Exercises

(1) Antenna array: LuMaMi's 100-element planar array. (2) User set: ArgosV3's support for 16--32 simultaneous UEs. (3) Real-time processing: the LuMaMi LabVIEW FPGA pipeline running at 20 MHz LTE-TDD. (4) Instrumentation: the HHI KIARA platform's raw I/Q logging at 20 Gb/s used for 3GPP TR 38.901 contributions.

A 5G NR slot contains 14 OFDM symbols independent of numerology. $\mu=1$: $T_{\rm slot} = 500~\mu$s, 14 symbols. $\mu=3$: $T_{\rm slot} = 125~\mu$s, 14 symbols. The FFT length scales inversely so that the sample rate stays constant at each bandwidth.

Both ZF and MMSE scale as $\Theta(N_tK^{2} + K^{3})$ per slot, plus $\Theta(N_tK)$ per data sample. Plugging $N_t=128$, $K=8$: the Gram step is $128 \cdot 64 = 8192$ complex multiplies and the inverse is $8^3 = 512$, giving $\sim 8700$ multiplies per slot for the once-per-slot work.

Per data sample the cost is $128 \cdot 8 = 1024$. Over a 300$\,\mu$s compute window with 13 OFDM data symbols and $N_{\rm sc} \sim 1024$ usable subcarriers, the per-slot application cost dominates at $\sim 1.4 \times 10^7$ complex multiplies.

The once-per-slot Gram and inverse fit naturally on a CPU core: low arithmetic intensity, moderate size. The per-sample application is better suited to an FPGA DSP fabric or GPU because of the uniform streaming structure. Hybrid FPGA+SoC splits exactly along this line.

Required $1 + \gamma_0 c_{\rm arch} 2^{-2b} \leq 1.122$, so $\gamma_0 c_{\rm arch} 2^{-2b} \leq 0.122$.

$\gamma_0 = 10^{2.5} \approx 316$, $c_{\rm arch} = 256 \cdot 16/4 = 1024$. Thus $2^{-2b} \leq 0.122/(316 \cdot 1024) = 3.77 \times 10^{-7}$.

$2b \geq -\log_2(3.77\times 10^{-7}) \approx 21.3$, so $b \geq 10.7$. Round to $b = 11$ mantissa bits. Add $\log_2(256) = 8$ bits of accumulator headroom; the implementation needs a 19-bit accumulator, rounding up to the next power of two supported by the FPGA fabric (typically 24 bits).

$\alpha_k' (\mathbf{H}_{k}^{\rm UL})^T \mathbf{C}' \mathbf{v}_k = (\alpha_k/\beta)(\mathbf{H}_{k}^{\rm UL})^T (\beta\mathbf{C}) \mathbf{v}_k = \alpha_k (\mathbf{H}_{k}^{\rm UL})^T \mathbf{C}\mathbf{v}_k$.

The product is invariant, so every user sees the same signal regardless of $\beta$. The UE's DMRS equalizer estimates the composite channel $\alpha_k (\mathbf{H}_{k}^{\rm UL})^T \mathbf{C} \mathbf{v}_k$ as a single scalar, and the $\beta$ ambiguity never leaks into any observable quantity.

For small noise $\hat r_n \approx r_n (1 + \delta_1/y_{0,n} - \delta_2/y_{n,0})$ with $\delta_1,\delta_2$ independent complex Gaussian. The relative error variance is $\sigma^2/|y_{0,n}|^2 + \sigma^2/|y_{n,0}|^2 \approx 2/\gamma_p$.

Repeating $M$ independent exchanges and averaging reduces the estimation noise variance as $2/(M\gamma_p)$, but the short-term drift between the first and last exchange is a deterministic change of $\mathbf{C}$ over time, which averaging does not cancel. The optimal number of repetitions is where the estimation error equals the drift error over the total observation window.

$|\alpha(0.01)|^2 \approx 1 - (\pi \cdot 0.01)^2/3 \approx 0.99967$. The denominator is $(\pi \cdot 0.01)^2/3 = 3.29 \times 10^{-4}$. The ceiling is $N_t \cdot 0.99967 / (3.29 \times 10^{-4}) \approx 3.9 \times 10^{5}$, about 55.9 dB.

The matched-filter bound at $N_t\text{SNR}$ exceeds this ceiling for any $\text{SNR} > 35$ dB. A 1% residual CFO therefore imposes a hard SINR ceiling near 56 dB on a 128-antenna array — well above typical operating SINR but close to the numerics of a mmWave link with strong LOS and good array gain.

At $N_t=64$, $K=8$, $N_{\rm RB} = 273$: once-per-slot cost $\approx 273 \cdot (64 \cdot 64 + 512) = 273 \cdot 4608 \approx 1.26 \times 10^6$; per-sample cost $\approx 273 \cdot 64 \cdot 8 \cdot 144 \approx 2.01 \times 10^7$. Total: $\sim 2.1 \times 10^7$ multiplies per slot.

At 2000 slots/s (30 kHz) the required throughput is $\sim 4.2 \times 10^{10}$ per second — below the stated $10^{11}$ ceiling, so the server can sustain this configuration. At $N_t=128, K=16$ the throughput roughly quadruples, crossing the ceiling. The practical limit is $(N_t, K) \approx (96, 12)$ on a single node.

The mean phase over the OFDM symbol is $\bar\phi = (1/T_{\rm OFDM})\int_0^{T_{\rm OFDM}}\phi(t)dt$, which has variance $T_{\rm OFDM}\sigma_\phi^2/3$ for Wiener noise. $\bar\phi$ appears as a per-symbol rotation common to all subcarriers; a one-tap tracking loop on DMRS estimates and removes it.

The deviation $\phi(t) - \bar\phi$ modulates the symbol across frequency, producing intercarrier leakage with total variance $\sim (T_{\rm OFDM}\sigma_\phi^2)^2/12$ after integration. This component is random per-subcarrier and uncorrelated across symbols, so a single tracking loop cannot cancel it — it looks to the receiver like a noise floor.

At FR2 the phase-noise ICI dominates the SINR ceiling; at sub-6 GHz the CPE is the larger raw effect but is eliminable, and the residual ICI is negligible. This is why phase-noise budgets enter FR2 link design and not sub-6 GHz.

60 ns is about 60/2300 $\approx 2.6\%$ of the cyclic prefix. Well within the CP budget for symbol-level sync — no inter-AP timing refinement needed for the OFDM receiver to avoid ISI.

However, at carrier 3.5 GHz, 60 ns corresponds to a phase of $2\pi f_c \cdot 60~\text{ns} \approx 750$ rad, which folds modulo $2\pi$ to an essentially uniform random phase. That means coherent inter-AP combining requires a separate phase calibration procedure — the GPSDO alone gives only timing, not the absolute carrier phase.

Set a target of 0.3 dB end-of-cycle penalty. Using the drift model $\rho_{\rm drift}^2 \propto T_{\rm cal}/\tau_{\rm lt}$, the 0.3 dB budget corresponds to $T_{\rm cal} \leq 0.02 \tau_{\rm lt} = 36$ s under pessimistic drift assumptions.

At calibration SNR $\sim 25$ dB and 64-sample pilots the estimation noise alone is well under 0.05 dB per cycle, so the drift dominates and the 36 s budget holds.

Run calibration every 10--30 seconds. Each cycle consumes about 1 ms across 2 slots, a $<0.01\%$ overhead. In practice deploy with an adaptive schedule that monitors the per-antenna temperature sensor and tightens the period if a thermal excursion is detected.

After substitution, the effective downlink channel at user $k$ factors as $\alpha_k (\mathbf{H}_{k}^{\rm UL})^T \mathbf{C}$, with $\alpha_k = v_k/u_k$ a per-user complex scalar. The UE measures $\alpha_k \mathbf{c}^T \mathbf{v}_k$ as part of its own DMRS channel estimate and equalizes it locally. The BS never needs to see $u_k$ or $v_k$; only the BS-internal ratio $\mathbf{C}$ is required. This is the critical separation that makes the calibration procedure BS-only.

A 12-bit input has full-scale amplitude $2^{11} = 2048$. A coherent sum over 256 complex samples reaches full-scale $256 \cdot 2048 = 5.24 \times 10^5$, which is $< 2^{19}$. The 24-bit accumulator has $2^{23} \approx 8.4 \times 10^6$ headroom — comfortable margin for coherent addition, 4 bits above the worst case.

Overflow happens if the input gain is set too high by AGC, pushing the 12-bit input above the nominal full-scale. The mitigation is a two-stage safety: (i) AGC configured with a peak-to-average headroom of at least 6 dB below the nominal scale, and (ii) a one-bit overflow detect on every accumulator with a reset on detect. The pipeline must trap the overflow event and flag the slot as invalid rather than silently wrapping.

Let ${N_t}_{\rm AP}$ antennas per AP, $K_{\rm AP}$ users per AP, $T_{\rm AP}$ per-AP compute throughput. The local ZF cost is $\Theta({N_t}_{\rm AP} K_{\rm AP}^2 + K_{\rm AP}^3)$, and the fronthaul adds $\Theta(K_{\rm AP} L_c)$ coefficient exchanges per slot.

Distributed processing fits in the slot iff $({N_t}_{\rm AP} K_{\rm AP}^2 + K_{\rm AP}^3)/T_{\rm AP} + K_{\rm AP} L_c / T_{\rm fh} < T_{\rm c}$, with $T_{\rm fh}$ the fronthaul link throughput and $T_{\rm c}$ the compute window. Centralized ZF has cost $({N_t}_{\rm total} K^2 + K^3)/T_{\rm cent}$ where ${N_t}_{\rm total} = L_c {N_t}_{\rm AP}$ and $K$ is total users. The distributed variant wins when $K_{\rm AP} \ll K$, i.e., when user-centric clustering sparsifies the per-AP load.

For LuMaMi-class APs with ${N_t}_{\rm AP} = 16$ and $K_{\rm AP} \leq 4$, per-AP cost is $\Theta(16 \cdot 16 + 64) = 320$ multiplies per slot — comfortably within budget and dominated entirely by fronthaul latency. This is the operating regime Massive Beams targets.

Failure mode: not logging the calibration state at capture time. Without knowing whether reciprocity was calibrated at each snapshot, downstream learning cannot distinguish channel variation from impairment variation.

Failure mode: missing outlier annotations. Unannotated blockage events get treated as ordinary channel samples; the trained model learns to overfit the pathology and fails in deployment.

Failure mode: random train/test split on a spatially correlated dataset, leaking information between train and test. Must split by position or scenario to evaluate real generalization.

The $1/f^2$ tail integrated from $f_0 = 1/T_{\rm OFDM} \approx 112$ kHz upward gives $\int_{f_0}^{\infty} 10^{-9} (1000/f)^2 df \approx 10^{-9} \cdot 10^6 / 112{,}000 \approx 9 \times 10^{-6}$ rad$^2$. Adding the flat region from 0 to $f_0$ contributes a similar amount.

Roughly half of this variance becomes CPE (correctable) and half becomes ICI (not correctable), i.e., an ICI-induced SINR floor of about $1/(9 \times 10^{-6}) \approx 50$ dB. At operating SINR 15 dB with $N_t=128$, the effective ceiling is 42 dB — well above the operating point, so the TCXO is adequate for sub-GHz compute but tight for higher-SINR operation.

A 0.5 dB penalty at 15 dB operating SINR requires the ICI floor to be $\geq$ 28 dB, which this TCXO provides. Margin is thin and an OCXO would give more comfort.

(1) Arrive, power up the array, warm up the front ends. (2) Verify GPSDO lock on all APs if cell-free. (3) Run reciprocity calibration; log the residual. (4) Inject a known test signal, verify baseline SNR at each antenna. (5) Run a short sounding sweep to confirm data logging. (6) Survey the planned Rx positions with a portable spectrum analyzer. (7) Execute the scheduled campaign with periodic calibration inserts. (8) Log temperature, time of day, visible occupants/vehicles, and any anomalies in a field notebook. (9) Verify data integrity (checksums, file counts) before teardown. (10) Back up raw data to two independent storage media before leaving the site. Any field measurement without step 10 is a gamble with the campaign budget.

First, annotate the measurement windows for blockage events. Removing the windows that correspond to heavy blockage from the analysis typically recovers 1--1.5 dB because the tail is grossly non-Gaussian. This does not change the algorithm; it changes the evaluation methodology.

Tighten the calibration schedule from once per second to every $\sim 200$ ms to better track outdoor thermal drift. This recovers $\sim 0.5$ dB in the steady-state regime without changing any hardware.

If the FPGA has headroom, widen the MAC mantissa from 12 to 16 bits. This is only worth doing if the measured quantization floor is visible in the post-calibration residual — the gain is small and the cost is significant.

The calibration ratio $r_n = y_{0,n}/y_{n,0}$ cancels the air-channel coupling $h_{n,0}$ exactly by electromagnetic reciprocity. Therefore the geometric distance from the reference antenna affects only the per-pair SNR (and thus how long each measurement must be integrated), not the value of the coefficient. Any antenna can serve as reference; a central one simply gives more uniform SNR across the array.

For a single-user link with channel $\mathbf{H}$, the ZF combiner is $\mathbf{H}^\dagger = (\mathbf{H}^{H}\mathbf{H})^{-1} \mathbf{H}^{H}$. Under $\mathbf{H} \to \mathbf{U}\mathbf{H}$ the Gram $\mathbf{H}^{H}\mathbf{U}^H\mathbf{U}\mathbf{H} = \mathbf{H}^{H} \mathbf{H}$ is unchanged, and the combiner becomes $(\mathbf{H}^{H}\mathbf{H})^{-1}\mathbf{H}^{H}\mathbf{U}^H$. Applied to the rotated observation, the result is the same.

This means the joint $\mathbf{U}\mathbf{C}$ is observable from data alone, but $\mathbf{C}$ individually is not — any unitary rotation of the BS antennas looks identical in the downlink data. The argos pilots break this invariance by injecting a BS-internal reference whose path is known to equal its reverse under reciprocity, pinning down $\mathbf{C}$ up to the harmless global scalar of Theorem 26.3.

Without internal calibration pilots, data-driven methods cannot recover $\mathbf{C}$ at all — they would only recover the product with an unknown unitary. Rogalin-Caire succeeded precisely because the argos pilots provide the missing reference.