s/Hz. Ratio: 16× — far beyond 35% due to compound cell-free + OTFS effects. 35% is the gain at the cell-free baseline.

UE receives $\sum_l \text{AP}_l$'s signal. Coherent sum: $|\sum_l A_l e^{j\phi_l}|$. At perfect sync: $|\sum_l A_l|$ (real-valued sum).

If $\phi_l \in \mathcal{N}(0, \sigma_\phi^2)$ i.i.d.: expected coherent gain = $L \cos(\sigma_\phi)$ (effective $L$ reduced). Effective $L_{\text{eff}} = L \cos(\sigma_\phi)$.

$\cos(10°) = 0.985$. Gain loss: 0.13 dB. Negligible.

For 1 dB loss: $\sigma_\phi \sim 27°$. For 3 dB loss: $45°$. Design target: $\leq 15°$ for sub-dB loss. GNSS-PPS + 1-dB margin achievable outdoors.

Per-UE rate is $\log_2(1 + \mathrm{SINR})$. SINR depends on UE position via $L$ nearest APs.

SINR at cell-center (fully covered): $\sim 25$ dB. SINR at cell-edge (~ farther from all APs): $\sim 15$ dB. Span: 10 dB.

$R_{50\%} = \log_2(1 + 10^{20/10}) = 6.7$ bits/s/Hz. $R_{95\%} = \log_2(1 + 10^{15/10}) = 5.1$ bits/s/Hz.

$\mathcal{U} = 5.1 / 6.7 = 0.76$. Compare cellular: $\mathcal{U} \approx 0.1$. Cell-free is much more uniform.

Mobility averages UE position. 95%-tile and 50%-tile converge slightly. $\mathcal{U}_{\text{mobile}} \approx 0.6$.

At 95%-tile, compared to cell-free OFDM: (i) DD-diversity $P$: ~20% gain. (ii) Pilot efficiency (superimposed): ~5% gain. (iii) ICI robustness: ~10% gain.

Effects nearly multiply: $1.20 \cdot 1.05 \cdot 1.10 = 1.39$. ~39%. In practice, some redundancy: ~35% is typical.

ICI robustness factor grows: 10% → 20-25% at 300 km/h. Total gain: ~50%.

ICI factor → 0% at 5 km/h. Total gain: ~20-25%.

The 35% average across 60-150 km/h mobility range.

$\|\mathbf{H}^{(l, k)}\|^2$ is a random variable with mean $\bar\beta_k$, variance $\sigma_\beta^2$.

$\sum_l \|\mathbf{H}^{(l, k)}\|^2 \to L \bar\beta_k$ almost surely (LLN). Variance: $L \sigma_\beta^2$. Coefficient of variation: $\sigma_\beta / (\bar\beta \sqrt{L}) \to 0$.

SINR fluctuation vanishes as $L \to \infty$. Effective channel gain is deterministic.

No small-scale fading effect at UE. Equivalent to LOS channel with fixed gain. Simplifies scheduling and adaptive modulation.

Select top-$L_k$ APs by channel magnitude $\|\mathbf{H}^{(l, k)}\|$. Typically $L_k = 5$-$10$.

Favor nearby APs (lower fronthaul latency). Weight magnitude by distance$^{-1/2}$.

Limit $L_k$ so per-AP fronthaul does not overrun. Typical: $L_k \leq K/10$ to share fronthaul among UEs.

Re-evaluate every 100 frames. Smooth transitions (add/drop one AP per update) to avoid oscillation.

def update_cluster(UE k): rank APs by effective channel strength select top-L_k add threshold/hysteresis for stability

$f$ fraction of APs fail i.i.d. Effective active APs: $L_{\text{eff}} = L(1 - f)$.

$\mathrm{SINR}_{\text{eff}} = L_{\text{eff}} \bar\beta / \sigma_w^2 = L(1 - f) \bar\beta / \sigma_w^2$. Rate: $R_{\text{eff}} = \log(1 + (1 - f) \cdot \mathrm{SINR}_0)$.

$R_{\text{eff}} < R_{\text{min}}$ when $f > 1 - 2^{R_{\text{min}}/1}/\mathrm{SINR}_0$. For $R_{\text{min}} = 1$ bit/s/Hz, $\mathrm{SINR}_0 = 20$: outage at $f > 0.9$.

Cell-free OTFS is extremely fault-tolerant. Up to 90% of APs can fail before link falls below 1 bit/s/Hz. 99.99% reliability achievable with modest redundancy ($f = 0.1$).

AP (Radio Unit, RU) handles RF + ADC/DAC + low-PHY. Edge CPU (Distributed Unit, DU) handles high-PHY including OTFS processing. Regional CPU (Central Unit, CU) handles scheduling and coordination.

Per-site edge CPU: $\sim 50$ APs. Regional CU: 100 sites. Total: $5000$ APs per region.

Low-rate UE-AP association updates between edge CPUs. High-rate data plane stays within edge CPU.

AI for user clustering, resource allocation, fault detection. Integrated at regional CPU (RIC — RAN Intelligent Controller).

Mature in 2028+ with 6G O-RAN. Current 5G O-RAN supports $\sim 50$ AP per edge CPU.

Single BS at 40 W Tx. Coverage area: 1 km². 100 UEs. Per-UE Tx power allocation: $\sim 0.4$ W. EE: rate/power.

$L = 64$ APs at 1 W Tx each. Total Tx: 64 W (vs 40 W cellular). Per-UE: distributed across nearby APs. Average path loss much lower (nearest-AP at $\sim 100$ m vs $\sim 500$ m for cellular).

Cellular: 0.4 W × 1/s / (0.5 bits/s/Hz × 100 MHz) = 8 nJ/bit. Cell-free: (64 W × 1/s) / (100 UEs × 2 bits/s/Hz × 100 MHz) = 3.2 nJ/bit. 2.5× more efficient.

Lower path loss to nearest APs + macro-diversity + better modulation robustness. Compound energy savings at the 95%-tile.

When two UEs share a pilot, each AP's estimate $\hat{h}^{(l, k)} = h^{(l, k)} + \epsilon \cdot h^{(l, k')}$ (contaminated with $k'$'s channel).

Conjugate BF with contaminated estimate steers some of UE $k$'s signal to UE $k'$. Creates "pilot contamination interference".

$\mathrm{SINR}_k = \frac{L \bar\beta_k}{\sigma_w^2 + (\text{contamination})^2 \cdot L \bar\beta_{k'}}$ At high $L$: ceiling at $\mathrm{SINR} \to 1/\text{contamination}^2$.

Longer pilot sequences, pilot reuse factor, precoded pilots, Bayesian channel estimation. Reduces contamination by $1/N$ pilot length factor.

For 35% gain: $K/L \leq 0.2$. Beyond this, contamination dominates. Operator must scale APs accordingly.