Two-Way Relaying under the Presence of Relay Transceiver Hardware Impairments

I. INTRODUCTION

• Two-way relaying lets two nodes communicate through a relay in two time slots, avoiding the two-slot source-to-destination exchange of classic half-duplex relaying.Both nodes transmit simultaneously to the relay in the first slot, and it broadcasts to them in the second.Source: §I · p. 1
• Relay transceiver impairments can impose a capacity ceiling that increasing transmit power cannot overcome, with particular impact in high-rate systems.The cited impairment sources include IQ imbalance, amplifier nonlinearities, and phase noise.Source: §I · p. 1
• Earlier impairment studies of relaying focused on one-way systems, analyzing SER and OP, respectively.The paper notes that relevant studies were few.Source: §I · p. 1
• The paper derives SNDR expressions at both transmitter nodes and exact and asymptotic OP/SER expressions for two-way AF relaying with relay impairments.The authors describe this as the first analysis of hardware impairments in a two-way relaying configuration.Source: §I · p. 1

II. SIGNAL AND SYSTEM MODEL

• Two nodes exchange information through a relay using two-way amplify-and-forward relaying.Source: §II · p. 1

A. Preliminaries on Distortion Noise from Impairments

• Transceiver impairments mismatch the intended and emitted symbol and distort the signal during reception.Source: A. Preliminaries on Distortion Noise… · p. 1
• The combined influence of hardware impairments is modeled using a generalized channel model with transmitter and receiver distortion-noise terms.The model treats the distortion noises as circularly symmetric complex Gaussian, justified by the aggregate effect of many impairments.Source: A. Preliminaries on Distortion Noise… · pp. 1–2
• The nonnegative parameters κt and κr represent transmitter and receiver impairment levels and can be interpreted as EVMs.EVM is defined as the ratio of distortion to signal magnitude; setting both parameters to zero recovers the classical ideal-hardware model.Source: A. Preliminaries on Distortion Noise… · p. 2
• The analysis considers impairments only at the relay for brevity, with κ-parameters reinterpretable as aggregate parameters for each link.Source: A. Preliminaries on Distortion Noise… · p. 2

B. Two-Way Relaying with Ideal and Non-Ideal Hardware

• In the first slot, T1 and T2 send symbols to the relay, which broadcasts an amplified version in the second slot.Source: B. Two-Way Relaying with Ideal and… · p. 2
• The links between each transmitter and relay are reciprocal independent Rayleigh-fading channels, and the information symbols and additive noise are modeled as complex Gaussian.The model assumes perfect instantaneous channel knowledge at all nodes, enabling variable-gain relaying; a mismatch in knowledge of relay receive impairment level can degrade performance.Source: B. Two-Way Relaying with Ideal and… · p. 2
• Relay receiver and transmitter hardware impairments contribute distortion noise to the received and rebroadcast signals.The ideal-hardware special case removes the distortion terms.Source: B. Two-Way Relaying with Ideal and… · p. 2
• Each destination transmitter cancels its own known transmitted-symbol term before detecting the other transmitter's symbol.Source: B. Two-Way Relaying with Ideal and… · p. 2
• The resulting effective SNDR is derived for each transmitter, and its ideal-hardware special case reduces to the expression in [2, Eqs. (8)–(9)].Source: B. Two-Way Relaying with Ideal and… · p. 2

• Outage probability at T_i is the probability that fading drives SNDR_i below the acceptable-quality threshold x.Source: A. Exact Outage Probability Analysis · p. 2
• For x≥1, the outage probability at T_i is 1.Source: A. Exact Outage Probability Analysis · p. 2
• The derivation conditions on ρ_i and evaluates the resulting integral using the channel-gain PDF and CDF.Source: A. Exact Outage Probability Analysis · p. 3
• Proposition 1 gives a tractable closed-form outage probability for impaired relay hardware, generalizing the ideal-hardware result in [2, Theorem 1].Source: A. Exact Outage Probability Analysis · p. 3

• In the high-power regime, the outage probability has a non-zero lower bound rather than converging to zero, unlike with ideal hardware.Source: B. Asymptotic Outage Probability… · p. 3
• The high-power SNDR has a finite ceiling characterized by the impairment level, and this ceiling disappears with ideal hardware.Source: B. Asymptotic Outage Probability… · p. 3
• The lower bound depends on impairment level, average channel gains Ω1 and Ω2, and threshold x; differing relay distortion-noise variances preserve randomness after second-hop amplification.Source: B. Asymptotic Outage Probability… · p. 3
• With a fixed total relay EVM constraint, equal transmitter and receiver hardware quality minimizes the asymptotic outage probability.Source: B. Asymptotic Outage Probability… · p. 3
• Because channel-estimation errors and other non-hardware impairments also degrade performance, these results are an upper bound on what is achievable in practice.Source: B. Asymptotic Outage Probability… · p. 3

C. Exact and Asymptotic Symbol Error Rate Analysis

• The SER can be computed by numerical integration, which is more efficient than Monte Carlo simulation.The resulting integral does not appear to have a closed-form solution.Source: C. Exact and Asymptotic Symbol Error… · p. 3
• For identical average channel gains, the high-power asymptotic SERs at T1 and T2 are identical.The result applies as P1 = P2 = τP3 → ∞.Source: C. Exact and Asymptotic Symbol Error… · p. 3
• With non-ideal hardware, the asymptotic SER is strictly positive, creating an irreducible error floor.The exception is ideal hardware, c = 0.Source: C. Exact and Asymptotic Symbol Error… · p. 3
• Inverting the asymptotic OP and SER expressions with respect to c gives impairment limits that can theoretically meet specified requirements.The design guidelines address a practical two-way relaying system with a given set of OP and SER requirements.Source: C. Exact and Asymptotic Symbol Error… · p. 3

IV. NUMERICAL RESULTS

• At κ = 0.2, the high-rate system is always in outage regardless of transmit power; at moderate impairment, OP saturates above zero, unlike ideal hardware.The moderate-impairment saturation is accurately predicted by Corollary 1; ideal-hardware OP tends asymptotically to zero.Source: §IV · p. 4
• With κ3t + κ3r = 0.2, equal relay hardware quality (κ3t = κ3r = 0.1) minimizes SER and asymptotically halves it versus κ3t = 0, κ3r = 0.2.The comparison uses BPSK, with exact curves obtained by numerical evaluation and high-power limits from Corollary 2.Source: §IV · p. 4

V. CONCLUSIONS

• The relay's receive and transmit hardware should be of the same quality to minimize both OP and SER.The paper's theoretical analysis identifies this as the favorable hardware balance.Source: §V · p. 4
• In high-rate systems and/or with high impairment levels, the system is always in full outage.Source: §V · p. 4
• Zero OP or SER generally cannot be achieved because the SNDR remains random in the high-power regime.Source: §V · p. 4
• The asymptotic expressions can guide selection of hardware that satisfies OP/SER requirements for a practical relaying topology.Source: §V · p. 4