Robust Interference Management for SISO Systems with Multiple Over-the-Air Computations
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In this paper, we consider the over-the-air computation of sums. Specifically, we wish to compute M≥ 2 sums s_m=∑_k∈Dmx_k over a shared complex-valued MAC at once with minimal mean-squared error (MSE). Finding appropriate Tx-Rx scaling factors balance between a low error in the computation of s_n and the interference induced by it in the computation of other sums s_m, m≠ n. In this paper, we are interested in designing an optimal Tx-Rx scaling policy that minimizes the mean-squared error max_m∈[1:M]MSE_m subject to a Tx power constraint with maximum power P. We show that an optimal design of the Tx-Rx scaling policy (a̅,b̅) involves optimizing (a) their phases and (b) their absolute values in order to (i) decompose the computation of M sums into, respectively, M_R and M_I (M=M_R+M_I) calculations over real and imaginary part of the Rx signal and (ii) to minimize the computation over each part – real and imaginary – individually. The primary focus of this paper is on (b). We derive conditions (i) on the feasibility of the optimization problem and (ii) on the Tx-Rx scaling policy of a local minimum for M_w=2 computations over the real (w=R) or the imaginary (w=I) part. Extensive simulations over a single Rx chain for M_w=2 show that the level of interference in terms of Δ D=|D_2|-|D_1| plays an important role on the ergodic worst-case MSE. At very high SNR, typically only the sensor with the weakest channel transmits with full power while all remaining sensors transmit with less to limit the interference. Interestingly, we observe that due to residual interference, the ergodic worst-case MSE is not vanishing; rather, it converges to |D_1||D_2|/K as SNR→∞.
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