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A New Look at Dual-Hop Relaying: Performance Limits with Hardware Impairments
Emil Björnson, Michail Matthaiou, Mérouane Debbah
TL;DR
Most relaying analyses assume ideal transceiver hardware, although practical impairments can affect high-rate performance. This paper models aggregate impairments in dual-hop AF and DF relaying and derives outage and capacity analyses. It shows that high-SNR performance is bounded by impairment-dependent SNDR and capacity ceilings.
Problem
Most relaying analyses assume ideal hardware, although practical transceivers retain impairments whose aggregate impact matters for system performance.
Source: §VII · p. 11Method
The paper models aggregate transceiver impairments in dual-hop AF and DF relaying and derives outage expressions and ergodic-capacity bounds under Nakagami-m fading.
Source: §I · p. 2, §VII · p. 11Results
At high SNR, dual-hop relaying has SNDR and capacity ceilings that cannot be crossed by increasing signal powers or changing fading conditions.
Source: §I · p. 2, §VII · p. 11Takeaways & Limitations
The design guidelines recommend equal hardware quality across transceivers, meeting necessary hardware-quality conditions, and favoring more robust protocols such as DF.
Source: §VII · p. 11
Abstract
from arXiv · showhide
Physical transceivers have hardware impairments that create distortions which degrade the performance of communication systems. The vast majority of technical contributions in the area of relaying neglect hardware impairments and, thus, assumes ideal hardware. Such approximations make sense in low-rate systems, but can lead to very misleading results when analyzing future high-rate systems. This paper quantifies the impact of hardware impairments on dual-hop relaying, for both amplify-and-forward and decode-and-forward protocols. The outage probability (OP) in these practical scenarios is a function of the effective end-to-end signal-to-noise-and-distortion ratio (SNDR). This paper derives new closed-form expressions for the exact and asymptotic OPs, accounting for hardware impairments at the source, relay, and destination. A similar analysis for the ergodic capacity is also pursued, resulting in new upper bounds. We assume that both hops are subject to independent but non-identically distributed Nakagami-m fading. This paper validates that the performance loss is small at low rates, but otherwise can be very substantial. In particular, it is proved that for high signal-to-noise ratio (SNR), the end-to-end SNDR converges to a deterministic constant, coined the SNDR ceiling, which is inversely proportional to the level of impairments. This stands in contrast to the ideal hardware case in which the end-to-end SNDR grows without bound in the high-SNR regime. Finally, we provide fundamental design guidelines for selecting hardware that satisfies the requirements of a practical relaying system.
I. INTRODUCTION
Most relaying analyses assume ideal hardware, although real transceivers retain impairments that matter especially at high rates. This paper analyzes aggregate impairments in dual-hop AF and DF relaying and derives outage, capacity, and high-SNR results.
I. INTRODUCTION
- Most relaying work assumes ideal transceiver hardware, despite residual impairments that can substantially affect performance, particularly in high-rate systems.Prior one-way-relaying work had examined bit- and symbol-error rates, with some analyses restricted to individual impairment types.Source: §I · p. 1
- The paper analyzes aggregate transceiver impairments in dual-hop AF and DF relaying, deriving exact outage expressions, ergodic-capacity upper bounds, and high-SNR limits.It also gives design guidelines for hardware-constrained systems.Source: §I · p. 2
A. Preliminaries on Distortion Noise from Impairments
- The generalized model represents transmitter and receiver impairments as distortion noises whose power depends on signal power and channel gain.The model uses aggregate Gaussian distortion noise and characterizes hardware quality through EVM parameters κ_t and κ_r.Source: A. Preliminaries on Distortion Noise… · pp. 2–3
- The analysis uses a single aggregate impairment parameter κ, combining transmitter and receiver contributions without loss of generality.The paper notes that the Gaussian assumption is particularly appropriate for residual distortion after compensation algorithms.Source: A. Preliminaries on Distortion Noise… · p. 3
B. System Model: Relaying with Non-Ideal Hardware
The relaying model applies aggregate hardware-distortion noise to both hops, with independent, non-identically distributed Nakagami-m fading. It derives distinct end-to-end SNDR formulations for AF relaying, including fixed- and variable-gain operation.
B. System Model: Relaying with Non-Ideal Hardware
- Impairment distortion noise at each hop passes through the same channel as the intended signal, distinguishing the model from a conventional multiple-access channel.The channel gains are independent but non-identically distributed Nakagami-m variates; the analysis also notes that much of its framework applies to other fading distributions.Source: B. System Model: Relaying with… · p. 3
- The analysis varies SNR by changing average fading power while holding signal power fixed; increasing signal power can raise impairment levels through amplifier nonlinearities.The high-SNR bounds and necessary conditions remain valid when signal power increases, but become optimistic and are no longer tight asymptotically.Source: B. System Model: Relaying with… · p. 4
C. End-to-End SNDR: Amplify-and-Forward Relaying
- AF relaying amplifies the relay's received signal, using variable gain with instantaneous first-hop channel knowledge or fixed gain with statistical channel information.The relay chooses its amplification factor to satisfy its power constraint.Source: C. End-to-End SNDR:… · p. 4
- With impaired hardware, AF end-to-end SNDR expressions have a more complicated form than in the ideal case because ρ1ρ2 appears in the denominator.This structure makes generalizing earlier Nakagami-m AF results to non-ideal hardware non-trivial.Source: C. End-to-End SNDR:… · p. 4
D. End-to-End SNDR: Decode-and-Forward Relaying
In DF relaying, the relay must decode the source signal before retransmitting it. With non-ideal hardware, end-to-end SNDR is determined by the weaker of the two hop SNDRs.
D. End-to-End SNDR: Decode-and-Forward Relaying
- DF end-to-end SNDR is the minimum of the source-to-relay and relay-to-destination SNDRs, since the relay must decode the signal for retransmission.The relay knows the first-hop channel and the destination knows the second-hop channel; the source requires no channel knowledge.Source: D. End-to-End SNDR: Decode-and-Forward… · p. 4
- Hardware impairments make the DF SNDR expression more complicated than the ideal-hardware form because its numerator and denominator are statistically dependent.Source: D. End-to-End SNDR: Decode-and-Forward… · p. 4
III. OUTAGE PROBABILITY ANALYSIS
The outage analysis derives expressions for the probability that end-to-end SNDR falls below a quality threshold. It first treats arbitrary fading distributions, then supports closed-form specialization to Nakagami-m and Rayleigh fading.
III. OUTAGE PROBABILITY ANALYSIS
- The paper derives exact outage-probability expressions for AF and DF relaying with transceiver impairments, extending analyses that assumed ideal hardware.Outage probability is the chance that fading drives effective end-to-end SNDR below the acceptable-quality threshold.Source: §III · p. 5
A. Arbitrary Channel Fading Distributions
- For arbitrary independent channel-gain distributions, the AF and DF outage expressions enable computation directly for DF and by numerical integration for AF.The expressions are then specialized to Nakagami-m and Rayleigh fading for closed-form results.Source: A. Arbitrary Channel Fading… · p. 5
- The analysis provides general outage expressions that account for channel gains appearing in both numerators and denominators of the impaired SNDRs.This structure is handled through a lemma for arbitrary fading distributions before deriving specialized closed forms.Source: A. Arbitrary Channel Fading… · p. 5
- For AF relaying, outage probability is 1 when x ≥ 1/d; for DF relaying, it is 1 when x ≥ 1/δ, regardless of fading distribution or SNR.These bounds mean certain SNDR thresholds cannot be crossed.Source: A. Arbitrary Channel Fading… · p. 5
B. Nakagami-m and Rayleigh Fading Channels
For Nakagami-m fading, this subsection derives closed-form outage probabilities for AF and DF relaying with non-ideal hardware, including Rayleigh special cases. The results generalize ideal-hardware analyses.
B. Nakagami-m and Rayleigh Fading Channels
- AF relaying has a closed-form non-ideal-hardware outage probability for x < 1/d, while the outage probability is 1 for x ≥ 1/d.The result assumes independent Gamma-distributed hop gains with integer shape parameters.Source: B. Nakagami-m and Rayleigh Fading… · p. 6
- DF relaying likewise has a closed-form outage probability for non-ideal hardware, with outage probability 1 for x ≥ 1/d.The theorem assumes independent Gamma-distributed hop gains with integer shape parameters and also gives a Rayleigh special case.Source: B. Nakagami-m and Rayleigh Fading… · p. 6
- The AF and DF results generalize prior ideal-hardware outage analyses, and the DF result extends straightforwardly to more than two hops.The DF theorem generalizes classical ideal-hardware results and extends to M > 2 hops by indexing all hops.Source: B. Nakagami-m and Rayleigh Fading… · p. 6
IV. ERGODIC CAPACITY ANALYSIS
The section develops tractable ergodic-capacity bounds and approximations for fixed-protocol AF and DF channels. For AF, it provides an upper bound and a simpler approximation that is nearly as tight and asymptotically exact.
IV. ERGODIC CAPACITY ANALYSIS
- For a fixed relaying protocol, the section treats ergodic capacity as an achievable rate of the end-to-end channel.This interpretation distinguishes fixed-protocol AF and DF rates from capacity with arbitrary relaying protocols.Source: §IV · p. 6
A. Capacity of AF Relaying
- Exact closed-form evaluation of the AF ergodic-capacity expression is tedious, if not impossible, so the analysis uses an upper bound and approximation.The capacity can instead be computed by numerical integration using the SNDR density derived from the outage CDF.Source: A. Capacity of AF Relaying · p. 6
- AF ergodic capacity with non-ideal hardware is upper bounded for Nakagami-m fading.The theorem covers fixed- and variable-gain relaying.Source: A. Capacity of AF Relaying · p. 6
- A simpler AF capacity approximation is also an upper bound, nearly as tight as the theorem's bound and asymptotically exact at high SNR.The passage reports these properties numerically for Nakagami-m fading.Source: A. Capacity of AF Relaying · p. 7
B. Capacity of DF Relaying
For strict DF with fixed power and equal time allocation, the paper derives an ergodic-capacity upper bound under hardware impairments. It uses the minimum capacity of the two hops as the bound's basis.
B. Capacity of DF Relaying
- The DF ergodic-capacity upper bound is based on the minimum of the two individual channel capacities.The analysis assumes strict DF, fixed power, and equal time allocation.Source: B. Capacity of DF Relaying · p. 7
- A closed-form upper bound for DF ergodic capacity is derived for any channel-fading distribution.The theorem states the bound for non-ideal hardware.Source: B. Capacity of DF Relaying · p. 7
- The bound does not grow without limit with SNR because distortion noise acts as an interference term proportional to SNR.This contrasts with the unbounded growth stated for ideal hardware.Source: B. Capacity of DF Relaying · p. 7
V. FUNDAMENTAL LIMITS: ASYMPTOTIC SNR ANALYSIS
At high SNR, impaired dual-hop relaying has a finite SNDR ceiling, and its outage and ergodic capacity saturate accordingly. The ceiling is inversely related to impairment levels, with the DF ceiling roughly twice the AF ceiling.
V. FUNDAMENTAL LIMITS: ASYMPTOTIC SNR ANALYSIS
- The asymptotic analysis assumes both hop SNRs grow with a finite, non-zero ratio and applies to independent, strictly positive fading gains.The stated capacity-ceiling result uses the same finite-ratio high-SNR condition.Source: §V · pp. 7–8
- The high-SNR outage probability tends to zero below the SNDR ceiling but remains one above it, independent of the hop fading distributions.With ideal hardware, outage tends to zero for any threshold as SNR increases.Source: §V · pp. 7–8
- The SNDR ceiling is inversely proportional to the squares of κ1 and κ2, and the DF ceiling is roughly twice the AF ceiling.The relative impact of κ1 and κ2 is otherwise similar for the two protocols.Source: §V · p. 8
- At high SNR, ergodic capacity saturates for both AF and DF, limiting high-rate performance with impaired hardware.For Nakagami-m fading, the approximate AF capacity expression becomes asymptotically exact.Source: §V · p. 8
A. Design Guidelines for Relaying Systems
The guidelines optimize relaying performance under a hardware-cost constraint by balancing impairment levels across transceiver chains. They also compare AF and DF hardware requirements for a target SNDR, while noting that sufficiency holds only asymptotically at high SNR.
A. Design Guidelines for Relaying Systems
- For a fixed hardware cost, the SNDR ceilings are maximized when transmitter and receiver EVMs are equal across all four transceiver chains.The cost model treats lower-quality hardware as less expensive and uses a convex function to represent diminishing returns from hardware improvements.Source: A. Design Guidelines for Relaying… · p. 8
- Equal impairment levels across chains imply that relay hardware should ideally match the source and destination hardware quality.The paper identifies four chains: source transmitter, relay receiver and transmitter, and destination receiver.Source: A. Design Guidelines for Relaying… · p. 8
- For a given SNDR threshold, the derived hardware requirements are looser for DF than AF; substituting x = 2^(2R) - 1 gives necessary conditions for capacity R.Source: A. Design Guidelines for Relaying… · p. 9
- The impairment guidelines are necessary, but sufficiency holds only asymptotically at high SNR, so finite-SNR systems should be designed more conservatively.The passage also calls for conservatism across different channel-fading conditions.Source: A. Design Guidelines for Relaying… · p. 9
VI. NUMERICAL ILLUSTRATIONS
The numerical illustrations validate the analysis and show that hardware impairments affect outage and capacity most strongly at high rates or SNRs. They also illustrate impairment-allocation and protocol choices for meeting outage targets.
A. Different Channel Fading Conditions
- At x = 31, hardware impairments cause around 5 dB SNR loss for AF and 2 dB for DF; at x = 3, the loss is minor.The impaired and ideal OP curves retain the same slope, so the loss appears as a rightward SNR shift.Source: A. Different Channel Fading Conditions · p. 9
- Higher Nakagami-m shape parameters reduce outage; with asymmetric SNRs, equal-strength hops perform better, and fixed-gain AF favors a stronger first hop.The first-hop advantage is attributed to noise amplification and disappears for variable-gain AF and DF.Source: A. Different Channel Fading Conditions · p. 9, Figure 4 · p. 10
B. SNDR and Capacity Ceilings
- At high thresholds, impaired AF and DF outage converges to SNDR ceilings, with DF’s ceiling roughly twice AF’s; ideal-hardware outage instead converges smoothly toward 1.At low thresholds, impairment-related degradation is slight.Source: B. SNDR and Capacity Ceilings · p. 10, Figure 5 · p. 10
- Variable-gain AF capacity saturates at an impairment-dependent ceiling; decreasing κ1 and κ2 raises it, and both shown capacity bounds are asymptotically exact at high SNR.Theorem 3’s upper bound is somewhat tighter, while the simpler approximation is easier to evaluate.Source: B. SNDR and Capacity Ceilings · p. 10, Figure 6 · p. 10
C. Design Guidelines
- With κ1 + κ2 = 0.3, AF outage is minimized by equal impairment levels, while DF favors slightly better hardware on the weaker hop.The AF guideline also works at finite SNR; the authors caution that equal impairment is not the true DF optimum.Source: Figure 7 · p. 10, C. Design Guidelines · p. 11
- Acceptable symmetric impairment limits are κ1 = κ2 ≤ 0.091 for fixed-gain AF, κ1 = κ2 ≤ 0.149 for variable-gain AF, and κ1 = κ2 ≤ 0.218 for DF; variable-gain AF and especially DF are more robust.The necessary conditions are upper bounds on impairment levels that may achieve OP below 1, providing a rough estimate despite not being sufficient.Source: C. Design Guidelines · p. 11
VII. CONCLUSIONS
- The conclusions recommend equal hardware quality across transceivers, using necessary conditions to select hardware, and favoring more robust protocols such as DF.The analysis applies to flat-fading dual-hop AF and DF with Nakagami-m fading; its ceilings cannot be overcome by increasing signal powers or changing fading conditions.Source: §VII · p. 11
APPENDIX A USEFUL LEMMAS
The appendix supplies lemmas used to derive outage probabilities under Nakagami-m fading. It develops a CDF for a two-variable SNDR-like quantity and evaluates it for gamma-distributed variables using an integral identity.
APPENDIX A USEFUL LEMMAS
- A preliminary lemma derives the CDF of SNDR-like expressions used to obtain outage probabilities under Nakagami-m fading.The appendix also summarizes an approach from for an integral evaluation.Source: Appendix A · pp. 11–12
- Lemma 3 gives the CDF for a random variable formed from two independent non-negative variables, then specializes the result to gamma variables with positive integer shape parameters.Its proof conditions on one variable and applies the law of total probability.Source: Appendix A · pp. 11–12
- For gamma-distributed variables, the remaining integral is evaluated using Lemma 4 and an integral identity from [39, Eq. (3.471.9)].The result uses the modified Bessel function of the second kind.Source: Appendix A · p. 12
APPENDIX B PROOF OF THEOREMS
The appendix proves the capacity bound by applying Jensen’s inequality to a concave function of the SNDR-related variable. It then evaluates the resulting expectations using a moment generating function.
APPENDIX B PROOF OF THEOREMS
- The remaining expectation is evaluated using the moment generating function derived in [32, Theorem 3].The authors note a sign typo in that reference’s equation and figure.Source: Proof of Theorem 3 · p. 12
- Theorem 3’s capacity expression is bounded using Jensen’s inequality, since the relevant logarithmic function is concave for nonnegative ψ.In the proof, ψ is set to Piρi and Ni = SNRi.Source: Proof of Theorem 3 · pp. 12–13
- Applying the bound to each expectation yields the capacity upper-bound expression.The proof identifies the expectations with the relevant terms in the capacity expression.Source: Proof of Theorem 3 · pp. 12–13