Ferkans — Interactive Telecom Tutor

Conjugate Beamforming: Simple and Near-Optimal

The precoder at each AP must steer energy toward its UEs in a distributed way — each AP acts on local channel knowledge, without global coordination at symbol level. Conjugate beamforming (sometimes called "matched filter precoding") applies the conjugate of the channel estimate as the precoder vector. It is locally computed, globally coherent when the APs are synchronized, and asymptotically optimal as $L \to \infty$ . This section develops it in the DD domain.

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Definition:
Conjugate Beamforming in the DD Domain

At AP $l$ , the conjugate beamforming vector for UE $k$ is $\mathbf{v}^{(l, k)}[\ell, m] \;=\; \frac{(\mathbf{H}^{(l, k)}[\ell, m])^H}{\|\mathbf{H}^{(l, k)}\|_F},$ applied per DD cell $(\ell, m)$ . The AP transmits $\mathbf{x}^{(l)}[\ell, m] \;=\; \sum_{k=1}^{K} \sqrt{\alpha_k} \mathbf{v}^{(l, k)}[\ell, m] s_k[\ell, m],$ where $\alpha_k$ is the per-UE power allocation ( $\sum_k \alpha_k = P_t/L$ ) and $s_k$ is UE $k$ 's DD data symbol.

Aggregate received signal at UE $k$ : $y_k[\ell, m] \;=\; \sum_{l=1}^{L} \sqrt{\alpha_k} \mathbf{H}^{(l, k)} \mathbf{v}^{(l, k)} s_k[\ell, m] + \underbrace{\text{multi-user interference}}_{\text{from } k' \neq k} + \mathbf{w}_{k}.$ The first term is the "channel-hardened" signal: $\sum_l |\mathbf{H}^{(l,k)}|^2 s_k$ — a real-valued positive sum proportional to the total channel energy.

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Theorem: Conjugate Beamforming Optimality

As $L \to \infty$ with $K$ fixed, conjugate beamforming in cell-free OTFS achieves the asymptotic SINR $\mathrm{SINR}_k \;\to\; \frac{\alpha_k \cdot L \cdot \bar\beta_k}{\sigma_w^2 / \bar\beta_k + \text{multi-user interference}},$ where $\bar\beta_k = \mathbb{E}_l[\|\mathbf{H}^{(l, k)}\|^2]$ is the average channel magnitude squared.

Interpretations:

Linear SINR scaling: $\mathrm{SINR}_k \propto L$ (signal scales with number of APs).
Channel hardening: the effective channel becomes deterministic at large $L$ . Fading variance $\to 0$ .
MU-MMSE near-optimal: adding multi-user interference cancellation at the CPU recovers a few additional dB.

For $L = 50$ , $K = 20$ , pilot contamination $\kappa = 0.3$ : $\mathrm{SINR}_k \approx 17$ dB. Compare cellular single-BS: $\sim 10$ dB. Cell-free advantage: $\sim 7$ dB — 30-40% in rate.

Conjugate beamforming is the distributed version of matched filtering. At each AP, the precoder is the complex conjugate of the channel — pointing signal energy back along the same path the channel brings it. When APs synchronize, the individual signal contributions add coherently at the UE, while interference averages out by the law of large numbers. At $L \to \infty$ : perfect beamforming, no interference. Finite $L$ : interference scales with pilot contamination and user separation.

Proof

Per-AP SINR

$\mathrm{SINR}_k^{(l)} = |(\mathbf{H}^{(l,k)})^H \mathbf{v}^{(l,k)}|^2 / (\text{interference terms})$ . With conjugate beamforming: numerator = $\|\mathbf{H}^{(l,k)}\|^4 / \|\mathbf{H}^{(l,k)}\|^2 = \|\mathbf{H}^{(l,k)}\|^2$ .

Aggregate signal

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

Noise aggregation

Per-AP noise $\sigma_w^2$ through the combining weights: $\sum_l |\mathbf{v}^{(l,k)}|^2 \sigma_w^2 = L \sigma_w^2$ (per unit norm). Per-signal SNR: $L \bar\beta_k / \sigma_w^2$ .

Multi-user interference

Leakage from UE $k' \neq k$ : correlated channels cause residual interference. Scales as $K / L$ asymptotically. Vanishes as $L \to \infty$ .

Asymptotic SINR

$\mathrm{SINR}_k \to L \bar\beta_k / (\sigma_w^2 + K \bar\beta_k \kappa / L)$ for pilot contamination $\kappa$ . Linear scaling in $L$ . $\blacksquare$

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Key Takeaway

Conjugate beamforming is both simple and near-optimal in cell- free. Each AP computes $\mathbf{v}^{(l, k)} = (\mathbf{H}^{(l, k)})^H / \|\mathbf{H}^{(l, k)}\|$ from its local estimate. No inter-AP coordination at signal level. Asymptotic SINR scales linearly with $L$ . This simplicity is why cell-free OTFS is deployable — the CPU only aggregates estimates, not per-symbol decisions.

Definition:
Regularized ZF for Finite $L$

For small-to-moderate $L$ ( $L \leq 100$ ), conjugate beamforming suffers from residual multi-user interference. Regularized ZF precoding reduces this: $\mathbf{V}_{\mathrm{RZF}} \;=\; \mathbf{H}_{\mathrm{DD}}^H (\mathbf{H}_{\mathrm{DD}} \mathbf{H}_{\mathrm{DD}}^H + \mu \mathbf{I})^{-1},$ where $\mathbf{H}_{\mathrm{DD}}$ stacks all UEs' DD channel vectors, and $\mu$ is a regularization parameter ( $\mu = K \sigma_w^2 / L$ near-optimal).

Tradeoff: RZF needs joint channel inversion ( $L \cdot K$ system), requiring CPU coordination. Conjugate BF is fully distributed; RZF gives $\sim 3$ dB gain at cost of centralization.

Practical rule: Use conjugate for $L \geq 100$ ; RZF for $L \leq 50$ .

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Theorem: Cell-Free OTFS BER Under Mobility

For cell-free OTFS with $L$ APs, $K$ UEs, conjugate BF, and Doppler spread $\nu_{\max}$ , the BER at target SNR $\gamma$ is $\mathrm{BER} \;\approx\; \binom{2 P - 1}{P} \cdot \frac{1}{\left(L \bar\beta / K / (\sigma_w^2/\gamma) \right)^P},$ where $P$ is the average number of resolvable paths per UE-AP link.

Consequence: The BER exponent is $P$ (full DD diversity) and the pre-factor is $L$ (macro-diversity). At high mobility, this vastly outperforms cellular. Example: $L = 50$ , $K = 20$ , $P = 8$ , $\gamma = 20$ dB:

Cellular (1 BS): BER $\sim 10^{-5}$ .
Cell-free OTFS: BER $\sim 10^{-13}$ — 8 orders of magnitude better.

Cell-free macro-diversity compounds with OTFS's DD-diversity ( $P$ ). The aggregate diversity is $L \cdot P$ , and the BER decay is exponential in this total. For realistic numbers, the effective diversity is so high that BER falls below $10^{-10}$ at 15-20 dB SNR — unheard of in classical MIMO. This is the reliability underpinning the 35% throughput gain.

Proof

Aggregate SINR

From Thm. 17.8: $\mathrm{SINR} \propto L \bar\beta / K / (\sigma_w^2/\gamma)$ .

Per-UE rate

$R_k = \log(1 + \mathrm{SINR}_k)$ .

Pairwise error

$P$ -path DD diversity: $P_e \propto \mathrm{SINR}^{-P}$ .

BER scaling

Plugging in: $\mathrm{BER} \propto (L \bar\beta / K)^{-P} / (\sigma_w^2/\gamma)^P$ . $\blacksquare$

Example: Cell-Free OTFS vs Cellular at High Mobility

Compare BER at 20 dB SNR, 120 km/h mobility, for: (a) Single-BS cellular OFDM. (b) Single-BS cellular OTFS. (c) Cell-free OFDM ( $L = 50$ ). (d) Cell-free OTFS ( $L = 50$ ).

Solution

Cellular OFDM

BER $\sim 10^{-2}$ (error floor from ICI at 120 km/h).

Cellular OTFS

BER $\sim 10^{-6}$ (full DD diversity with single BS).

Cell-free OFDM

Macro-diversity helps, but ICI still limits. BER $\sim 10^{-4}$ .

Cell-free OTFS

Compounded diversity $L \cdot P$ . BER $\sim 10^{-13}$ .

Summary

Cellular OFDM → cell-free OTFS: 11 orders of magnitude BER improvement at high mobility. This is the quantitative case for the CommIT cell-free OTFS architecture.

Cell-Free OTFS BER vs Mobility

Plot BER vs UE velocity (0-300 km/h) for four configurations. Sliders: $L$ , $K$ , $N_a$ .

Parameters

L

50

K

20

N_a

4

🎓CommIT Contribution(2023)

Conjugate Beamforming in the DD Domain for Cell-Free OTFS

M. Mohammadi, H. Q. Ngo, M. Matthaiou, G. Caire — IEEE Trans. Wireless Communications

The CommIT contribution extends conjugate beamforming — the workhorse of cellular massive MIMO — to the DD domain for cell- free architectures. Three key results:

Distributed DD-conjugate BF: each AP computes its precoder locally from its DD channel estimate. No symbol-level inter-AP coordination needed.
Asymptotic SINR analysis: derives the exact scaling $\mathrm{SINR} \propto L \bar\beta$ for the DD setting, accounting for Doppler phase coherence across APs.
Quantitative performance: at $L = 50$ , $K = 20$ , 120 km/h, 20 dB SNR: $\sim 7$ dB SINR gain over cellular OTFS, $\sim 10$ dB over cellular OFDM.

Combined with the embedded-pilot estimation (§2), this yields the 35% improvement in 95%-likely per-user throughput. The DD-domain framework is essential: without it, conjugate BF at distributed APs cannot maintain Doppler-coherent combining.

commitcell-freeconjugate-bf

🔧Engineering Note

CPU Compute Scaling

CPU processing requirements in cell-free OTFS:

Channel aggregation: CPU receives $L$ per-AP DD estimates per UE per frame. Aggregation: $\mathcal{O}(L K MN)$ per frame.
Precoder computation (RZF if used): $\mathcal{O}(L^3)$ for full system, $\mathcal{O}(L_k^3)$ per UE for user-centric.
Resource allocation: $\mathcal{O}(L K)$ per frame.
Detection coordination: $\mathcal{O}(L K)$ per frame.

Total per frame: $\sim 10^6$ - $10^7$ ops for $L = 100$ , $K = 200$ . At 100 Hz frame rate: $10^8$ - $10^9$ ops/sec — well within a modern server CPU (2024-era Intel Xeon: 100 GFLOPS per core, 10+ cores).

Scaling to 1000 APs: conjugate BF scales linearly; RZF cubically. At $L = 1000$ : user-centric clustering ( $L_k = 10$ ) keeps it tractable. Without clustering: need GPU acceleration.

Practical Constraints

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Conjugate BF: O(LK) per frame
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RZF: O(L³) — needs user-centric clustering
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Modern server CPU handles L=100, K=200
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L=1000+: requires user-centric + GPU

Common Mistake: Conjugate BF Fails Without Phase Sync

Mistake:

Running conjugate beamforming with unsynchronized APs. If AP phases are random, the coherent combining at the UE is lost — signals add non-coherently, and gain drops from $L$ to $\sqrt{L}$ (a $\sqrt{L}$ factor of lost rate).

Correction:

Phase synchronization across APs is mandatory for conjugate BF. Options:

GNSS-PPS: $\pm 50$ ns phase accuracy. Works for sub-6 GHz.
PTP-1588v2 over fiber: $\pm 10$ ns. Works for mmWave.
Bi-directional calibration: bootstrap phases at deployment, refresh periodically.

Deployment checklist: verify cross-AP phase lock (coherence) at center frequency before turning on conjugate BF. Automatic fallback to MRT or per-AP-independent beamforming if sync fails.

Conjugate Beamforming in the DD Domain