A Semi-Closed-Form Solution to Optimal Distributed Beamforming for Two-Way Relay Networks
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Bibliographic record
Abstract
In this correspondence, we present a computationally simple semi-closed-form solution to the problem of designing distributed beamformer for two-way (bi-directional) multi-relay networks. In such a network, the relay nodes use amplify-and-forward relaying protocol to help two transceivers exchange information in a bidirectional manner. We consider a total power minimization approach to optimally find the relay beamforming weights and the transceiver transmit powers. This approach is based on the minimization of the total transmit power, consumed in the whole network, subject to SNR constraints at the two transceivers. We show that as far as the relay beamforming weight vector is concerned, this minimization problem is equivalent to the minimization of the total transmit power for a one-way relay network where the target SNR of the receiving transceiver is equal to the sum of the target SNRs of the two transceivers in the original two-way relay network. Based on this observation, we show that the relay beamforming weight vector can be obtained in a closed from given that an intermediate parameter, namely the transmit power of the transmitter in the equivalent one-way relay network, is available. This intermediate parameter is shown to be the solution to a one-dimensional optimization problem, and thus, it can be obtained using a simple bisection method. Our semi-closed-form solution not only reveals the structure of the optimal beamforming weight vector, but also leads to a one-dimensional search regardless of the number of relays. This provides the computational advantage over the gradient based numerical method of Havary-Nassab <etal xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> , where the gradient dimension reflects the number of relays.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it