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In this work, we study a classic robust design problem in two-hop one-wayrelay system. We are particularly interested in the scenario where channeluncertainty exists in both the transmitter-to-relay and relay-to-receiverlinks. By considering the problem design that minimizes the averageamplify-and-forward power budget at the relay side while satisfying SNR outagerequirements, an outage-constrained robust design problem involving quarticperturbations is formulated to guarantee the robustness during transmission.This problem is in general difficult as it involves constraints on the tailprobability of a high-order polynomial. Herein, we resort to moment inequalityand Bernstein-type inequality to tackle this problem, which provide convexrestrictions, or safe approximations, of the original design. We also analyzethe relative tightness of the two safe approximations for a quadraticperturbation-based outage constrained problem. Our analysis shows that theBernstein-type inequality approach is less conservative than the momentinequality approach when the outage rate is within some prescribed regime. Toour best knowledge, this is the first provable tightness result for these twosafe approximations. Our numerical simulations verify the superiority of therobust design and corroborate the tightness results.

Quartic Perturbation-based Outage-constrained Robust Design in Two-hop One-way Relay Networks